phanerozoic/Agda-UniMath · Datasets at Hugging Face

fact stringlengths 9 34.3k	type stringclasses 3 values	library stringclasses 2 values	imports listlengths 0 227	filename stringlengths 22 99	symbolic_name stringlengths 1 57
concept : ...	function	docs	[ "open import foundation-core.template public", "open import ..." ]	docs/TEMPLATE.lagda.md	concept
satisfies-property-concept : ...	function	docs	[ "open import foundation-core.template public", "open import ..." ]	docs/TEMPLATE.lagda.md	satisfies-property-concept
concept-subconcept : ...	function	docs	[ "open import foundation-core.template public", "open import ..." ]	docs/TEMPLATE.lagda.md	concept-subconcept
is-complete-prop-Metric-Ab : {l1 l2 : Level} → Metric-Ab l1 l2 → Prop (l1 ⊔ l2)	function	src	[ "open import analysis.metric-abelian-groups", "open import foundation.dependent-pair-types", "open import foundation.propositions", "open import foundation.subtypes", "open import foundation.universe-levels", "open import metric-spaces.complete-metric-spaces", "open import metric-spaces.metric-spaces" ]	src/analysis/complete-metric-abelian-groups.lagda.md	is-complete-prop-Metric-Ab
is-complete-Metric-Ab : {l1 l2 : Level} → Metric-Ab l1 l2 → UU (l1 ⊔ l2)	function	src	[ "open import analysis.metric-abelian-groups", "open import foundation.dependent-pair-types", "open import foundation.propositions", "open import foundation.subtypes", "open import foundation.universe-levels", "open import metric-spaces.complete-metric-spaces", "open import metric-spaces.metric-spaces" ]	src/analysis/complete-metric-abelian-groups.lagda.md	is-complete-Metric-Ab
Complete-Metric-Ab : (l1 l2 : Level) → UU (lsuc l1 ⊔ lsuc l2)	function	src	[ "open import analysis.metric-abelian-groups", "open import foundation.dependent-pair-types", "open import foundation.propositions", "open import foundation.subtypes", "open import foundation.universe-levels", "open import metric-spaces.complete-metric-spaces", "open import metric-spaces.metric-spaces" ]	src/analysis/complete-metric-abelian-groups.lagda.md	Complete-Metric-Ab
convergent-series-Complete-Metric-Ab : {l1 l2 : Level} (G : Complete-Metric-Ab l1 l2) → UU (l1 ⊔ l2)	function	src	[ "open import analysis.complete-metric-abelian-groups", "open import analysis.convergent-series-metric-abelian-groups", "open import analysis.series-complete-metric-abelian-groups", "open import foundation.dependent-pair-types", "open import foundation.inhabited-types", "open import foundation.propositions", "open import foundation.subtypes", "open import foundation.universe-levels", "open import metric-spaces.cauchy-sequences-complete-metric-spaces", "open import metric-spaces.cauchy-sequences-metric-spaces" ]	src/analysis/convergent-series-complete-metric-abelian-groups.lagda.md	convergent-series-Complete-Metric-Ab
convergent-series-Metric-Ab : {l1 l2 : Level} (G : Metric-Ab l1 l2) → UU (l1 ⊔ l2)	function	src	[ "open import analysis.metric-abelian-groups", "open import analysis.series-metric-abelian-groups", "open import elementary-number-theory.addition-natural-numbers", "open import elementary-number-theory.inequality-natural-numbers", "open import elementary-number-theory.natural-numbers", "open import foundation.action-on-identifications-functions", "open import foundation.binary-transport", "open import foundation.dependent-pair-types", "open import foundation.functoriality-propositional-truncation", "open import foundation.identity-types", "open import foundation.logical-equivalences", "open import foundation.propositions", "open import foundation.subtypes", "open import foundation.transport-along-identifications", "open import foundation.universe-levels", "open import lists.sequences", "open import metric-spaces.convergent-sequences-metric-spaces", "open import metric-spaces.limits-of-sequences-metric-spaces" ]	src/analysis/convergent-series-metric-abelian-groups.lagda.md	convergent-series-Metric-Ab
convergent-series-ℝ : (l : Level) → UU (lsuc l)	function	src	[ "open import analysis.convergent-series-complete-metric-abelian-groups", "open import analysis.convergent-series-metric-abelian-groups", "open import analysis.series-real-numbers", "open import foundation.dependent-pair-types", "open import foundation.propositions", "open import foundation.subtypes", "open import foundation.universe-levels", "open import lists.sequences", "open import real-numbers.cauchy-sequences-real-numbers", "open import real-numbers.dedekind-real-numbers", "open import real-numbers.metric-additive-group-of-real-numbers" ]	src/analysis/convergent-series-real-numbers.lagda.md	convergent-series-ℝ
Metric-Ab : (l1 l2 : Level) → UU (lsuc l1 ⊔ lsuc l2)	function	src	[ "open import elementary-number-theory.positive-rational-numbers", "open import foundation.action-on-identifications-binary-functions", "open import foundation.binary-relations", "open import foundation.cartesian-product-types", "open import foundation.conjunction", "open import foundation.dependent-pair-types", "open import foundation.identity-types", "open import foundation.propositions", "open import foundation.universe-levels", "open import group-theory.abelian-groups", "open import metric-spaces.extensionality-pseudometric-spaces", "open import metric-spaces.isometries-metric-spaces", "open import metric-spaces.isometries-pseudometric-spaces", "open import metric-spaces.metric-spaces", "open import metric-spaces.pseudometric-spaces", "open import metric-spaces.rational-neighborhood-relations" ]	src/analysis/metric-abelian-groups.lagda.md	Metric-Ab
is-nonnegative-prop-series-ℝ : {l : Level} → series-ℝ l → Prop l	function	src	[ "open import analysis.series-real-numbers", "open import elementary-number-theory.natural-numbers", "open import foundation.dependent-pair-types", "open import foundation.function-types", "open import foundation.propositions", "open import foundation.universe-levels", "open import order-theory.increasing-sequences-posets", "open import real-numbers.absolute-value-real-numbers", "open import real-numbers.addition-nonnegative-real-numbers", "open import real-numbers.inequality-real-numbers", "open import real-numbers.nonnegative-real-numbers" ]	src/analysis/nonnegative-series-real-numbers.lagda.md	is-nonnegative-prop-series-ℝ
is-nonnegative-series-ℝ : {l : Level} → series-ℝ l → UU l	function	src	[ "open import analysis.series-real-numbers", "open import elementary-number-theory.natural-numbers", "open import foundation.dependent-pair-types", "open import foundation.function-types", "open import foundation.propositions", "open import foundation.universe-levels", "open import order-theory.increasing-sequences-posets", "open import real-numbers.absolute-value-real-numbers", "open import real-numbers.addition-nonnegative-real-numbers", "open import real-numbers.inequality-real-numbers", "open import real-numbers.nonnegative-real-numbers" ]	src/analysis/nonnegative-series-real-numbers.lagda.md	is-nonnegative-series-ℝ
series-Metric-Ab {l1 l2 : Level} (G : Metric-Ab l1 l2) : UU l1 where constructor series-terms-Metric-Ab field term-series-Metric-Ab : sequence (type-Metric-Ab G)	record	src	[ "open import analysis.metric-abelian-groups", "open import elementary-number-theory.addition-natural-numbers", "open import elementary-number-theory.natural-numbers", "open import foundation.action-on-identifications-functions", "open import foundation.dependent-pair-types", "open import foundation.equivalences", "open import foundation.function-extensionality", "open import foundation.function-types", "open import foundation.homotopies", "open import foundation.identity-types", "open import foundation.universe-levels", "open import group-theory.abelian-groups", "open import group-theory.sums-of-finite-sequences-of-elements-abelian-groups", "open import lists.sequences", "open import univalent-combinatorics.coproduct-types", "open import univalent-combinatorics.standard-finite-types" ]	src/analysis/series-metric-abelian-groups.lagda.md	series-Metric-Ab
series-ℝ : (l : Level) → UU (lsuc l)	function	src	[ "open import analysis.series-complete-metric-abelian-groups", "open import analysis.series-metric-abelian-groups", "open import elementary-number-theory.addition-natural-numbers", "open import elementary-number-theory.natural-numbers", "open import foundation.dependent-pair-types", "open import foundation.function-types", "open import foundation.identity-types", "open import foundation.propositions", "open import foundation.universe-levels", "open import lists.sequences", "open import order-theory.increasing-sequences-posets", "open import real-numbers.absolute-value-real-numbers", "open import real-numbers.addition-nonnegative-real-numbers", "open import real-numbers.dedekind-real-numbers", "open import real-numbers.difference-real-numbers", "open import real-numbers.inequality-real-numbers", "open import real-numbers.metric-additive-group-of-real-numbers", "open import real-numbers.nonnegative-real-numbers" ]	src/analysis/series-real-numbers.lagda.md	series-ℝ
series-terms-ℝ : {l : Level} → sequence (ℝ l) → series-ℝ l	function	src	[ "open import analysis.series-complete-metric-abelian-groups", "open import analysis.series-metric-abelian-groups", "open import elementary-number-theory.addition-natural-numbers", "open import elementary-number-theory.natural-numbers", "open import foundation.dependent-pair-types", "open import foundation.function-types", "open import foundation.identity-types", "open import foundation.propositions", "open import foundation.universe-levels", "open import lists.sequences", "open import order-theory.increasing-sequences-posets", "open import real-numbers.absolute-value-real-numbers", "open import real-numbers.addition-nonnegative-real-numbers", "open import real-numbers.dedekind-real-numbers", "open import real-numbers.difference-real-numbers", "open import real-numbers.inequality-real-numbers", "open import real-numbers.metric-additive-group-of-real-numbers", "open import real-numbers.nonnegative-real-numbers" ]	src/analysis/series-real-numbers.lagda.md	series-terms-ℝ
term-series-ℝ : {l : Level} → series-ℝ l → sequence (ℝ l)	function	src	[ "open import analysis.series-complete-metric-abelian-groups", "open import analysis.series-metric-abelian-groups", "open import elementary-number-theory.addition-natural-numbers", "open import elementary-number-theory.natural-numbers", "open import foundation.dependent-pair-types", "open import foundation.function-types", "open import foundation.identity-types", "open import foundation.propositions", "open import foundation.universe-levels", "open import lists.sequences", "open import order-theory.increasing-sequences-posets", "open import real-numbers.absolute-value-real-numbers", "open import real-numbers.addition-nonnegative-real-numbers", "open import real-numbers.dedekind-real-numbers", "open import real-numbers.difference-real-numbers", "open import real-numbers.inequality-real-numbers", "open import real-numbers.metric-additive-group-of-real-numbers", "open import real-numbers.nonnegative-real-numbers" ]	src/analysis/series-real-numbers.lagda.md	term-series-ℝ
partial-sum-series-ℝ : {l : Level} → series-ℝ l → sequence (ℝ l)	function	src	[ "open import analysis.series-complete-metric-abelian-groups", "open import analysis.series-metric-abelian-groups", "open import elementary-number-theory.addition-natural-numbers", "open import elementary-number-theory.natural-numbers", "open import foundation.dependent-pair-types", "open import foundation.function-types", "open import foundation.identity-types", "open import foundation.propositions", "open import foundation.universe-levels", "open import lists.sequences", "open import order-theory.increasing-sequences-posets", "open import real-numbers.absolute-value-real-numbers", "open import real-numbers.addition-nonnegative-real-numbers", "open import real-numbers.dedekind-real-numbers", "open import real-numbers.difference-real-numbers", "open import real-numbers.inequality-real-numbers", "open import real-numbers.metric-additive-group-of-real-numbers", "open import real-numbers.nonnegative-real-numbers" ]	src/analysis/series-real-numbers.lagda.md	partial-sum-series-ℝ
obj-augmented-simplex-Category : UU lzero	function	src	[ "open import category-theory.composition-operations-on-binary-families-of-sets", "open import category-theory.precategories", "open import elementary-number-theory.inequality-standard-finite-types", "open import elementary-number-theory.natural-numbers", "open import foundation.dependent-pair-types", "open import foundation.identity-types", "open import foundation.sets", "open import foundation.strictly-involutive-identity-types", "open import foundation.universe-levels", "open import order-theory.order-preserving-maps-posets" ]	src/category-theory/augmented-simplex-category.lagda.md	obj-augmented-simplex-Category
hom-set-augmented-simplex-Category : obj-augmented-simplex-Category → obj-augmented-simplex-Category → Set lzero	function	src	[ "open import category-theory.composition-operations-on-binary-families-of-sets", "open import category-theory.precategories", "open import elementary-number-theory.inequality-standard-finite-types", "open import elementary-number-theory.natural-numbers", "open import foundation.dependent-pair-types", "open import foundation.identity-types", "open import foundation.sets", "open import foundation.strictly-involutive-identity-types", "open import foundation.universe-levels", "open import order-theory.order-preserving-maps-posets" ]	src/category-theory/augmented-simplex-category.lagda.md	hom-set-augmented-simplex-Category
hom-augmented-simplex-Category : obj-augmented-simplex-Category → obj-augmented-simplex-Category → UU lzero	function	src	[ "open import category-theory.composition-operations-on-binary-families-of-sets", "open import category-theory.precategories", "open import elementary-number-theory.inequality-standard-finite-types", "open import elementary-number-theory.natural-numbers", "open import foundation.dependent-pair-types", "open import foundation.identity-types", "open import foundation.sets", "open import foundation.strictly-involutive-identity-types", "open import foundation.universe-levels", "open import order-theory.order-preserving-maps-posets" ]	src/category-theory/augmented-simplex-category.lagda.md	hom-augmented-simplex-Category
id-hom-augmented-simplex-Category : (n : obj-augmented-simplex-Category) → hom-augmented-simplex-Category n n	function	src	[ "open import category-theory.composition-operations-on-binary-families-of-sets", "open import category-theory.precategories", "open import elementary-number-theory.inequality-standard-finite-types", "open import elementary-number-theory.natural-numbers", "open import foundation.dependent-pair-types", "open import foundation.identity-types", "open import foundation.sets", "open import foundation.strictly-involutive-identity-types", "open import foundation.universe-levels", "open import order-theory.order-preserving-maps-posets" ]	src/category-theory/augmented-simplex-category.lagda.md	id-hom-augmented-simplex-Category
augmented-simplex-Precategory : Precategory lzero lzero	function	src	[ "open import category-theory.composition-operations-on-binary-families-of-sets", "open import category-theory.precategories", "open import elementary-number-theory.inequality-standard-finite-types", "open import elementary-number-theory.natural-numbers", "open import foundation.dependent-pair-types", "open import foundation.identity-types", "open import foundation.sets", "open import foundation.strictly-involutive-identity-types", "open import foundation.universe-levels", "open import order-theory.order-preserving-maps-posets" ]	src/category-theory/augmented-simplex-category.lagda.md	augmented-simplex-Precategory
Category : (l1 l2 : Level) → UU (lsuc l1 ⊔ lsuc l2)	function	src	[ "open import category-theory.composition-operations-on-binary-families-of-sets", "open import category-theory.isomorphisms-in-precategories", "open import category-theory.nonunital-precategories", "open import category-theory.precategories", "open import category-theory.preunivalent-categories", "open import category-theory.strongly-preunivalent-categories", "open import foundation.1-types", "open import foundation.cartesian-product-types", "open import foundation.dependent-pair-types", "open import foundation.equivalences", "open import foundation.fundamental-theorem-of-identity-types", "open import foundation.identity-types", "open import foundation.logical-equivalences", "open import foundation.propositions", "open import foundation.sets", "open import foundation.strictly-involutive-identity-types", "open import foundation.surjective-maps", "open import foundation.universe-levels" ]	src/category-theory/categories.lagda.md	Category
total-hom-Category : {l1 l2 : Level} (C : Category l1 l2) → UU (l1 ⊔ l2)	function	src	[ "open import category-theory.composition-operations-on-binary-families-of-sets", "open import category-theory.isomorphisms-in-precategories", "open import category-theory.nonunital-precategories", "open import category-theory.precategories", "open import category-theory.preunivalent-categories", "open import category-theory.strongly-preunivalent-categories", "open import foundation.1-types", "open import foundation.cartesian-product-types", "open import foundation.dependent-pair-types", "open import foundation.equivalences", "open import foundation.fundamental-theorem-of-identity-types", "open import foundation.identity-types", "open import foundation.logical-equivalences", "open import foundation.propositions", "open import foundation.sets", "open import foundation.strictly-involutive-identity-types", "open import foundation.surjective-maps", "open import foundation.universe-levels" ]	src/category-theory/categories.lagda.md	total-hom-Category
