problem_id stringdate 1-01-01 00:00:00 100-01-01 00:00:00 | domain listlengths 1 1 | level stringclasses 3
values | tag listlengths 1 1 | reference stringlengths 0 147 | declaration listlengths 0 0 | problem_path stringlengths 16 16 | lean_path stringlengths 25 25 | natural_language_statement stringlengths 62 1.03k | formal_statement stringlengths 109 5.38k |
|---|---|---|---|---|---|---|---|---|---|
0001 | [
"Category"
] | Easy | [
"Basic"
] | [] | problems/0001.md | CAT_statement/S_0001.lean | Theorem: Let $\mathcal{C}$ be a category and $\mathrm{Id}_{\mathcal{C}}$ the identity functor. Then monoid of natural transformations $\mathrm{End}(\mathrm{Id}_{\mathcal{C}})$ is commutative. | import Mathlib
open CategoryTheory
variable {C : Type*} [Category.{v} C]
theorem id_comm (Ξ± Ξ² : (π C) βΆ (π C)) : Ξ± β« Ξ² = Ξ² β« Ξ± := by
sorry | |
0002 | [
"Category"
] | Easy | [
"Basic"
] | [] | problems/0002.md | CAT_statement/S_0002.lean | Theorem: Let $\mathcal{C}$ be a category and let $f,g$ be morphisms in $\mathcal{C}$ such that $f\circ g$ is monic. Then $g$ is monic. | import Mathlib
open CategoryTheory
variable {C : Type*} [Category C]
theorem monic_of_comp_monic {X Y Z : C} (g : X βΆ Y) (f : Y βΆ Z)
[Mono (g β« f)] : Mono g := by
sorry | |
0003 | [
"Category"
] | Medium | [
"Basic"
] | [] | problems/0003.md | CAT_statement/S_0003.lean | Theorem: The forgetful functor $\mathcal{T}\mathrm{op}\to\mathcal{S}\mathrm{et}$, $\mathcal{G}\mathrm{rp}\to\mathcal{S}\mathrm{et}$, $\mathcal{R}\mathrm{ing}\to\mathcal{A}\mathrm{b}$, $\mathcal{T}\mathrm{op}_*\to\mathcal{T}\mathrm{op}$ are faithful but not full.\nomenclature{$\mathcal{S}\mathrm{et}$}{the category of se... | import Mathlib
open CategoryTheory Limits
theorem forget_Top_faithful_not_full :
(forget TopCat).Faithful β§ Β¬ (forget TopCat).Full := by
sorry
theorem forget_Grp_faithful_not_full :
(forget Grp).Faithful β§ Β¬ (forget Grp).Full := by
sorry
theorem forget_Ring_Ab_faithful_not_full :
(forgetβ RingCat... | |
0004 | [
"Category"
] | Medium | [
"Basic"
] | [] | problems/0004.md | CAT_statement/S_0004.lean | Theorem: Let $\{*\}\in\mathcal{S}\mathrm{et}$ be the terminal object in $\mathcal{S}\mathrm{et}$.Then $\hom_{\mathcal{S}\mathrm{et}}(\{*\},-):\mathcal{S}\mathrm{et}\to\mathcal{S}\mathrm{et}$ is an equivalence of categories. | import Mathlib
open CategoryTheory
universe u
def fromTerminalFunctor : Type u β₯€ Type u where
obj Ξ± := PUnit.{u} β Ξ±
map {Ξ± Ξ²} (f : Ξ± β Ξ²) := fun g => f β g
map_id := by
intro Ξ±
funext g x
rfl
map_comp := by
intro Ξ± Ξ² Ξ³ f g
funext h x
rfl
theorem fromTerminalEquivalence : fromTermin... | |
0005 | [
"Category"
] | Easy | [
"Basic"
] | [] | problems/0005.md | CAT_statement/S_0005.lean | Theorem: Let $\mathcal{C}$ be a category, if every idempotent in $\mathcal{C}$ can be factored into an epimorhisms followed by a monomorphism, then all idempotents split in $\mathcal{C}$. | import Mathlib
open CategoryTheory Idempotents
variable {C : Type*} [Category.{v} C]
theorem idempotent_splitting_from_epi_mono_factorization
(h : β (X : C) (p : X βΆ X) (hpp : p β« p = p),
β (Y : C) (e : X βΆ Y) (he : Epi e) (m : Y βΆ X) (hm : Mono m), p = e β« m) :
IsIdempotentComplete C := by
sorry | |
0006 | [
"Category"
] | Medium | [
"Basic"
] | [] | problems/0006.md | CAT_statement/S_0006.lean | Theorem: Let $\mathcal{C}$ and $\mathcal{D}$ be two categories.
