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
| docstring
stringclasses 1
value |
|---|---|---|---|---|---|---|
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
|
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