sSet-Large-Precategory : Large-Precategory lsuc (_⊔_)	function	src	[ "open import category-theory.categories", "open import category-theory.large-categories", "open import category-theory.large-precategories", "open import category-theory.precategories", "open import category-theory.presheaf-categories", "open import category-theory.simplex-category", "open import foundation.commuting-squares-of-maps", "open import foundation.function-types", "open import foundation.homotopies", "open import foundation.identity-types", "open import foundation.sets", "open import foundation.strictly-involutive-identity-types", "open import foundation.universe-levels" ]	src/category-theory/category-of-simplicial-sets.lagda.md	sSet-Large-Precategory
is-large-category-sSet-Large-Category : is-large-category-Large-Precategory sSet-Large-Precategory	function	src	[ "open import category-theory.categories", "open import category-theory.large-categories", "open import category-theory.large-precategories", "open import category-theory.precategories", "open import category-theory.presheaf-categories", "open import category-theory.simplex-category", "open import foundation.commuting-squares-of-maps", "open import foundation.function-types", "open import foundation.homotopies", "open import foundation.identity-types", "open import foundation.sets", "open import foundation.strictly-involutive-identity-types", "open import foundation.universe-levels" ]	src/category-theory/category-of-simplicial-sets.lagda.md	is-large-category-sSet-Large-Category
sSet-Large-Category : Large-Category lsuc (_⊔_)	function	src	[ "open import category-theory.categories", "open import category-theory.large-categories", "open import category-theory.large-precategories", "open import category-theory.precategories", "open import category-theory.presheaf-categories", "open import category-theory.simplex-category", "open import foundation.commuting-squares-of-maps", "open import foundation.function-types", "open import foundation.homotopies", "open import foundation.identity-types", "open import foundation.sets", "open import foundation.strictly-involutive-identity-types", "open import foundation.universe-levels" ]	src/category-theory/category-of-simplicial-sets.lagda.md	sSet-Large-Category
sSet : (l : Level) → UU (lsuc l)	function	src	[ "open import category-theory.categories", "open import category-theory.large-categories", "open import category-theory.large-precategories", "open import category-theory.precategories", "open import category-theory.presheaf-categories", "open import category-theory.simplex-category", "open import foundation.commuting-squares-of-maps", "open import foundation.function-types", "open import foundation.homotopies", "open import foundation.identity-types", "open import foundation.sets", "open import foundation.strictly-involutive-identity-types", "open import foundation.universe-levels" ]	src/category-theory/category-of-simplicial-sets.lagda.md	sSet
hom-set-sSet : {l1 l2 : Level} (X : sSet l1) (Y : sSet l2) → Set (l1 ⊔ l2)	function	src	[ "open import category-theory.categories", "open import category-theory.large-categories", "open import category-theory.large-precategories", "open import category-theory.precategories", "open import category-theory.presheaf-categories", "open import category-theory.simplex-category", "open import foundation.commuting-squares-of-maps", "open import foundation.function-types", "open import foundation.homotopies", "open import foundation.identity-types", "open import foundation.sets", "open import foundation.strictly-involutive-identity-types", "open import foundation.universe-levels" ]	src/category-theory/category-of-simplicial-sets.lagda.md	hom-set-sSet
hom-sSet : {l1 l2 : Level} (X : sSet l1) (Y : sSet l2) → UU (l1 ⊔ l2)	function	src	[ "open import category-theory.categories", "open import category-theory.large-categories", "open import category-theory.large-precategories", "open import category-theory.precategories", "open import category-theory.presheaf-categories", "open import category-theory.simplex-category", "open import foundation.commuting-squares-of-maps", "open import foundation.function-types", "open import foundation.homotopies", "open import foundation.identity-types", "open import foundation.sets", "open import foundation.strictly-involutive-identity-types", "open import foundation.universe-levels" ]	src/category-theory/category-of-simplicial-sets.lagda.md	hom-sSet
id-hom-sSet : {l1 : Level} (X : sSet l1) → hom-sSet X X	function	src	[ "open import category-theory.categories", "open import category-theory.large-categories", "open import category-theory.large-precategories", "open import category-theory.precategories", "open import category-theory.presheaf-categories", "open import category-theory.simplex-category", "open import foundation.commuting-squares-of-maps", "open import foundation.function-types", "open import foundation.homotopies", "open import foundation.identity-types", "open import foundation.sets", "open import foundation.strictly-involutive-identity-types", "open import foundation.universe-levels" ]	src/category-theory/category-of-simplicial-sets.lagda.md	id-hom-sSet
sSet-Precategory : (l : Level) → Precategory (lsuc l) l	function	src	[ "open import category-theory.categories", "open import category-theory.large-categories", "open import category-theory.large-precategories", "open import category-theory.precategories", "open import category-theory.presheaf-categories", "open import category-theory.simplex-category", "open import foundation.commuting-squares-of-maps", "open import foundation.function-types", "open import foundation.homotopies", "open import foundation.identity-types", "open import foundation.sets", "open import foundation.strictly-involutive-identity-types", "open import foundation.universe-levels" ]	src/category-theory/category-of-simplicial-sets.lagda.md	sSet-Precategory
sSet-Category : (l : Level) → Category (lsuc l) l	function	src	[ "open import category-theory.categories", "open import category-theory.large-categories", "open import category-theory.large-precategories", "open import category-theory.precategories", "open import category-theory.presheaf-categories", "open import category-theory.simplex-category", "open import foundation.commuting-squares-of-maps", "open import foundation.function-types", "open import foundation.homotopies", "open import foundation.identity-types", "open import foundation.sets", "open import foundation.strictly-involutive-identity-types", "open import foundation.universe-levels" ]	src/category-theory/category-of-simplicial-sets.lagda.md	sSet-Category
extensions : the identity map on `D` trivially gives a right extension of `D`	function	src	[ "open import category-theory.algebras-monads-on-precategories", "open import category-theory.functors-precategories", "open import category-theory.monads-on-precategories", "open import category-theory.natural-transformations-functors-precategories", "open import category-theory.precategories", "open import category-theory.right-extensions-precategories", "open import category-theory.right-kan-extensions-precategories", "open import foundation.action-on-identifications-functions", "open import foundation.dependent-pair-types", "open import foundation.equivalences", "open import foundation.identity-types", "open import foundation.universe-levels" ]	src/category-theory/codensity-monads-on-precategories.lagda.md	extensions
core-precategory-Category : {l1 l2 : Level} (C : Category l1 l2) → Precategory l1 l2	function	src	[ "open import category-theory.categories", "open import category-theory.cores-precategories", "open import category-theory.groupoids", "open import category-theory.isomorphisms-in-categories", "open import category-theory.precategories", "open import category-theory.pregroupoids", "open import category-theory.subcategories", "open import category-theory.wide-subcategories", "open import foundation.dependent-pair-types", "open import foundation.equivalences", "open import foundation.universe-levels" ]	src/category-theory/cores-categories.lagda.md	core-precategory-Category
core-category-Category : {l1 l2 : Level} (C : Category l1 l2) → Category l1 l2	function	src	[ "open import category-theory.categories", "open import category-theory.cores-precategories", "open import category-theory.groupoids", "open import category-theory.isomorphisms-in-categories", "open import category-theory.precategories", "open import category-theory.pregroupoids", "open import category-theory.subcategories", "open import category-theory.wide-subcategories", "open import foundation.dependent-pair-types", "open import foundation.equivalences", "open import foundation.universe-levels" ]	src/category-theory/cores-categories.lagda.md	core-category-Category
core-precategory-Precategory : {l1 l2 : Level} (C : Precategory l1 l2) → Precategory l1 l2	function	src	[ "open import category-theory.categories", "open import category-theory.isomorphisms-in-categories", "open import category-theory.isomorphisms-in-precategories", "open import category-theory.precategories", "open import category-theory.pregroupoids", "open import category-theory.replete-subprecategories", "open import category-theory.subprecategories", "open import category-theory.wide-subprecategories", "open import foundation.contractible-types", "open import foundation.dependent-pair-types", "open import foundation.equivalences", "open import foundation.functoriality-dependent-pair-types", "open import foundation.fundamental-theorem-of-identity-types", "open import foundation.subtypes", "open import foundation.torsorial-type-families", "open import foundation.unit-type", "open import foundation.universe-levels" ]	src/category-theory/cores-precategories.lagda.md	core-precategory-Precategory
Full-Subcategory : {l1 l2 : Level} (l3 : Level) (C : Category l1 l2) → UU (l1 ⊔ lsuc l3)	function	src	[ "open import category-theory.categories", "open import category-theory.composition-operations-on-binary-families-of-sets", "open import category-theory.embeddings-precategories", "open import category-theory.full-subprecategories", "open import category-theory.fully-faithful-functors-precategories", "open import category-theory.functors-categories", "open import category-theory.maps-categories", "open import category-theory.precategories", "open import foundation.dependent-pair-types", "open import foundation.embeddings", "open import foundation.identity-types", "open import foundation.propositions", "open import foundation.sets", "open import foundation.strictly-involutive-identity-types", "open import foundation.subtypes", "open import foundation.universe-levels" ]	src/category-theory/full-subcategories.lagda.md	Full-Subcategory
Full-Subprecategory : {l1 l2 : Level} (l3 : Level) (C : Precategory l1 l2) → UU (l1 ⊔ lsuc l3)	function	src	[ "open import category-theory.categories", "open import category-theory.composition-operations-on-binary-families-of-sets", "open import category-theory.embeddings-precategories", "open import category-theory.fully-faithful-functors-precategories", "open import category-theory.functors-precategories", "open import category-theory.isomorphisms-in-categories", "open import category-theory.isomorphisms-in-precategories", "open import category-theory.maps-precategories", "open import category-theory.precategories", "open import foundation.dependent-pair-types", "open import foundation.embeddings", "open import foundation.equivalences", "open import foundation.function-types", "open import foundation.fundamental-theorem-of-identity-types", "open import foundation.identity-types", "open import foundation.propositions", "open import foundation.sets", "open import foundation.strictly-involutive-identity-types", "open import foundation.subtype-identity-principle", "open import foundation.subtypes", "open import foundation.universe-levels" ]	src/category-theory/full-subprecategories.lagda.md	Full-Subprecategory
id-functor-Category : {l1 l2 : Level} (C : Category l1 l2) → functor-Category C C	function	src	[ "open import category-theory.categories", "open import category-theory.functors-precategories", "open import category-theory.isomorphisms-in-categories", "open import category-theory.maps-categories", "open import foundation.equivalences", "open import foundation.function-types", "open import foundation.homotopies", "open import foundation.identity-types", "open import foundation.propositions", "open import foundation.universe-levels" ]	src/category-theory/functors-categories.lagda.md	id-functor-Category
of : - a map `F₀ : C → D` on objects at some chosen universe level `γ`,	function	src	[ "open import category-theory.functors-precategories", "open import category-theory.large-precategories", "open import category-theory.maps-from-small-to-large-precategories", "open import category-theory.precategories", "open import foundation.cartesian-product-types", "open import foundation.dependent-pair-types", "open import foundation.equivalences", "open import foundation.function-types", "open import foundation.homotopies", "open import foundation.identity-types", "open import foundation.universe-levels" ]	src/category-theory/functors-from-small-to-large-precategories.lagda.md	of
id-functor-Precategory : {l1 l2 : Level} (C : Precategory l1 l2) → functor-Precategory C C	function	src	[ "open import category-theory.functors-set-magmoids", "open import category-theory.isomorphisms-in-precategories", "open import category-theory.maps-precategories", "open import category-theory.opposite-precategories", "open import category-theory.precategories", "open import foundation.action-on-identifications-functions", "open import foundation.dependent-pair-types", "open import foundation.embeddings", "open import foundation.equivalences", "open import foundation.function-types", "open import foundation.homotopies", "open import foundation.identity-types", "open import foundation.propositions", "open import foundation.subtypes", "open import foundation.type-arithmetic-dependent-pair-types", "open import foundation.universe-levels" ]	src/category-theory/functors-precategories.lagda.md	id-functor-Precategory
Gaunt-Category : (l1 l2 : Level) → UU (lsuc l1 ⊔ lsuc l2)	function	src	[ "open import category-theory.categories", "open import category-theory.composition-operations-on-binary-families-of-sets", "open import category-theory.isomorphism-induction-categories", "open import category-theory.isomorphisms-in-categories", "open import category-theory.isomorphisms-in-precategories", "open import category-theory.nonunital-precategories", "open import category-theory.precategories", "open import category-theory.rigid-objects-categories", "open import category-theory.strict-categories", "open import category-theory.strongly-preunivalent-categories", "open import foundation.cartesian-product-types", "open import foundation.dependent-pair-types", "open import foundation.identity-types", "open import foundation.propositions", "open import foundation.sets", "open import foundation.strictly-involutive-identity-types", "open import foundation.surjective-maps", "open import foundation.universe-levels" ]	src/category-theory/gaunt-categories.lagda.md	Gaunt-Category
nonunital-precategory-Gaunt-Category : {l1 l2 : Level} → Gaunt-Category l1 l2 → Nonunital-Precategory l1 l2	function	src	[ "open import category-theory.categories", "open import category-theory.composition-operations-on-binary-families-of-sets", "open import category-theory.isomorphism-induction-categories", "open import category-theory.isomorphisms-in-categories", "open import category-theory.isomorphisms-in-precategories", "open import category-theory.nonunital-precategories", "open import category-theory.precategories", "open import category-theory.rigid-objects-categories", "open import category-theory.strict-categories", "open import category-theory.strongly-preunivalent-categories", "open import foundation.cartesian-product-types", "open import foundation.dependent-pair-types", "open import foundation.identity-types", "open import foundation.propositions", "open import foundation.sets", "open import foundation.strictly-involutive-identity-types", "open import foundation.surjective-maps", "open import foundation.universe-levels" ]	src/category-theory/gaunt-categories.lagda.md	nonunital-precategory-Gaunt-Category
precategory-Gaunt-Category : {l1 l2 : Level} → Gaunt-Category l1 l2 → Precategory l1 l2	function	src	[ "open import category-theory.categories", "open import category-theory.composition-operations-on-binary-families-of-sets", "open import category-theory.isomorphism-induction-categories", "open import category-theory.isomorphisms-in-categories", "open import category-theory.isomorphisms-in-precategories", "open import category-theory.nonunital-precategories", "open import category-theory.precategories", "open import category-theory.rigid-objects-categories", "open import category-theory.strict-categories", "open import category-theory.strongly-preunivalent-categories", "open import foundation.cartesian-product-types", "open import foundation.dependent-pair-types", "open import foundation.identity-types", "open import foundation.propositions", "open import foundation.sets", "open import foundation.strictly-involutive-identity-types", "open import foundation.surjective-maps", "open import foundation.universe-levels" ]	src/category-theory/gaunt-categories.lagda.md	precategory-Gaunt-Category
strongly-preunivalent-category-Gaunt-Category : {l1 l2 : Level} → Gaunt-Category l1 l2 → Strongly-Preunivalent-Category l1 l2	function	src	[ "open import category-theory.categories", "open import category-theory.composition-operations-on-binary-families-of-sets", "open import category-theory.isomorphism-induction-categories", "open import category-theory.isomorphisms-in-categories", "open import category-theory.isomorphisms-in-precategories", "open import category-theory.nonunital-precategories", "open import category-theory.precategories", "open import category-theory.rigid-objects-categories", "open import category-theory.strict-categories", "open import category-theory.strongly-preunivalent-categories", "open import foundation.cartesian-product-types", "open import foundation.dependent-pair-types", "open import foundation.identity-types", "open import foundation.propositions", "open import foundation.sets", "open import foundation.strictly-involutive-identity-types", "open import foundation.surjective-maps", "open import foundation.universe-levels" ]	src/category-theory/gaunt-categories.lagda.md	strongly-preunivalent-category-Gaunt-Category
total-hom-Gaunt-Category : {l1 l2 : Level} (C : Gaunt-Category l1 l2) → UU (l1 ⊔ l2)	function	src	[ "open import category-theory.categories", "open import category-theory.composition-operations-on-binary-families-of-sets", "open import category-theory.isomorphism-induction-categories", "open import category-theory.isomorphisms-in-categories", "open import category-theory.isomorphisms-in-precategories", "open import category-theory.nonunital-precategories", "open import category-theory.precategories", "open import category-theory.rigid-objects-categories", "open import category-theory.strict-categories", "open import category-theory.strongly-preunivalent-categories", "open import foundation.cartesian-product-types", "open import foundation.dependent-pair-types", "open import foundation.identity-types", "open import foundation.propositions", "open import foundation.sets", "open import foundation.strictly-involutive-identity-types", "open import foundation.surjective-maps", "open import foundation.universe-levels" ]	src/category-theory/gaunt-categories.lagda.md	total-hom-Gaunt-Category
is-groupoid-prop-Category : {l1 l2 : Level} (C : Category l1 l2) → Prop (l1 ⊔ l2)	function	src	[ "open import category-theory.categories", "open import category-theory.functors-categories", "open import category-theory.isomorphisms-in-categories", "open import category-theory.isomorphisms-in-precategories", "open import category-theory.precategories", "open import category-theory.pregroupoids", "open import foundation.1-types", "open import foundation.contractible-types", "open import foundation.dependent-pair-types", "open import foundation.equivalences", "open import foundation.function-types", "open import foundation.functoriality-dependent-pair-types", "open import foundation.fundamental-theorem-of-identity-types", "open import foundation.identity-types", "open import foundation.iterated-dependent-pair-types", "open import foundation.propositions", "open import foundation.sets", "open import foundation.strictly-involutive-identity-types", "open import foundation.torsorial-type-families", "open import foundation.type-arithmetic-dependent-pair-types", "open import foundation.universe-levels" ]	src/category-theory/groupoids.lagda.md	is-groupoid-prop-Category
is-groupoid-Category : {l1 l2 : Level} (C : Category l1 l2) → UU (l1 ⊔ l2)	function	src	[ "open import category-theory.categories", "open import category-theory.functors-categories", "open import category-theory.isomorphisms-in-categories", "open import category-theory.isomorphisms-in-precategories", "open import category-theory.precategories", "open import category-theory.pregroupoids", "open import foundation.1-types", "open import foundation.contractible-types", "open import foundation.dependent-pair-types", "open import foundation.equivalences", "open import foundation.function-types", "open import foundation.functoriality-dependent-pair-types", "open import foundation.fundamental-theorem-of-identity-types", "open import foundation.identity-types", "open import foundation.iterated-dependent-pair-types", "open import foundation.propositions", "open import foundation.sets", "open import foundation.strictly-involutive-identity-types", "open import foundation.torsorial-type-families", "open import foundation.type-arithmetic-dependent-pair-types", "open import foundation.universe-levels" ]	src/category-theory/groupoids.lagda.md	is-groupoid-Category
Groupoid : (l1 l2 : Level) → UU (lsuc l1 ⊔ lsuc l2)	function	src	[ "open import category-theory.categories", "open import category-theory.functors-categories", "open import category-theory.isomorphisms-in-categories", "open import category-theory.isomorphisms-in-precategories", "open import category-theory.precategories", "open import category-theory.pregroupoids", "open import foundation.1-types", "open import foundation.contractible-types", "open import foundation.dependent-pair-types", "open import foundation.equivalences", "open import foundation.function-types", "open import foundation.functoriality-dependent-pair-types", "open import foundation.fundamental-theorem-of-identity-types", "open import foundation.identity-types", "open import foundation.iterated-dependent-pair-types", "open import foundation.propositions", "open import foundation.sets", "open import foundation.strictly-involutive-identity-types", "open import foundation.torsorial-type-families", "open import foundation.type-arithmetic-dependent-pair-types", "open import foundation.universe-levels" ]	src/category-theory/groupoids.lagda.md	Groupoid
is-section-indiscrete-Precategory : {l : Level} → obj-Precategory ∘ indiscrete-Precategory {l} ~ id	function	src	[ "open import category-theory.isomorphisms-in-precategories", "open import category-theory.precategories", "open import category-theory.pregroupoids", "open import category-theory.preunivalent-categories", "open import category-theory.strict-categories", "open import category-theory.subterminal-precategories", "open import foundation.contractible-types", "open import foundation.dependent-pair-types", "open import foundation.equivalences", "open import foundation.function-types", "open import foundation.homotopies", "open import foundation.identity-types", "open import foundation.iterated-dependent-product-types", "open import foundation.propositions", "open import foundation.sets", "open import foundation.unit-type", "open import foundation.universe-levels" ]	src/category-theory/indiscrete-precategories.lagda.md	is-section-indiscrete-Precategory
obj-initial-Category : UU lzero	function	src	[ "open import category-theory.categories", "open import category-theory.functors-precategories", "open import category-theory.gaunt-categories", "open import category-theory.indiscrete-precategories", "open import category-theory.precategories", "open import category-theory.strict-categories", "open import category-theory.strongly-preunivalent-categories", "open import foundation.contractible-types", "open import foundation.dependent-pair-types", "open import foundation.empty-types", "open import foundation.identity-types", "open import foundation.sets", "open import foundation.unit-type", "open import foundation.universe-levels" ]	src/category-theory/initial-category.lagda.md	obj-initial-Category
hom-set-initial-Category : obj-initial-Category → obj-initial-Category → Set lzero	function	src	[ "open import category-theory.categories", "open import category-theory.functors-precategories", "open import category-theory.gaunt-categories", "open import category-theory.indiscrete-precategories", "open import category-theory.precategories", "open import category-theory.strict-categories", "open import category-theory.strongly-preunivalent-categories", "open import foundation.contractible-types", "open import foundation.dependent-pair-types", "open import foundation.empty-types", "open import foundation.identity-types", "open import foundation.sets", "open import foundation.unit-type", "open import foundation.universe-levels" ]	src/category-theory/initial-category.lagda.md	hom-set-initial-Category
hom-initial-Category : obj-initial-Category → obj-initial-Category → UU lzero	function	src	[ "open import category-theory.categories", "open import category-theory.functors-precategories", "open import category-theory.gaunt-categories", "open import category-theory.indiscrete-precategories", "open import category-theory.precategories", "open import category-theory.strict-categories", "open import category-theory.strongly-preunivalent-categories", "open import foundation.contractible-types", "open import foundation.dependent-pair-types", "open import foundation.empty-types", "open import foundation.identity-types", "open import foundation.sets", "open import foundation.unit-type", "open import foundation.universe-levels" ]	src/category-theory/initial-category.lagda.md	hom-initial-Category
id-hom-initial-Category : {x : obj-initial-Category} → hom-initial-Category x x	function	src	[ "open import category-theory.categories", "open import category-theory.functors-precategories", "open import category-theory.gaunt-categories", "open import category-theory.indiscrete-precategories", "open import category-theory.precategories", "open import category-theory.strict-categories", "open import category-theory.strongly-preunivalent-categories", "open import foundation.contractible-types", "open import foundation.dependent-pair-types", "open import foundation.empty-types", "open import foundation.identity-types", "open import foundation.sets", "open import foundation.unit-type", "open import foundation.universe-levels" ]	src/category-theory/initial-category.lagda.md	id-hom-initial-Category
initial-Precategory : Precategory lzero lzero	function	src	[ "open import category-theory.categories", "open import category-theory.functors-precategories", "open import category-theory.gaunt-categories", "open import category-theory.indiscrete-precategories", "open import category-theory.precategories", "open import category-theory.strict-categories", "open import category-theory.strongly-preunivalent-categories", "open import foundation.contractible-types", "open import foundation.dependent-pair-types", "open import foundation.empty-types", "open import foundation.identity-types", "open import foundation.sets", "open import foundation.unit-type", "open import foundation.universe-levels" ]	src/category-theory/initial-category.lagda.md	initial-Precategory
is-category-initial-Category : is-category-Precategory initial-Precategory	function	src	[ "open import category-theory.categories", "open import category-theory.functors-precategories", "open import category-theory.gaunt-categories", "open import category-theory.indiscrete-precategories", "open import category-theory.precategories", "open import category-theory.strict-categories", "open import category-theory.strongly-preunivalent-categories", "open import foundation.contractible-types", "open import foundation.dependent-pair-types", "open import foundation.empty-types", "open import foundation.identity-types", "open import foundation.sets", "open import foundation.unit-type", "open import foundation.universe-levels" ]	src/category-theory/initial-category.lagda.md	is-category-initial-Category
initial-Category : Category lzero lzero	function	src	[ "open import category-theory.categories", "open import category-theory.functors-precategories", "open import category-theory.gaunt-categories", "open import category-theory.indiscrete-precategories", "open import category-theory.precategories", "open import category-theory.strict-categories", "open import category-theory.strongly-preunivalent-categories", "open import foundation.contractible-types", "open import foundation.dependent-pair-types", "open import foundation.empty-types", "open import foundation.identity-types", "open import foundation.sets", "open import foundation.unit-type", "open import foundation.universe-levels" ]	src/category-theory/initial-category.lagda.md	initial-Category
is-strongly-preunivalent-initial-Category : is-strongly-preunivalent-Precategory initial-Precategory	function	src	[ "open import category-theory.categories", "open import category-theory.functors-precategories", "open import category-theory.gaunt-categories", "open import category-theory.indiscrete-precategories", "open import category-theory.precategories", "open import category-theory.strict-categories", "open import category-theory.strongly-preunivalent-categories", "open import foundation.contractible-types", "open import foundation.dependent-pair-types", "open import foundation.empty-types", "open import foundation.identity-types", "open import foundation.sets", "open import foundation.unit-type", "open import foundation.universe-levels" ]	src/category-theory/initial-category.lagda.md	is-strongly-preunivalent-initial-Category
is-strict-category-initial-Category : is-strict-category-Precategory initial-Precategory	function	src	[ "open import category-theory.categories", "open import category-theory.functors-precategories", "open import category-theory.gaunt-categories", "open import category-theory.indiscrete-precategories", "open import category-theory.precategories", "open import category-theory.strict-categories", "open import category-theory.strongly-preunivalent-categories", "open import foundation.contractible-types", "open import foundation.dependent-pair-types", "open import foundation.empty-types", "open import foundation.identity-types", "open import foundation.sets", "open import foundation.unit-type", "open import foundation.universe-levels" ]	src/category-theory/initial-category.lagda.md	is-strict-category-initial-Category
initial-Strict-Category : Strict-Category lzero lzero	function	src	[ "open import category-theory.categories", "open import category-theory.functors-precategories", "open import category-theory.gaunt-categories", "open import category-theory.indiscrete-precategories", "open import category-theory.precategories", "open import category-theory.strict-categories", "open import category-theory.strongly-preunivalent-categories", "open import foundation.contractible-types", "open import foundation.dependent-pair-types", "open import foundation.empty-types", "open import foundation.identity-types", "open import foundation.sets", "open import foundation.unit-type", "open import foundation.universe-levels" ]	src/category-theory/initial-category.lagda.md	initial-Strict-Category
is-gaunt-initial-Category : is-gaunt-Category initial-Category	function	src	[ "open import category-theory.categories", "open import category-theory.functors-precategories", "open import category-theory.gaunt-categories", "open import category-theory.indiscrete-precategories", "open import category-theory.precategories", "open import category-theory.strict-categories", "open import category-theory.strongly-preunivalent-categories", "open import foundation.contractible-types", "open import foundation.dependent-pair-types", "open import foundation.empty-types", "open import foundation.identity-types", "open import foundation.sets", "open import foundation.unit-type", "open import foundation.universe-levels" ]	src/category-theory/initial-category.lagda.md	is-gaunt-initial-Category
initial-Gaunt-Category : Gaunt-Category lzero lzero	function	src	[ "open import category-theory.categories", "open import category-theory.functors-precategories", "open import category-theory.gaunt-categories", "open import category-theory.indiscrete-precategories", "open import category-theory.precategories", "open import category-theory.strict-categories", "open import category-theory.strongly-preunivalent-categories", "open import foundation.contractible-types", "open import foundation.dependent-pair-types", "open import foundation.empty-types", "open import foundation.identity-types", "open import foundation.sets", "open import foundation.unit-type", "open import foundation.universe-levels" ]	src/category-theory/initial-category.lagda.md	initial-Gaunt-Category
initial-obj-Precategory : {l1 l2 : Level} (C : Precategory l1 l2) → UU (l1 ⊔ l2)	function	src	[ "open import category-theory.precategories", "open import foundation.contractible-types", "open import foundation.dependent-pair-types", "open import foundation.function-types", "open import foundation.propositions", "open import foundation.universe-levels", "open import foundation-core.identity-types" ]	src/category-theory/initial-objects-precategories.lagda.md	initial-obj-Precategory
Large-Category (α : Level → Level) (β : Level → Level → Level) : UUω where constructor make-Large-Category field large-precategory-Large-Category : Large-Precategory α β is-large-category-Large-Category : is-large-category-Large-Precategory large-precategory-Large-Category	record	src	[ "open import category-theory.categories", "open import category-theory.isomorphisms-in-large-precategories", "open import category-theory.large-precategories", "open import category-theory.precategories", "open import foundation.action-on-identifications-binary-functions", "open import foundation.dependent-pair-types", "open import foundation.equivalences", "open import foundation.homotopies", "open import foundation.identity-types", "open import foundation.sets", "open import foundation.strictly-involutive-identity-types", "open import foundation.universe-levels" ]	src/category-theory/large-categories.lagda.md	Large-Category
Large-Precategory (α : Level → Level) (β : Level → Level → Level) : UUω where field obj-Large-Precategory : (l : Level) → UU (α l) hom-set-Large-Precategory : {l1 l2 : Level} → obj-Large-Precategory l1 → obj-Large-Precategory l2 → Set (β l1 l2) hom-Large-Precategory : {l1 l2 : Level} → obj-Large-Precategory l1 → obj-Large-Precategory l2 → UU (β l1 l2) hom-Large-Precategory X Y = type-Set (hom-set-Large-Precategory X Y) is-set-hom-Large-Precategory : {l1 l2 : Level} (X : obj-Large-Precategory l1) (Y : obj-Large-Precategory l2) → is-set (hom-Large-Precategory X Y) is-set-hom-Large-Precategory X Y = is-set-type-Set (hom-set-Large-Precategory X Y) field comp-hom-Large-Precategory : {l1 l2 l3 : Level} {X : obj-Large-Precategory l1} {Y : obj-Large-Precategory l2} {Z : obj-Large-Precategory l3} → hom-Large-Precategory Y Z → hom-Large-Precategory X Y → hom-Large-Precategory X Z id-hom-Large-Precategory : {l1 : Level} {X : obj-Large-Precategory l1} → hom-Large-Precategory X X involutive-eq-associative-comp-hom-Large-Precategory : {l1 l2 l3 l4 : Level} {X : obj-Large-Precategory l1} {Y : obj-Large-Precategory l2} {Z : obj-Large-Precategory l3} {W : obj-Large-Precategory l4} → (h : hom-Large-Precategory Z W) (g : hom-Large-Precategory Y Z) (f : hom-Large-Precategory X Y) → ( comp-hom-Large-Precategory (comp-hom-Large-Precategory h g) f) ＝ⁱ ( comp-hom-Large-Precategory h (comp-hom-Large-Precategory g f)) left-unit-law-comp-hom-Large-Precategory : {l1 l2 : Level} {X : obj-Large-Precategory l1} {Y : obj-Large-Precategory l2} (f : hom-Large-Precategory X Y) → ( comp-hom-Large-Precategory id-hom-Large-Precategory f) ＝ f right-unit-law-comp-hom-Large-Precategory : {l1 l2 : Level} {X : obj-Large-Precategory l1} {Y : obj-Large-Precategory l2} (f : hom-Large-Precategory X Y) → ( comp-hom-Large-Precategory f id-hom-Large-Precategory) ＝ f associative-comp-hom-Large-Precategory : {l1 l2 l3 l4 : Level} {X : obj-Large-Precategory l1} {Y : obj-Large-Precategory l2} {Z : obj-Large-Precategory l3} {W : obj-Large-Precategory l4} → (h : hom-Large-Precategory Z W) (g : hom-Large-Precategory Y Z) (f : hom-Large-Precategory X Y) → ( comp-hom-Large-Precategory (comp-hom-Large-Precategory h g) f) ＝ ( comp-hom-Large-Precategory h (comp-hom-Large-Precategory g f)) associative-comp-hom-Large-Precategory h g f = eq-involutive-eq ( involutive-eq-associative-comp-hom-Large-Precategory h g f)	record	src	[ "open import category-theory.precategories", "open import foundation.action-on-identifications-binary-functions", "open import foundation.dependent-pair-types", "open import foundation.function-types", "open import foundation.homotopies", "open import foundation.identity-types", "open import foundation.sets", "open import foundation.strictly-involutive-identity-types", "open import foundation.universe-levels" ]	src/category-theory/large-precategories.lagda.md	Large-Precategory
Large-Subprecategory {α : Level → Level} {β : Level → Level → Level} (γ : Level → Level) (δ : Level → Level → Level) (C : Large-Precategory α β) : UUω where field subtype-obj-Large-Subprecategory : (l : Level) → subtype (γ l) (obj-Large-Precategory C l) is-in-obj-Large-Subprecategory : {l : Level} → obj-Large-Precategory C l → UU (γ l) is-in-obj-Large-Subprecategory {l} = is-in-subtype (subtype-obj-Large-Subprecategory l) obj-Large-Subprecategory : (l : Level) → UU (α l ⊔ γ l) obj-Large-Subprecategory l = type-subtype (subtype-obj-Large-Subprecategory l) field subtype-hom-Large-Subprecategory : {l1 l2 : Level} (X : obj-Large-Precategory C l1) (Y : obj-Large-Precategory C l2) → is-in-obj-Large-Subprecategory X → is-in-obj-Large-Subprecategory Y → subtype (δ l1 l2) (hom-Large-Precategory C X Y) is-in-hom-is-in-obj-Large-Subprecategory : {l1 l2 : Level} {X : obj-Large-Precategory C l1} {Y : obj-Large-Precategory C l2} (x : is-in-obj-Large-Subprecategory X) (y : is-in-obj-Large-Subprecategory Y) → hom-Large-Precategory C X Y → UU (δ l1 l2) is-in-hom-is-in-obj-Large-Subprecategory {l1} {l2} {X} {Y} x y = is-in-subtype (subtype-hom-Large-Subprecategory X Y x y) field contains-id-Large-Subprecategory : {l : Level} (X : obj-Large-Precategory C l) → (H : is-in-obj-Large-Subprecategory X) → is-in-hom-is-in-obj-Large-Subprecategory H H (id-hom-Large-Precategory C) is-closed-under-composition-Large-Subprecategory : {l1 l2 l3 : Level} (X : obj-Large-Precategory C l1) (Y : obj-Large-Precategory C l2) (Z : obj-Large-Precategory C l3) (g : hom-Large-Precategory C Y Z) (f : hom-Large-Precategory C X Y) → (K : is-in-obj-Large-Subprecategory X) → (L : is-in-obj-Large-Subprecategory Y) → (M : is-in-obj-Large-Subprecategory Z) → is-in-hom-is-in-obj-Large-Subprecategory L M g → is-in-hom-is-in-obj-Large-Subprecategory K L f → is-in-hom-is-in-obj-Large-Subprecategory K M ( comp-hom-Large-Precategory C g f) hom-Large-Subprecategory : {l1 l2 : Level} (X : obj-Large-Subprecategory l1) (Y : obj-Large-Subprecategory l2) → UU (β l1 l2 ⊔ δ l1 l2) hom-Large-Subprecategory (X , x) (Y , y) = type-subtype (subtype-hom-Large-Subprecategory X Y x y) hom-set-Large-Subprecategory : {l1 l2 : Level} (X : obj-Large-Subprecategory l1) (Y : obj-Large-Subprecategory l2) → Set (β l1 l2 ⊔ δ l1 l2) hom-set-Large-Subprecategory (X , x) (Y , y) = set-subset ( hom-set-Large-Precategory C X Y) ( subtype-hom-Large-Subprecategory X Y x y) is-set-hom-Large-Subprecategory : {l1 l2 : Level} (X : obj-Large-Subprecategory l1) (Y : obj-Large-Subprecategory l2) → is-set (hom-Large-Subprecategory X Y) is-set-hom-Large-Subprecategory X Y = is-set-type-Set (hom-set-Large-Subprecategory X Y) id-hom-Large-Subprecategory : {l : Level} (X : obj-Large-Subprecategory l) → hom-Large-Subprecategory X X id-hom-Large-Subprecategory (X , x) = ( id-hom-Large-Precategory C , contains-id-Large-Subprecategory X x) comp-hom-Large-Subprecategory : {l1 l2 l3 : Level} (X : obj-Large-Subprecategory l1) (Y : obj-Large-Subprecategory l2) (Z : obj-Large-Subprecategory l3) → hom-Large-Subprecategory Y Z → hom-Large-Subprecategory X Y → hom-Large-Subprecategory X Z comp-hom-Large-Subprecategory (X , x) (Y , y) (Z , z) (G , g) (F , f) = ( comp-hom-Large-Precategory C G F , is-closed-under-composition-Large-Subprecategory X Y Z G F x y z g f) associative-comp-hom-Large-Subprecategory : {l1 l2 l3 l4 : Level} (X : obj-Large-Subprecategory l1) (Y : obj-Large-Subprecategory l2) (Z : obj-Large-Subprecategory l3) (W : obj-Large-Subprecategory l4) (h : hom-Large-Subprecategory Z W) (g : hom-Large-Subprecategory Y Z) (f : hom-Large-Subprecategory X Y) → comp-hom-Large-Subprecategory X Y W ( comp-hom-Large-Subprecategory Y Z W h g) ( f) ＝ comp-hom-Large-Subprecategory X Z W ( h) ( comp-hom-Large-Subprecategory X Y Z g f) associative-comp-hom-Large-Subprecategory ( X , x) (Y , y) (Z , z) (W , w) (H , h) (G , g) (F , f) = eq-type-subtype ( subtype-hom-Large-Subprecategory X W x w) ( associative-comp-hom-Large-Precategory C H G F) involutive-eq-associative-comp-hom-Large-Subprecategory : {l1 l2 l3 l4 : Level} (X : obj-Large-Subprecategory l1) (Y : obj-Large-Subprecategory l2) (Z : obj-Large-Subprecategory l3) (W : obj-Large-Subprecategory l4) (h : hom-Large-Subprecategory Z W) (g : hom-Large-Subprecategory Y Z) (f : hom-Large-Subprecategory X Y) → comp-hom-Large-Subprecategory X Y W ( comp-hom-Large-Subprecategory Y Z W h g) ( f) ＝ⁱ comp-hom-Large-Subprecategory X Z W ( h) ( comp-hom-Large-Subprecategory X Y Z g f) involutive-eq-associative-comp-hom-Large-Subprecategory X Y Z W h g f = involutive-eq-eq (associative-comp-hom-Large-Subprecategory X Y Z W h g f) left-unit-law-comp-hom-Large-Subprecategory : {l1 l2 : Level} (X : obj-Large-Subprecategory l1) (Y : obj-Large-Subprecategory l2) (f : hom-Large-Subprecategory X Y) → comp-hom-Large-Subprecategory X Y Y (id-hom-Large-Subprecategory Y) f ＝ f left-unit-law-comp-hom-Large-Subprecategory (X , x) (Y , y) (F , f) = eq-type-subtype ( subtype-hom-Large-Subprecategory X Y x y) ( left-unit-law-comp-hom-Large-Precategory C F) right-unit-law-comp-hom-Large-Subprecategory : {l1 l2 : Level} (X : obj-Large-Subprecategory l1) (Y : obj-Large-Subprecategory l2) (f : hom-Large-Subprecategory X Y) → comp-hom-Large-Subprecategory X X Y f (id-hom-Large-Subprecategory X) ＝ f right-unit-law-comp-hom-Large-Subprecategory (X , x) (Y , y) (F , f) = eq-type-subtype ( subtype-hom-Large-Subprecategory X Y x y) ( right-unit-law-comp-hom-Large-Precategory C F) ``` ### The underlying large precategory of a large subprecategory ```agda large-precategory-Large-Subprecategory : Large-Precategory (λ l → α l ⊔ γ l) (λ l1 l2 → β l1 l2 ⊔ δ l1 l2) large-precategory-Large-Subprecategory = λ where .obj-Large-Precategory → obj-Large-Subprecategory .hom-set-Large-Precategory → hom-set-Large-Subprecategory .comp-hom-Large-Precategory {X = X} {Y} {Z} → comp-hom-Large-Subprecategory X Y Z .id-hom-Large-Precategory {X = X} → id-hom-Large-Subprecategory X .involutive-eq-associative-comp-hom-Large-Precategory {X = X} {Y} {Z} {W} → involutive-eq-associative-comp-hom-Large-Subprecategory X Y Z W .left-unit-law-comp-hom-Large-Precategory {X = X} {Y} → left-unit-law-comp-hom-Large-Subprecategory X Y .right-unit-law-comp-hom-Large-Precategory {X = X} {Y} → right-unit-law-comp-hom-Large-Subprecategory X Y	record	src	[ "open import category-theory.large-precategories", "open import foundation.dependent-pair-types", "open import foundation.identity-types", "open import foundation.sets", "open import foundation.strictly-involutive-identity-types", "open import foundation.subtypes", "open import foundation.universe-levels" ]	src/category-theory/large-subprecategories.lagda.md	Large-Subprecategory
extensions : the identity map gives a left extension (with the identity natural	function	src	[ "open import category-theory.functors-precategories", "open import category-theory.natural-transformations-functors-precategories", "open import category-theory.precategories", "open import foundation.action-on-equivalences-functions", "open import foundation.action-on-identifications-functions", "open import foundation.contractible-types", "open import foundation.dependent-identifications", "open import foundation.dependent-pair-types", "open import foundation.equality-dependent-function-types", "open import foundation.equality-dependent-pair-types", "open import foundation.equivalences", "open import foundation.function-extensionality", "open import foundation.function-types", "open import foundation.functoriality-dependent-pair-types", "open import foundation.fundamental-theorem-of-identity-types", "open import foundation.homotopies", "open import foundation.homotopy-induction", "open import foundation.identity-types", "open import foundation.multivariable-homotopies", "open import foundation.propositions", "open import foundation.retractions", "open import foundation.sections", "open import foundation.sets", "open import foundation.structure-identity-principle", "open import foundation.torsorial-type-families", "open import foundation.unit-type", "open import foundation.universe-levels", "open import foundation-core.functoriality-dependent-function-types" ]	src/category-theory/left-extensions-precategories.lagda.md	extensions
Nonunital-Precategory : (l1 l2 : Level) → UU (lsuc l1 ⊔ lsuc l2)	function	src	[ "open import category-theory.composition-operations-on-binary-families-of-sets", "open import category-theory.set-magmoids", "open import foundation.cartesian-product-types", "open import foundation.dependent-pair-types", "open import foundation.identity-types", "open import foundation.propositions", "open import foundation.sets", "open import foundation.strictly-involutive-identity-types", "open import foundation.truncated-types", "open import foundation.truncation-levels", "open import foundation.universe-levels" ]	src/category-theory/nonunital-precategories.lagda.md	Nonunital-Precategory
total-hom-Nonunital-Precategory : {l1 l2 : Level} (C : Nonunital-Precategory l1 l2) → UU (l1 ⊔ l2)	function	src	[ "open import category-theory.composition-operations-on-binary-families-of-sets", "open import category-theory.set-magmoids", "open import foundation.cartesian-product-types", "open import foundation.dependent-pair-types", "open import foundation.identity-types", "open import foundation.propositions", "open import foundation.sets", "open import foundation.strictly-involutive-identity-types", "open import foundation.truncated-types", "open import foundation.truncation-levels", "open import foundation.universe-levels" ]	src/category-theory/nonunital-precategories.lagda.md	total-hom-Nonunital-Precategory
is-one-object-prop-Precategory : {l1 l2 : Level} → Precategory l1 l2 → Prop l1	function	src	[ "open import category-theory.endomorphisms-in-precategories", "open import category-theory.precategories", "open import foundation.contractible-types", "open import foundation.dependent-pair-types", "open import foundation.identity-types", "open import foundation.propositions", "open import foundation.sets", "open import foundation.unit-type", "open import foundation.universe-levels", "open import group-theory.monoids" ]	src/category-theory/one-object-precategories.lagda.md	is-one-object-prop-Precategory
is-one-object-Precategory : {l1 l2 : Level} → Precategory l1 l2 → UU l1	function	src	[ "open import category-theory.endomorphisms-in-precategories", "open import category-theory.precategories", "open import foundation.contractible-types", "open import foundation.dependent-pair-types", "open import foundation.identity-types", "open import foundation.propositions", "open import foundation.sets", "open import foundation.unit-type", "open import foundation.universe-levels", "open import group-theory.monoids" ]	src/category-theory/one-object-precategories.lagda.md	is-one-object-Precategory
One-Object-Precategory : (l1 l2 : Level) → UU (lsuc l1 ⊔ lsuc l2)	function	src	[ "open import category-theory.endomorphisms-in-precategories", "open import category-theory.precategories", "open import foundation.contractible-types", "open import foundation.dependent-pair-types", "open import foundation.identity-types", "open import foundation.propositions", "open import foundation.sets", "open import foundation.unit-type", "open import foundation.universe-levels", "open import group-theory.monoids" ]	src/category-theory/one-object-precategories.lagda.md	One-Object-Precategory
monoid-One-Object-Precategory : {l1 l2 : Level} → One-Object-Precategory l1 l2 → Monoid l2	function	src	[ "open import category-theory.endomorphisms-in-precategories", "open import category-theory.precategories", "open import foundation.contractible-types", "open import foundation.dependent-pair-types", "open import foundation.identity-types", "open import foundation.propositions", "open import foundation.sets", "open import foundation.unit-type", "open import foundation.universe-levels", "open import group-theory.monoids" ]	src/category-theory/one-object-precategories.lagda.md	monoid-One-Object-Precategory
is-involution-opposite-Category : {l1 l2 : Level} → is-involution (opposite-Category {l1} {l2})	function	src	[ "open import category-theory.categories", "open import category-theory.isomorphisms-in-precategories", "open import category-theory.opposite-precategories", "open import category-theory.precategories", "open import foundation.dependent-pair-types", "open import foundation.equivalences", "open import foundation.identity-types", "open import foundation.involutions", "open import foundation.sets", "open import foundation.strictly-involutive-identity-types", "open import foundation.subtypes", "open import foundation.universe-levels" ]	src/category-theory/opposite-categories.lagda.md	is-involution-opposite-Category
involution-opposite-Category : (l1 l2 : Level) → involution (Category l1 l2)	function	src	[ "open import category-theory.categories", "open import category-theory.isomorphisms-in-precategories", "open import category-theory.opposite-precategories", "open import category-theory.precategories", "open import foundation.dependent-pair-types", "open import foundation.equivalences", "open import foundation.identity-types", "open import foundation.involutions", "open import foundation.sets", "open import foundation.strictly-involutive-identity-types", "open import foundation.subtypes", "open import foundation.universe-levels" ]	src/category-theory/opposite-categories.lagda.md	involution-opposite-Category
is-equiv-opposite-Category : {l1 l2 : Level} → is-equiv (opposite-Category {l1} {l2})	function	src	[ "open import category-theory.categories", "open import category-theory.isomorphisms-in-precategories", "open import category-theory.opposite-precategories", "open import category-theory.precategories", "open import foundation.dependent-pair-types", "open import foundation.equivalences", "open import foundation.identity-types", "open import foundation.involutions", "open import foundation.sets", "open import foundation.strictly-involutive-identity-types", "open import foundation.subtypes", "open import foundation.universe-levels" ]	src/category-theory/opposite-categories.lagda.md	is-equiv-opposite-Category
equiv-opposite-Category : (l1 l2 : Level) → Category l1 l2 ≃ Category l1 l2	function	src	[ "open import category-theory.categories", "open import category-theory.isomorphisms-in-precategories", "open import category-theory.opposite-precategories", "open import category-theory.precategories", "open import foundation.dependent-pair-types", "open import foundation.equivalences", "open import foundation.identity-types", "open import foundation.involutions", "open import foundation.sets", "open import foundation.strictly-involutive-identity-types", "open import foundation.subtypes", "open import foundation.universe-levels" ]	src/category-theory/opposite-categories.lagda.md	equiv-opposite-Category
is-involution-opposite-Precategory : {l1 l2 : Level} → is-involution (opposite-Precategory {l1} {l2})	function	src	[ "open import category-theory.isomorphisms-in-precategories", "open import category-theory.precategories", "open import foundation.dependent-pair-types", "open import foundation.equivalences", "open import foundation.homotopies", "open import foundation.identity-types", "open import foundation.involutions", "open import foundation.sets", "open import foundation.strictly-involutive-identity-types", "open import foundation.universe-levels" ]	src/category-theory/opposite-precategories.lagda.md	is-involution-opposite-Precategory
involution-opposite-Precategory : (l1 l2 : Level) → involution (Precategory l1 l2)	function	src	[ "open import category-theory.isomorphisms-in-precategories", "open import category-theory.precategories", "open import foundation.dependent-pair-types", "open import foundation.equivalences", "open import foundation.homotopies", "open import foundation.identity-types", "open import foundation.involutions", "open import foundation.sets", "open import foundation.strictly-involutive-identity-types", "open import foundation.universe-levels" ]	src/category-theory/opposite-precategories.lagda.md	involution-opposite-Precategory
is-equiv-opposite-Precategory : {l1 l2 : Level} → is-equiv (opposite-Precategory {l1} {l2})	function	src	[ "open import category-theory.isomorphisms-in-precategories", "open import category-theory.precategories", "open import foundation.dependent-pair-types", "open import foundation.equivalences", "open import foundation.homotopies", "open import foundation.identity-types", "open import foundation.involutions", "open import foundation.sets", "open import foundation.strictly-involutive-identity-types", "open import foundation.universe-levels" ]	src/category-theory/opposite-precategories.lagda.md	is-equiv-opposite-Precategory
equiv-opposite-Precategory : (l1 l2 : Level) → Precategory l1 l2 ≃ Precategory l1 l2	function	src	[ "open import category-theory.isomorphisms-in-precategories", "open import category-theory.precategories", "open import foundation.dependent-pair-types", "open import foundation.equivalences", "open import foundation.homotopies", "open import foundation.identity-types", "open import foundation.involutions", "open import foundation.sets", "open import foundation.strictly-involutive-identity-types", "open import foundation.universe-levels" ]	src/category-theory/opposite-precategories.lagda.md	equiv-opposite-Precategory
is-involution-opposite-Preunivalent-Category : {l1 l2 : Level} → is-involution (opposite-Preunivalent-Category {l1} {l2})	function	src	[ "open import category-theory.isomorphisms-in-precategories", "open import category-theory.opposite-precategories", "open import category-theory.precategories", "open import category-theory.preunivalent-categories", "open import foundation.dependent-pair-types", "open import foundation.embeddings", "open import foundation.equivalences", "open import foundation.identity-types", "open import foundation.involutions", "open import foundation.sets", "open import foundation.strictly-involutive-identity-types", "open import foundation.subtypes", "open import foundation.universe-levels" ]	src/category-theory/opposite-preunivalent-categories.lagda.md	is-involution-opposite-Preunivalent-Category
involution-opposite-Preunivalent-Category : (l1 l2 : Level) → involution (Preunivalent-Category l1 l2)	function	src	[ "open import category-theory.isomorphisms-in-precategories", "open import category-theory.opposite-precategories", "open import category-theory.precategories", "open import category-theory.preunivalent-categories", "open import foundation.dependent-pair-types", "open import foundation.embeddings", "open import foundation.equivalences", "open import foundation.identity-types", "open import foundation.involutions", "open import foundation.sets", "open import foundation.strictly-involutive-identity-types", "open import foundation.subtypes", "open import foundation.universe-levels" ]	src/category-theory/opposite-preunivalent-categories.lagda.md	involution-opposite-Preunivalent-Category
is-equiv-opposite-Preunivalent-Category : {l1 l2 : Level} → is-equiv (opposite-Preunivalent-Category {l1} {l2})	function	src	[ "open import category-theory.isomorphisms-in-precategories", "open import category-theory.opposite-precategories", "open import category-theory.precategories", "open import category-theory.preunivalent-categories", "open import foundation.dependent-pair-types", "open import foundation.embeddings", "open import foundation.equivalences", "open import foundation.identity-types", "open import foundation.involutions", "open import foundation.sets", "open import foundation.strictly-involutive-identity-types", "open import foundation.subtypes", "open import foundation.universe-levels" ]	src/category-theory/opposite-preunivalent-categories.lagda.md	is-equiv-opposite-Preunivalent-Category
equiv-opposite-Preunivalent-Category : (l1 l2 : Level) → Preunivalent-Category l1 l2 ≃ Preunivalent-Category l1 l2	function	src	[ "open import category-theory.isomorphisms-in-precategories", "open import category-theory.opposite-precategories", "open import category-theory.precategories", "open import category-theory.preunivalent-categories", "open import foundation.dependent-pair-types", "open import foundation.embeddings", "open import foundation.equivalences", "open import foundation.identity-types", "open import foundation.involutions", "open import foundation.sets", "open import foundation.strictly-involutive-identity-types", "open import foundation.subtypes", "open import foundation.universe-levels" ]	src/category-theory/opposite-preunivalent-categories.lagda.md	equiv-opposite-Preunivalent-Category
involution-opposite-Strongly-Preunivalent-Category : (l1 l2 : Level) → involution (Strongly-Preunivalent-Category l1 l2)	function	src	[ "open import category-theory.isomorphisms-in-precategories", "open import category-theory.opposite-precategories", "open import category-theory.precategories", "open import category-theory.strongly-preunivalent-categories", "open import foundation.dependent-pair-types", "open import foundation.embeddings", "open import foundation.equivalences", "open import foundation.functoriality-dependent-pair-types", "open import foundation.identity-types", "open import foundation.involutions", "open import foundation.sets", "open import foundation.strictly-involutive-identity-types", "open import foundation.subtypes", "open import foundation.universe-levels" ]	src/category-theory/opposite-strongly-preunivalent-categories.lagda.md	involution-opposite-Strongly-Preunivalent-Category
is-equiv-opposite-Strongly-Preunivalent-Category : {l1 l2 : Level} → is-equiv (opposite-Strongly-Preunivalent-Category {l1} {l2})	function	src	[ "open import category-theory.isomorphisms-in-precategories", "open import category-theory.opposite-precategories", "open import category-theory.precategories", "open import category-theory.strongly-preunivalent-categories", "open import foundation.dependent-pair-types", "open import foundation.embeddings", "open import foundation.equivalences", "open import foundation.functoriality-dependent-pair-types", "open import foundation.identity-types", "open import foundation.involutions", "open import foundation.sets", "open import foundation.strictly-involutive-identity-types", "open import foundation.subtypes", "open import foundation.universe-levels" ]	src/category-theory/opposite-strongly-preunivalent-categories.lagda.md	is-equiv-opposite-Strongly-Preunivalent-Category
Precategory : (l1 l2 : Level) → UU (lsuc l1 ⊔ lsuc l2)	function	src	[ "open import category-theory.composition-operations-on-binary-families-of-sets", "open import category-theory.nonunital-precategories", "open import category-theory.set-magmoids", "open import foundation.action-on-identifications-functions", "open import foundation.cartesian-product-types", "open import foundation.dependent-pair-types", "open import foundation.function-types", "open import foundation.identity-types", "open import foundation.propositions", "open import foundation.sets", "open import foundation.strictly-involutive-identity-types", "open import foundation.truncated-types", "open import foundation.truncation-levels", "open import foundation.universe-levels" ]	src/category-theory/precategories.lagda.md	Precategory
total-hom-Precategory : {l1 l2 : Level} (C : Precategory l1 l2) → UU (l1 ⊔ l2)	function	src	[ "open import category-theory.composition-operations-on-binary-families-of-sets", "open import category-theory.nonunital-precategories", "open import category-theory.set-magmoids", "open import foundation.action-on-identifications-functions", "open import foundation.cartesian-product-types", "open import foundation.dependent-pair-types", "open import foundation.function-types", "open import foundation.identity-types", "open import foundation.propositions", "open import foundation.sets", "open import foundation.strictly-involutive-identity-types", "open import foundation.truncated-types", "open import foundation.truncation-levels", "open import foundation.universe-levels" ]	src/category-theory/precategories.lagda.md	total-hom-Precategory
Pregroupoid : (l1 l2 : Level) → UU (lsuc l1 ⊔ lsuc l2)	function	src	[ "open import category-theory.isomorphisms-in-precategories", "open import category-theory.precategories", "open import foundation.dependent-pair-types", "open import foundation.equivalences", "open import foundation.identity-types", "open import foundation.iterated-dependent-product-types", "open import foundation.propositions", "open import foundation.sets", "open import foundation.strictly-involutive-identity-types", "open import foundation.type-arithmetic-dependent-pair-types", "open import foundation.universe-levels" ]	src/category-theory/pregroupoids.lagda.md	Pregroupoid
Preunivalent-Category : (l1 l2 : Level) → UU (lsuc l1 ⊔ lsuc l2)	function	src	[ "open import category-theory.composition-operations-on-binary-families-of-sets", "open import category-theory.isomorphisms-in-precategories", "open import category-theory.precategories", "open import foundation.1-types", "open import foundation.cartesian-product-types", "open import foundation.dependent-pair-types", "open import foundation.embeddings", "open import foundation.identity-types", "open import foundation.propositions", "open import foundation.sets", "open import foundation.strictly-involutive-identity-types", "open import foundation.universe-levels" ]	src/category-theory/preunivalent-categories.lagda.md	Preunivalent-Category
total-hom-Preunivalent-Category : {l1 l2 : Level} (𝒞 : Preunivalent-Category l1 l2) → UU (l1 ⊔ l2)	function	src	[ "open import category-theory.composition-operations-on-binary-families-of-sets", "open import category-theory.isomorphisms-in-precategories", "open import category-theory.precategories", "open import foundation.1-types", "open import foundation.cartesian-product-types", "open import foundation.dependent-pair-types", "open import foundation.embeddings", "open import foundation.identity-types", "open import foundation.propositions", "open import foundation.sets", "open import foundation.strictly-involutive-identity-types", "open import foundation.universe-levels" ]	src/category-theory/preunivalent-categories.lagda.md	total-hom-Preunivalent-Category
that : - sends an object `x` of `C` to the [set](foundation-core.sets.md) `hom c x` and	function	src	[ "open import category-theory.functors-large-precategories", "open import category-theory.large-precategories", "open import category-theory.natural-transformations-functors-large-precategories", "open import foundation.category-of-sets", "open import foundation.function-extensionality", "open import foundation.function-types", "open import foundation.identity-types", "open import foundation.sets", "open import foundation.universe-levels" ]	src/category-theory/representable-functors-large-precategories.lagda.md	that
that : - sends an object `x` of `C` to the [set](foundation-core.sets.md) `hom c x` and	function	src	[ "open import category-theory.copresheaf-categories", "open import category-theory.functors-precategories", "open import category-theory.maps-precategories", "open import category-theory.natural-transformations-functors-precategories", "open import category-theory.opposite-precategories", "open import category-theory.precategories", "open import foundation.category-of-sets", "open import foundation.dependent-pair-types", "open import foundation.function-extensionality", "open import foundation.homotopies", "open import foundation.identity-types", "open import foundation.sets", "open import foundation.universe-levels" ]	src/category-theory/representable-functors-precategories.lagda.md	that
obj-representing-arrow-Category : UU lzero	function	src	[ "open import category-theory.categories", "open import category-theory.isomorphisms-in-precategories", "open import category-theory.precategories", "open import foundation.booleans", "open import foundation.decidable-propositions", "open import foundation.dependent-pair-types", "open import foundation.empty-types", "open import foundation.identity-types", "open import foundation.inequality-booleans", "open import foundation.logical-equivalences", "open import foundation.logical-operations-booleans", "open import foundation.propositions", "open import foundation.sets", "open import foundation.subtypes", "open import foundation.unit-type", "open import foundation.universe-levels", "open import order-theory.posets" ]	src/category-theory/representing-arrow-category.lagda.md	obj-representing-arrow-Category
hom-set-representing-arrow-Category : obj-representing-arrow-Category → obj-representing-arrow-Category → Set lzero	function	src	[ "open import category-theory.categories", "open import category-theory.isomorphisms-in-precategories", "open import category-theory.precategories", "open import foundation.booleans", "open import foundation.decidable-propositions", "open import foundation.dependent-pair-types", "open import foundation.empty-types", "open import foundation.identity-types", "open import foundation.inequality-booleans", "open import foundation.logical-equivalences", "open import foundation.logical-operations-booleans", "open import foundation.propositions", "open import foundation.sets", "open import foundation.subtypes", "open import foundation.unit-type", "open import foundation.universe-levels", "open import order-theory.posets" ]	src/category-theory/representing-arrow-category.lagda.md	hom-set-representing-arrow-Category
hom-representing-arrow-Category : obj-representing-arrow-Category → obj-representing-arrow-Category → UU lzero	function	src	[ "open import category-theory.categories", "open import category-theory.isomorphisms-in-precategories", "open import category-theory.precategories", "open import foundation.booleans", "open import foundation.decidable-propositions", "open import foundation.dependent-pair-types", "open import foundation.empty-types", "open import foundation.identity-types", "open import foundation.inequality-booleans", "open import foundation.logical-equivalences", "open import foundation.logical-operations-booleans", "open import foundation.propositions", "open import foundation.sets", "open import foundation.subtypes", "open import foundation.unit-type", "open import foundation.universe-levels", "open import order-theory.posets" ]	src/category-theory/representing-arrow-category.lagda.md	hom-representing-arrow-Category
id-hom-representing-arrow-Category : {x : obj-representing-arrow-Category} → hom-representing-arrow-Category x x	function	src	[ "open import category-theory.categories", "open import category-theory.isomorphisms-in-precategories", "open import category-theory.precategories", "open import foundation.booleans", "open import foundation.decidable-propositions", "open import foundation.dependent-pair-types", "open import foundation.empty-types", "open import foundation.identity-types", "open import foundation.inequality-booleans", "open import foundation.logical-equivalences", "open import foundation.logical-operations-booleans", "open import foundation.propositions", "open import foundation.sets", "open import foundation.subtypes", "open import foundation.unit-type", "open import foundation.universe-levels", "open import order-theory.posets" ]	src/category-theory/representing-arrow-category.lagda.md	id-hom-representing-arrow-Category
representing-arrow-Precategory : Precategory lzero lzero	function	src	[ "open import category-theory.categories", "open import category-theory.isomorphisms-in-precategories", "open import category-theory.precategories", "open import foundation.booleans", "open import foundation.decidable-propositions", "open import foundation.dependent-pair-types", "open import foundation.empty-types", "open import foundation.identity-types", "open import foundation.inequality-booleans", "open import foundation.logical-equivalences", "open import foundation.logical-operations-booleans", "open import foundation.propositions", "open import foundation.sets", "open import foundation.subtypes", "open import foundation.unit-type", "open import foundation.universe-levels", "open import order-theory.posets" ]	src/category-theory/representing-arrow-category.lagda.md	representing-arrow-Precategory

concept : ...

function

docs

[ "open import foundation-core.template public", "open import ..." ]

docs/TEMPLATE.lagda.md

concept

satisfies-property-concept : ...

function

docs

[ "open import foundation-core.template public", "open import ..." ]

docs/TEMPLATE.lagda.md

satisfies-property-concept

concept-subconcept : ...

function

docs

[ "open import foundation-core.template public", "open import ..." ]

docs/TEMPLATE.lagda.md

concept-subconcept

is-complete-prop-Metric-Ab : {l1 l2 : Level} → Metric-Ab l1 l2 → Prop (l1 ⊔ l2)

function

src

[ "open import analysis.metric-abelian-groups", "open import foundation.dependent-pair-types", "open import foundation.propositions", "open import foundation.subtypes", "open import foundation.universe-levels", "open import metric-spaces.complete-metric-spaces", "open import metric-spaces.metric-spaces" ]

src/analysis/complete-metric-abelian-groups.lagda.md

is-complete-prop-Metric-Ab

is-complete-Metric-Ab : {l1 l2 : Level} → Metric-Ab l1 l2 → UU (l1 ⊔ l2)

function

src

[ "open import analysis.metric-abelian-groups", "open import foundation.dependent-pair-types", "open import foundation.propositions", "open import foundation.subtypes", "open import foundation.universe-levels", "open import metric-spaces.complete-metric-spaces", "open import metric-spaces.metric-spaces" ]

src/analysis/complete-metric-abelian-groups.lagda.md

is-complete-Metric-Ab

Complete-Metric-Ab : (l1 l2 : Level) → UU (lsuc l1 ⊔ lsuc l2)

function

src

[ "open import analysis.metric-abelian-groups", "open import foundation.dependent-pair-types", "open import foundation.propositions", "open import foundation.subtypes", "open import foundation.universe-levels", "open import metric-spaces.complete-metric-spaces", "open import metric-spaces.metric-spaces" ]

src/analysis/complete-metric-abelian-groups.lagda.md

Complete-Metric-Ab

convergent-series-Complete-Metric-Ab : {l1 l2 : Level} (G : Complete-Metric-Ab l1 l2) → UU (l1 ⊔ l2)

function

src

[ "open import analysis.complete-metric-abelian-groups", "open import analysis.convergent-series-metric-abelian-groups", "open import analysis.series-complete-metric-abelian-groups", "open import foundation.dependent-pair-types", "open import foundation.inhabited-types", "open import foundation.propositions", "open import foundation.subtypes", "open import foundation.universe-levels", "open import metric-spaces.cauchy-sequences-complete-metric-spaces", "open import metric-spaces.cauchy-sequences-metric-spaces" ]

src/analysis/convergent-series-complete-metric-abelian-groups.lagda.md

convergent-series-Complete-Metric-Ab

convergent-series-Metric-Ab : {l1 l2 : Level} (G : Metric-Ab l1 l2) → UU (l1 ⊔ l2)

function

src

[ "open import analysis.metric-abelian-groups", "open import analysis.series-metric-abelian-groups", "open import elementary-number-theory.addition-natural-numbers", "open import elementary-number-theory.inequality-natural-numbers", "open import elementary-number-theory.natural-numbers", "open import foundation.action-on-identifications-functions", "open import foundation.binary-transport", "open import foundation.dependent-pair-types", "open import foundation.functoriality-propositional-truncation", "open import foundation.identity-types", "open import foundation.logical-equivalences", "open import foundation.propositions", "open import foundation.subtypes", "open import foundation.transport-along-identifications", "open import foundation.universe-levels", "open import lists.sequences", "open import metric-spaces.convergent-sequences-metric-spaces", "open import metric-spaces.limits-of-sequences-metric-spaces" ]

src/analysis/convergent-series-metric-abelian-groups.lagda.md

convergent-series-Metric-Ab

convergent-series-ℝ : (l : Level) → UU (lsuc l)

function

src

[ "open import analysis.convergent-series-complete-metric-abelian-groups", "open import analysis.convergent-series-metric-abelian-groups", "open import analysis.series-real-numbers", "open import foundation.dependent-pair-types", "open import foundation.propositions", "open import foundation.subtypes", "open import foundation.universe-levels", "open import lists.sequences", "open import real-numbers.cauchy-sequences-real-numbers", "open import real-numbers.dedekind-real-numbers", "open import real-numbers.metric-additive-group-of-real-numbers" ]

src/analysis/convergent-series-real-numbers.lagda.md

convergent-series-ℝ

Metric-Ab : (l1 l2 : Level) → UU (lsuc l1 ⊔ lsuc l2)

function

src

[ "open import elementary-number-theory.positive-rational-numbers", "open import foundation.action-on-identifications-binary-functions", "open import foundation.binary-relations", "open import foundation.cartesian-product-types", "open import foundation.conjunction", "open import foundation.dependent-pair-types", "open import foundation.identity-types", "open import foundation.propositions", "open import foundation.universe-levels", "open import group-theory.abelian-groups", "open import metric-spaces.extensionality-pseudometric-spaces", "open import metric-spaces.isometries-metric-spaces", "open import metric-spaces.isometries-pseudometric-spaces", "open import metric-spaces.metric-spaces", "open import metric-spaces.pseudometric-spaces", "open import metric-spaces.rational-neighborhood-relations" ]

src/analysis/metric-abelian-groups.lagda.md

Metric-Ab

is-nonnegative-prop-series-ℝ : {l : Level} → series-ℝ l → Prop l

function

src

[ "open import analysis.series-real-numbers", "open import elementary-number-theory.natural-numbers", "open import foundation.dependent-pair-types", "open import foundation.function-types", "open import foundation.propositions", "open import foundation.universe-levels", "open import order-theory.increasing-sequences-posets", "open import real-numbers.absolute-value-real-numbers", "open import real-numbers.addition-nonnegative-real-numbers", "open import real-numbers.inequality-real-numbers", "open import real-numbers.nonnegative-real-numbers" ]

src/analysis/nonnegative-series-real-numbers.lagda.md

is-nonnegative-prop-series-ℝ

is-nonnegative-series-ℝ : {l : Level} → series-ℝ l → UU l

function

src

[ "open import analysis.series-real-numbers", "open import elementary-number-theory.natural-numbers", "open import foundation.dependent-pair-types", "open import foundation.function-types", "open import foundation.propositions", "open import foundation.universe-levels", "open import order-theory.increasing-sequences-posets", "open import real-numbers.absolute-value-real-numbers", "open import real-numbers.addition-nonnegative-real-numbers", "open import real-numbers.inequality-real-numbers", "open import real-numbers.nonnegative-real-numbers" ]

src/analysis/nonnegative-series-real-numbers.lagda.md

is-nonnegative-series-ℝ

series-Metric-Ab {l1 l2 : Level} (G : Metric-Ab l1 l2) : UU l1 where constructor series-terms-Metric-Ab field term-series-Metric-Ab : sequence (type-Metric-Ab G)

record

src

[ "open import analysis.metric-abelian-groups", "open import elementary-number-theory.addition-natural-numbers", "open import elementary-number-theory.natural-numbers", "open import foundation.action-on-identifications-functions", "open import foundation.dependent-pair-types", "open import foundation.equivalences", "open import foundation.function-extensionality", "open import foundation.function-types", "open import foundation.homotopies", "open import foundation.identity-types", "open import foundation.universe-levels", "open import group-theory.abelian-groups", "open import group-theory.sums-of-finite-sequences-of-elements-abelian-groups", "open import lists.sequences", "open import univalent-combinatorics.coproduct-types", "open import univalent-combinatorics.standard-finite-types" ]

src/analysis/series-metric-abelian-groups.lagda.md

series-Metric-Ab

series-ℝ : (l : Level) → UU (lsuc l)

function

src

[ "open import analysis.series-complete-metric-abelian-groups", "open import analysis.series-metric-abelian-groups", "open import elementary-number-theory.addition-natural-numbers", "open import elementary-number-theory.natural-numbers", "open import foundation.dependent-pair-types", "open import foundation.function-types", "open import foundation.identity-types", "open import foundation.propositions", "open import foundation.universe-levels", "open import lists.sequences", "open import order-theory.increasing-sequences-posets", "open import real-numbers.absolute-value-real-numbers", "open import real-numbers.addition-nonnegative-real-numbers", "open import real-numbers.dedekind-real-numbers", "open import real-numbers.difference-real-numbers", "open import real-numbers.inequality-real-numbers", "open import real-numbers.metric-additive-group-of-real-numbers", "open import real-numbers.nonnegative-real-numbers" ]

src/analysis/series-real-numbers.lagda.md

series-ℝ

series-terms-ℝ : {l : Level} → sequence (ℝ l) → series-ℝ l

function

src

[ "open import analysis.series-complete-metric-abelian-groups", "open import analysis.series-metric-abelian-groups", "open import elementary-number-theory.addition-natural-numbers", "open import elementary-number-theory.natural-numbers", "open import foundation.dependent-pair-types", "open import foundation.function-types", "open import foundation.identity-types", "open import foundation.propositions", "open import foundation.universe-levels", "open import lists.sequences", "open import order-theory.increasing-sequences-posets", "open import real-numbers.absolute-value-real-numbers", "open import real-numbers.addition-nonnegative-real-numbers", "open import real-numbers.dedekind-real-numbers", "open import real-numbers.difference-real-numbers", "open import real-numbers.inequality-real-numbers", "open import real-numbers.metric-additive-group-of-real-numbers", "open import real-numbers.nonnegative-real-numbers" ]

src/analysis/series-real-numbers.lagda.md

series-terms-ℝ

term-series-ℝ : {l : Level} → series-ℝ l → sequence (ℝ l)

function

src

[ "open import analysis.series-complete-metric-abelian-groups", "open import analysis.series-metric-abelian-groups", "open import elementary-number-theory.addition-natural-numbers", "open import elementary-number-theory.natural-numbers", "open import foundation.dependent-pair-types", "open import foundation.function-types", "open import foundation.identity-types", "open import foundation.propositions", "open import foundation.universe-levels", "open import lists.sequences", "open import order-theory.increasing-sequences-posets", "open import real-numbers.absolute-value-real-numbers", "open import real-numbers.addition-nonnegative-real-numbers", "open import real-numbers.dedekind-real-numbers", "open import real-numbers.difference-real-numbers", "open import real-numbers.inequality-real-numbers", "open import real-numbers.metric-additive-group-of-real-numbers", "open import real-numbers.nonnegative-real-numbers" ]

src/analysis/series-real-numbers.lagda.md

term-series-ℝ

partial-sum-series-ℝ : {l : Level} → series-ℝ l → sequence (ℝ l)

function

src

[ "open import analysis.series-complete-metric-abelian-groups", "open import analysis.series-metric-abelian-groups", "open import elementary-number-theory.addition-natural-numbers", "open import elementary-number-theory.natural-numbers", "open import foundation.dependent-pair-types", "open import foundation.function-types", "open import foundation.identity-types", "open import foundation.propositions", "open import foundation.universe-levels", "open import lists.sequences", "open import order-theory.increasing-sequences-posets", "open import real-numbers.absolute-value-real-numbers", "open import real-numbers.addition-nonnegative-real-numbers", "open import real-numbers.dedekind-real-numbers", "open import real-numbers.difference-real-numbers", "open import real-numbers.inequality-real-numbers", "open import real-numbers.metric-additive-group-of-real-numbers", "open import real-numbers.nonnegative-real-numbers" ]

src/analysis/series-real-numbers.lagda.md

partial-sum-series-ℝ

obj-augmented-simplex-Category : UU lzero

function

src

[ "open import category-theory.composition-operations-on-binary-families-of-sets", "open import category-theory.precategories", "open import elementary-number-theory.inequality-standard-finite-types", "open import elementary-number-theory.natural-numbers", "open import foundation.dependent-pair-types", "open import foundation.identity-types", "open import foundation.sets", "open import foundation.strictly-involutive-identity-types", "open import foundation.universe-levels", "open import order-theory.order-preserving-maps-posets" ]

src/category-theory/augmented-simplex-category.lagda.md

obj-augmented-simplex-Category

hom-set-augmented-simplex-Category : obj-augmented-simplex-Category → obj-augmented-simplex-Category → Set lzero

function

src

[ "open import category-theory.composition-operations-on-binary-families-of-sets", "open import category-theory.precategories", "open import elementary-number-theory.inequality-standard-finite-types", "open import elementary-number-theory.natural-numbers", "open import foundation.dependent-pair-types", "open import foundation.identity-types", "open import foundation.sets", "open import foundation.strictly-involutive-identity-types", "open import foundation.universe-levels", "open import order-theory.order-preserving-maps-posets" ]

src/category-theory/augmented-simplex-category.lagda.md

hom-set-augmented-simplex-Category

hom-augmented-simplex-Category : obj-augmented-simplex-Category → obj-augmented-simplex-Category → UU lzero

function

src

[ "open import category-theory.composition-operations-on-binary-families-of-sets", "open import category-theory.precategories", "open import elementary-number-theory.inequality-standard-finite-types", "open import elementary-number-theory.natural-numbers", "open import foundation.dependent-pair-types", "open import foundation.identity-types", "open import foundation.sets", "open import foundation.strictly-involutive-identity-types", "open import foundation.universe-levels", "open import order-theory.order-preserving-maps-posets" ]

src/category-theory/augmented-simplex-category.lagda.md

hom-augmented-simplex-Category

id-hom-augmented-simplex-Category : (n : obj-augmented-simplex-Category) → hom-augmented-simplex-Category n n

function

src

[ "open import category-theory.composition-operations-on-binary-families-of-sets", "open import category-theory.precategories", "open import elementary-number-theory.inequality-standard-finite-types", "open import elementary-number-theory.natural-numbers", "open import foundation.dependent-pair-types", "open import foundation.identity-types", "open import foundation.sets", "open import foundation.strictly-involutive-identity-types", "open import foundation.universe-levels", "open import order-theory.order-preserving-maps-posets" ]

src/category-theory/augmented-simplex-category.lagda.md

id-hom-augmented-simplex-Category

augmented-simplex-Precategory : Precategory lzero lzero

function

src

[ "open import category-theory.composition-operations-on-binary-families-of-sets", "open import category-theory.precategories", "open import elementary-number-theory.inequality-standard-finite-types", "open import elementary-number-theory.natural-numbers", "open import foundation.dependent-pair-types", "open import foundation.identity-types", "open import foundation.sets", "open import foundation.strictly-involutive-identity-types", "open import foundation.universe-levels", "open import order-theory.order-preserving-maps-posets" ]

src/category-theory/augmented-simplex-category.lagda.md

augmented-simplex-Precategory

Category : (l1 l2 : Level) → UU (lsuc l1 ⊔ lsuc l2)

function

src

[ "open import category-theory.composition-operations-on-binary-families-of-sets", "open import category-theory.isomorphisms-in-precategories", "open import category-theory.nonunital-precategories", "open import category-theory.precategories", "open import category-theory.preunivalent-categories", "open import category-theory.strongly-preunivalent-categories", "open import foundation.1-types", "open import foundation.cartesian-product-types", "open import foundation.dependent-pair-types", "open import foundation.equivalences", "open import foundation.fundamental-theorem-of-identity-types", "open import foundation.identity-types", "open import foundation.logical-equivalences", "open import foundation.propositions", "open import foundation.sets", "open import foundation.strictly-involutive-identity-types", "open import foundation.surjective-maps", "open import foundation.universe-levels" ]

src/category-theory/categories.lagda.md

Category

total-hom-Category : {l1 l2 : Level} (C : Category l1 l2) → UU (l1 ⊔ l2)

function

src

[ "open import category-theory.composition-operations-on-binary-families-of-sets", "open import category-theory.isomorphisms-in-precategories", "open import category-theory.nonunital-precategories", "open import category-theory.precategories", "open import category-theory.preunivalent-categories", "open import category-theory.strongly-preunivalent-categories", "open import foundation.1-types", "open import foundation.cartesian-product-types", "open import foundation.dependent-pair-types", "open import foundation.equivalences", "open import foundation.fundamental-theorem-of-identity-types", "open import foundation.identity-types", "open import foundation.logical-equivalences", "open import foundation.propositions", "open import foundation.sets", "open import foundation.strictly-involutive-identity-types", "open import foundation.surjective-maps", "open import foundation.universe-levels" ]

src/category-theory/categories.lagda.md

total-hom-Category

sSet-Large-Precategory : Large-Precategory lsuc (_⊔_)

function

src

[ "open import category-theory.categories", "open import category-theory.large-categories", "open import category-theory.large-precategories", "open import category-theory.precategories", "open import category-theory.presheaf-categories", "open import category-theory.simplex-category", "open import foundation.commuting-squares-of-maps", "open import foundation.function-types", "open import foundation.homotopies", "open import foundation.identity-types", "open import foundation.sets", "open import foundation.strictly-involutive-identity-types", "open import foundation.universe-levels" ]

src/category-theory/category-of-simplicial-sets.lagda.md

sSet-Large-Precategory

is-large-category-sSet-Large-Category : is-large-category-Large-Precategory sSet-Large-Precategory

function

src

[ "open import category-theory.categories", "open import category-theory.large-categories", "open import category-theory.large-precategories", "open import category-theory.precategories", "open import category-theory.presheaf-categories", "open import category-theory.simplex-category", "open import foundation.commuting-squares-of-maps", "open import foundation.function-types", "open import foundation.homotopies", "open import foundation.identity-types", "open import foundation.sets", "open import foundation.strictly-involutive-identity-types", "open import foundation.universe-levels" ]

src/category-theory/category-of-simplicial-sets.lagda.md

is-large-category-sSet-Large-Category

sSet-Large-Category : Large-Category lsuc (_⊔_)

function

src

[ "open import category-theory.categories", "open import category-theory.large-categories", "open import category-theory.large-precategories", "open import category-theory.precategories", "open import category-theory.presheaf-categories", "open import category-theory.simplex-category", "open import foundation.commuting-squares-of-maps", "open import foundation.function-types", "open import foundation.homotopies", "open import foundation.identity-types", "open import foundation.sets", "open import foundation.strictly-involutive-identity-types", "open import foundation.universe-levels" ]

src/category-theory/category-of-simplicial-sets.lagda.md

sSet-Large-Category

sSet : (l : Level) → UU (lsuc l)

function

src

[ "open import category-theory.categories", "open import category-theory.large-categories", "open import category-theory.large-precategories", "open import category-theory.precategories", "open import category-theory.presheaf-categories", "open import category-theory.simplex-category", "open import foundation.commuting-squares-of-maps", "open import foundation.function-types", "open import foundation.homotopies", "open import foundation.identity-types", "open import foundation.sets", "open import foundation.strictly-involutive-identity-types", "open import foundation.universe-levels" ]

src/category-theory/category-of-simplicial-sets.lagda.md

sSet

hom-set-sSet : {l1 l2 : Level} (X : sSet l1) (Y : sSet l2) → Set (l1 ⊔ l2)

function

src

[ "open import category-theory.categories", "open import category-theory.large-categories", "open import category-theory.large-precategories", "open import category-theory.precategories", "open import category-theory.presheaf-categories", "open import category-theory.simplex-category", "open import foundation.commuting-squares-of-maps", "open import foundation.function-types", "open import foundation.homotopies", "open import foundation.identity-types", "open import foundation.sets", "open import foundation.strictly-involutive-identity-types", "open import foundation.universe-levels" ]

src/category-theory/category-of-simplicial-sets.lagda.md

hom-set-sSet

hom-sSet : {l1 l2 : Level} (X : sSet l1) (Y : sSet l2) → UU (l1 ⊔ l2)

function

src

[ "open import category-theory.categories", "open import category-theory.large-categories", "open import category-theory.large-precategories", "open import category-theory.precategories", "open import category-theory.presheaf-categories", "open import category-theory.simplex-category", "open import foundation.commuting-squares-of-maps", "open import foundation.function-types", "open import foundation.homotopies", "open import foundation.identity-types", "open import foundation.sets", "open import foundation.strictly-involutive-identity-types", "open import foundation.universe-levels" ]

src/category-theory/category-of-simplicial-sets.lagda.md

hom-sSet

id-hom-sSet : {l1 : Level} (X : sSet l1) → hom-sSet X X

function

src

[ "open import category-theory.categories", "open import category-theory.large-categories", "open import category-theory.large-precategories", "open import category-theory.precategories", "open import category-theory.presheaf-categories", "open import category-theory.simplex-category", "open import foundation.commuting-squares-of-maps", "open import foundation.function-types", "open import foundation.homotopies", "open import foundation.identity-types", "open import foundation.sets", "open import foundation.strictly-involutive-identity-types", "open import foundation.universe-levels" ]

src/category-theory/category-of-simplicial-sets.lagda.md

id-hom-sSet

sSet-Precategory : (l : Level) → Precategory (lsuc l) l

function

src

[ "open import category-theory.categories", "open import category-theory.large-categories", "open import category-theory.large-precategories", "open import category-theory.precategories", "open import category-theory.presheaf-categories", "open import category-theory.simplex-category", "open import foundation.commuting-squares-of-maps", "open import foundation.function-types", "open import foundation.homotopies", "open import foundation.identity-types", "open import foundation.sets", "open import foundation.strictly-involutive-identity-types", "open import foundation.universe-levels" ]

src/category-theory/category-of-simplicial-sets.lagda.md

sSet-Precategory

sSet-Category : (l : Level) → Category (lsuc l) l

function

src

[ "open import category-theory.categories", "open import category-theory.large-categories", "open import category-theory.large-precategories", "open import category-theory.precategories", "open import category-theory.presheaf-categories", "open import category-theory.simplex-category", "open import foundation.commuting-squares-of-maps", "open import foundation.function-types", "open import foundation.homotopies", "open import foundation.identity-types", "open import foundation.sets", "open import foundation.strictly-involutive-identity-types", "open import foundation.universe-levels" ]

src/category-theory/category-of-simplicial-sets.lagda.md

sSet-Category

extensions : the identity map on `D` trivially gives a right extension of `D`

function

src

[ "open import category-theory.algebras-monads-on-precategories", "open import category-theory.functors-precategories", "open import category-theory.monads-on-precategories", "open import category-theory.natural-transformations-functors-precategories", "open import category-theory.precategories", "open import category-theory.right-extensions-precategories", "open import category-theory.right-kan-extensions-precategories", "open import foundation.action-on-identifications-functions", "open import foundation.dependent-pair-types", "open import foundation.equivalences", "open import foundation.identity-types", "open import foundation.universe-levels" ]

src/category-theory/codensity-monads-on-precategories.lagda.md

extensions

core-precategory-Category : {l1 l2 : Level} (C : Category l1 l2) → Precategory l1 l2

function

src

[ "open import category-theory.categories", "open import category-theory.cores-precategories", "open import category-theory.groupoids", "open import category-theory.isomorphisms-in-categories", "open import category-theory.precategories", "open import category-theory.pregroupoids", "open import category-theory.subcategories", "open import category-theory.wide-subcategories", "open import foundation.dependent-pair-types", "open import foundation.equivalences", "open import foundation.universe-levels" ]

src/category-theory/cores-categories.lagda.md

core-precategory-Category

core-category-Category : {l1 l2 : Level} (C : Category l1 l2) → Category l1 l2

function

src

[ "open import category-theory.categories", "open import category-theory.cores-precategories", "open import category-theory.groupoids", "open import category-theory.isomorphisms-in-categories", "open import category-theory.precategories", "open import category-theory.pregroupoids", "open import category-theory.subcategories", "open import category-theory.wide-subcategories", "open import foundation.dependent-pair-types", "open import foundation.equivalences", "open import foundation.universe-levels" ]

src/category-theory/cores-categories.lagda.md

core-category-Category

core-precategory-Precategory : {l1 l2 : Level} (C : Precategory l1 l2) → Precategory l1 l2

function

src

[ "open import category-theory.categories", "open import category-theory.isomorphisms-in-categories", "open import category-theory.isomorphisms-in-precategories", "open import category-theory.precategories", "open import category-theory.pregroupoids", "open import category-theory.replete-subprecategories", "open import category-theory.subprecategories", "open import category-theory.wide-subprecategories", "open import foundation.contractible-types", "open import foundation.dependent-pair-types", "open import foundation.equivalences", "open import foundation.functoriality-dependent-pair-types", "open import foundation.fundamental-theorem-of-identity-types", "open import foundation.subtypes", "open import foundation.torsorial-type-families", "open import foundation.unit-type", "open import foundation.universe-levels" ]

src/category-theory/cores-precategories.lagda.md

core-precategory-Precategory

Full-Subcategory : {l1 l2 : Level} (l3 : Level) (C : Category l1 l2) → UU (l1 ⊔ lsuc l3)

function

src

[ "open import category-theory.categories", "open import category-theory.composition-operations-on-binary-families-of-sets", "open import category-theory.embeddings-precategories", "open import category-theory.full-subprecategories", "open import category-theory.fully-faithful-functors-precategories", "open import category-theory.functors-categories", "open import category-theory.maps-categories", "open import category-theory.precategories", "open import foundation.dependent-pair-types", "open import foundation.embeddings", "open import foundation.identity-types", "open import foundation.propositions", "open import foundation.sets", "open import foundation.strictly-involutive-identity-types", "open import foundation.subtypes", "open import foundation.universe-levels" ]

src/category-theory/full-subcategories.lagda.md

Full-Subcategory

Full-Subprecategory : {l1 l2 : Level} (l3 : Level) (C : Precategory l1 l2) → UU (l1 ⊔ lsuc l3)

function

src

[ "open import category-theory.categories", "open import category-theory.composition-operations-on-binary-families-of-sets", "open import category-theory.embeddings-precategories", "open import category-theory.fully-faithful-functors-precategories", "open import category-theory.functors-precategories", "open import category-theory.isomorphisms-in-categories", "open import category-theory.isomorphisms-in-precategories", "open import category-theory.maps-precategories", "open import category-theory.precategories", "open import foundation.dependent-pair-types", "open import foundation.embeddings", "open import foundation.equivalences", "open import foundation.function-types", "open import foundation.fundamental-theorem-of-identity-types", "open import foundation.identity-types", "open import foundation.propositions", "open import foundation.sets", "open import foundation.strictly-involutive-identity-types", "open import foundation.subtype-identity-principle", "open import foundation.subtypes", "open import foundation.universe-levels" ]

src/category-theory/full-subprecategories.lagda.md

Full-Subprecategory

id-functor-Category : {l1 l2 : Level} (C : Category l1 l2) → functor-Category C C

function

src

[ "open import category-theory.categories", "open import category-theory.functors-precategories", "open import category-theory.isomorphisms-in-categories", "open import category-theory.maps-categories", "open import foundation.equivalences", "open import foundation.function-types", "open import foundation.homotopies", "open import foundation.identity-types", "open import foundation.propositions", "open import foundation.universe-levels" ]

src/category-theory/functors-categories.lagda.md

id-functor-Category

of : - a map `F₀ : C → D` on objects at some chosen universe level `γ`,

function

src

[ "open import category-theory.functors-precategories", "open import category-theory.large-precategories", "open import category-theory.maps-from-small-to-large-precategories", "open import category-theory.precategories", "open import foundation.cartesian-product-types", "open import foundation.dependent-pair-types", "open import foundation.equivalences", "open import foundation.function-types", "open import foundation.homotopies", "open import foundation.identity-types", "open import foundation.universe-levels" ]

src/category-theory/functors-from-small-to-large-precategories.lagda.md

of

id-functor-Precategory : {l1 l2 : Level} (C : Precategory l1 l2) → functor-Precategory C C

function

src

[ "open import category-theory.functors-set-magmoids", "open import category-theory.isomorphisms-in-precategories", "open import category-theory.maps-precategories", "open import category-theory.opposite-precategories", "open import category-theory.precategories", "open import foundation.action-on-identifications-functions", "open import foundation.dependent-pair-types", "open import foundation.embeddings", "open import foundation.equivalences", "open import foundation.function-types", "open import foundation.homotopies", "open import foundation.identity-types", "open import foundation.propositions", "open import foundation.subtypes", "open import foundation.type-arithmetic-dependent-pair-types", "open import foundation.universe-levels" ]

src/category-theory/functors-precategories.lagda.md

id-functor-Precategory

Gaunt-Category : (l1 l2 : Level) → UU (lsuc l1 ⊔ lsuc l2)

function

src

[ "open import category-theory.categories", "open import category-theory.composition-operations-on-binary-families-of-sets", "open import category-theory.isomorphism-induction-categories", "open import category-theory.isomorphisms-in-categories", "open import category-theory.isomorphisms-in-precategories", "open import category-theory.nonunital-precategories", "open import category-theory.precategories", "open import category-theory.rigid-objects-categories", "open import category-theory.strict-categories", "open import category-theory.strongly-preunivalent-categories", "open import foundation.cartesian-product-types", "open import foundation.dependent-pair-types", "open import foundation.identity-types", "open import foundation.propositions", "open import foundation.sets", "open import foundation.strictly-involutive-identity-types", "open import foundation.surjective-maps", "open import foundation.universe-levels" ]

src/category-theory/gaunt-categories.lagda.md

Gaunt-Category

nonunital-precategory-Gaunt-Category : {l1 l2 : Level} → Gaunt-Category l1 l2 → Nonunital-Precategory l1 l2

function

src

[ "open import category-theory.categories", "open import category-theory.composition-operations-on-binary-families-of-sets", "open import category-theory.isomorphism-induction-categories", "open import category-theory.isomorphisms-in-categories", "open import category-theory.isomorphisms-in-precategories", "open import category-theory.nonunital-precategories", "open import category-theory.precategories", "open import category-theory.rigid-objects-categories", "open import category-theory.strict-categories", "open import category-theory.strongly-preunivalent-categories", "open import foundation.cartesian-product-types", "open import foundation.dependent-pair-types", "open import foundation.identity-types", "open import foundation.propositions", "open import foundation.sets", "open import foundation.strictly-involutive-identity-types", "open import foundation.surjective-maps", "open import foundation.universe-levels" ]

src/category-theory/gaunt-categories.lagda.md

nonunital-precategory-Gaunt-Category

precategory-Gaunt-Category : {l1 l2 : Level} → Gaunt-Category l1 l2 → Precategory l1 l2

function

src

[ "open import category-theory.categories", "open import category-theory.composition-operations-on-binary-families-of-sets", "open import category-theory.isomorphism-induction-categories", "open import category-theory.isomorphisms-in-categories", "open import category-theory.isomorphisms-in-precategories", "open import category-theory.nonunital-precategories", "open import category-theory.precategories", "open import category-theory.rigid-objects-categories", "open import category-theory.strict-categories", "open import category-theory.strongly-preunivalent-categories", "open import foundation.cartesian-product-types", "open import foundation.dependent-pair-types", "open import foundation.identity-types", "open import foundation.propositions", "open import foundation.sets", "open import foundation.strictly-involutive-identity-types", "open import foundation.surjective-maps", "open import foundation.universe-levels" ]

src/category-theory/gaunt-categories.lagda.md

precategory-Gaunt-Category

strongly-preunivalent-category-Gaunt-Category : {l1 l2 : Level} → Gaunt-Category l1 l2 → Strongly-Preunivalent-Category l1 l2

function

src

[ "open import category-theory.categories", "open import category-theory.composition-operations-on-binary-families-of-sets", "open import category-theory.isomorphism-induction-categories", "open import category-theory.isomorphisms-in-categories", "open import category-theory.isomorphisms-in-precategories", "open import category-theory.nonunital-precategories", "open import category-theory.precategories", "open import category-theory.rigid-objects-categories", "open import category-theory.strict-categories", "open import category-theory.strongly-preunivalent-categories", "open import foundation.cartesian-product-types", "open import foundation.dependent-pair-types", "open import foundation.identity-types", "open import foundation.propositions", "open import foundation.sets", "open import foundation.strictly-involutive-identity-types", "open import foundation.surjective-maps", "open import foundation.universe-levels" ]

src/category-theory/gaunt-categories.lagda.md

strongly-preunivalent-category-Gaunt-Category

total-hom-Gaunt-Category : {l1 l2 : Level} (C : Gaunt-Category l1 l2) → UU (l1 ⊔ l2)

function

src

[ "open import category-theory.categories", "open import category-theory.composition-operations-on-binary-families-of-sets", "open import category-theory.isomorphism-induction-categories", "open import category-theory.isomorphisms-in-categories", "open import category-theory.isomorphisms-in-precategories", "open import category-theory.nonunital-precategories", "open import category-theory.precategories", "open import category-theory.rigid-objects-categories", "open import category-theory.strict-categories", "open import category-theory.strongly-preunivalent-categories", "open import foundation.cartesian-product-types", "open import foundation.dependent-pair-types", "open import foundation.identity-types", "open import foundation.propositions", "open import foundation.sets", "open import foundation.strictly-involutive-identity-types", "open import foundation.surjective-maps", "open import foundation.universe-levels" ]

src/category-theory/gaunt-categories.lagda.md

total-hom-Gaunt-Category

is-groupoid-prop-Category : {l1 l2 : Level} (C : Category l1 l2) → Prop (l1 ⊔ l2)

function

src

[ "open import category-theory.categories", "open import category-theory.functors-categories", "open import category-theory.isomorphisms-in-categories", "open import category-theory.isomorphisms-in-precategories", "open import category-theory.precategories", "open import category-theory.pregroupoids", "open import foundation.1-types", "open import foundation.contractible-types", "open import foundation.dependent-pair-types", "open import foundation.equivalences", "open import foundation.function-types", "open import foundation.functoriality-dependent-pair-types", "open import foundation.fundamental-theorem-of-identity-types", "open import foundation.identity-types", "open import foundation.iterated-dependent-pair-types", "open import foundation.propositions", "open import foundation.sets", "open import foundation.strictly-involutive-identity-types", "open import foundation.torsorial-type-families", "open import foundation.type-arithmetic-dependent-pair-types", "open import foundation.universe-levels" ]

src/category-theory/groupoids.lagda.md

is-groupoid-prop-Category

is-groupoid-Category : {l1 l2 : Level} (C : Category l1 l2) → UU (l1 ⊔ l2)

function

src

[ "open import category-theory.categories", "open import category-theory.functors-categories", "open import category-theory.isomorphisms-in-categories", "open import category-theory.isomorphisms-in-precategories", "open import category-theory.precategories", "open import category-theory.pregroupoids", "open import foundation.1-types", "open import foundation.contractible-types", "open import foundation.dependent-pair-types", "open import foundation.equivalences", "open import foundation.function-types", "open import foundation.functoriality-dependent-pair-types", "open import foundation.fundamental-theorem-of-identity-types", "open import foundation.identity-types", "open import foundation.iterated-dependent-pair-types", "open import foundation.propositions", "open import foundation.sets", "open import foundation.strictly-involutive-identity-types", "open import foundation.torsorial-type-families", "open import foundation.type-arithmetic-dependent-pair-types", "open import foundation.universe-levels" ]

src/category-theory/groupoids.lagda.md

is-groupoid-Category

Groupoid : (l1 l2 : Level) → UU (lsuc l1 ⊔ lsuc l2)

function

src

[ "open import category-theory.categories", "open import category-theory.functors-categories", "open import category-theory.isomorphisms-in-categories", "open import category-theory.isomorphisms-in-precategories", "open import category-theory.precategories", "open import category-theory.pregroupoids", "open import foundation.1-types", "open import foundation.contractible-types", "open import foundation.dependent-pair-types", "open import foundation.equivalences", "open import foundation.function-types", "open import foundation.functoriality-dependent-pair-types", "open import foundation.fundamental-theorem-of-identity-types", "open import foundation.identity-types", "open import foundation.iterated-dependent-pair-types", "open import foundation.propositions", "open import foundation.sets", "open import foundation.strictly-involutive-identity-types", "open import foundation.torsorial-type-families", "open import foundation.type-arithmetic-dependent-pair-types", "open import foundation.universe-levels" ]

src/category-theory/groupoids.lagda.md

Groupoid

is-section-indiscrete-Precategory : {l : Level} → obj-Precategory ∘ indiscrete-Precategory {l} ~ id

function

src

[ "open import category-theory.isomorphisms-in-precategories", "open import category-theory.precategories", "open import category-theory.pregroupoids", "open import category-theory.preunivalent-categories", "open import category-theory.strict-categories", "open import category-theory.subterminal-precategories", "open import foundation.contractible-types", "open import foundation.dependent-pair-types", "open import foundation.equivalences", "open import foundation.function-types", "open import foundation.homotopies", "open import foundation.identity-types", "open import foundation.iterated-dependent-product-types", "open import foundation.propositions", "open import foundation.sets", "open import foundation.unit-type", "open import foundation.universe-levels" ]

src/category-theory/indiscrete-precategories.lagda.md

is-section-indiscrete-Precategory

obj-initial-Category : UU lzero

function

src

[ "open import category-theory.categories", "open import category-theory.functors-precategories", "open import category-theory.gaunt-categories", "open import category-theory.indiscrete-precategories", "open import category-theory.precategories", "open import category-theory.strict-categories", "open import category-theory.strongly-preunivalent-categories", "open import foundation.contractible-types", "open import foundation.dependent-pair-types", "open import foundation.empty-types", "open import foundation.identity-types", "open import foundation.sets", "open import foundation.unit-type", "open import foundation.universe-levels" ]