Let $F:\mathcal{C}\to\mathcal{D}$ be a functor.
Then $F$ has a quasi-inverse if and only if
\begin{enumerate}
\item $F$ is fully faithful;
\item $F$ is essentially surjective.
\end{enumerate} | import Mathlib
open CategoryTheory
theorem funtor_has_quasi_inverse_iff {C D : Type*} [Category C] [Category D] (F : C β₯€ D):
(β G : D β₯€ C, (Nonempty (Functor.id C β
F.comp G)) β§ (Nonempty (G.comp F β
Functor.id D)))
β F.IsEquivalence := by
sorry | |
0007 | [
"Category"
] | Medium | [
"Basic"
] | [] | problems/0007.md | CAT_statement/S_0007.lean | Theorem: Let $\mathcal{C}$ be a category and $\mathrm{Kar}(\mathcal{C})$ be its idempotent completion. Let $I:\mathcal{C}\to \mathrm{Kar}(\mathcal{C})$ be the inclusion functor. Then for any category $\mathcal{D}$ in which idempotents split and any functor $F:\mathcal{C}\to \mathcal{D}$, there is a unique (up to isomor... | import Mathlib
open CategoryTheory
variable {C D : Type*} [Category C] [Category D]
theorem karoubi_universal_property [IsIdempotentComplete D] (F : C β₯€ D) :
β (F' : Idempotents.Karoubi C β₯€ D), Nonempty ((Idempotents.toKaroubi C) β F' β
F) := by
sorry
theorem karoubi_universal_property_unique_up_to_iso [IsIde... | |
0008 | [
"Category"
] | Medium | [
"Basic"
] | [] | problems/0008.md | CAT_statement/S_0008.lean | Theorem: Let $G_1$ and $G_2$ be two objects in the category $\mathcal{G}\mathrm{rp}$ of groups.\nomenclature{$\mathcal{G}\mathrm{rp}$}{the category of groups}
The coproduct of $G_1$ and $G_2$ in $\mathcal{G}\mathrm{rp}$ is equivalent to the free product of $G_1$ and $G_2$. | import Mathlib
open CategoryTheory Limits
universe u
variable {G H : Grp.{u}}
theorem freeProdGrp_iso_coprod [HasBinaryCoproduct G H] :
Nonempty (Monoid.Coprod G H β
coprod G H) := by
sorry | |
0009 | [
"Category"
] | Medium | [
"Basic"
] | [] | problems/0009.md | CAT_statement/S_0009.lean | Theorem: There exists a morphism in $\mathcal{R}\mathrm{ing}$ such that it is epic but not surjective.\nomenclature{$\mathcal{R}\mathrm{ing}$}{the category of rings} | import Mathlib
open CategoryTheory
theorem exists_epic_not_surjective_in_Ring :
β (A B : RingCat) (f : A βΆ B), Epi f β§ Β¬ Function.Surjective f := by
sorry | |
0010 | [
"Category"
] | Medium | [
"Basic"
] | [] | problems/0010.md | CAT_statement/S_0010.lean | Theorem: Let $F:\mathcal{G}\mathrm{rp}\to \mathcal{S}\mathrm{et}$ be the functor that $G\mapsto \{g\in G\mid g^2=1\}$.
Then $F$ is representable. | import Mathlib
open CategoryTheory
def functor_involution : Grp.{u} β₯€ Type u where
obj := fun G => { g : G.carrier | g * g = 1 }
map := fun {G H} f x => β¨f.hom x.val, by
refine Set.mem_setOf.mpr ?_
rcases x with β¨g, hgβ©
simp only [Set.mem_setOf_eq] at hg
rw [β f.hom.map_mul, hg]
simp only [ma... | |
0011 | [
"Category"
] | High | [
"Basic"
] | Exercise 9D, AdΓ‘mek, J., Herrlich, H. and Strecker, G., 1990. Abstract and concrete categories. Wiley-Interscience. | [] | problems/0011.md | CAT_statement/S_0011.lean | Definition: Let $(\mathcal C,U)$ be a concrete category over $\mathcal B$.
A morphism $f: x\to y$ in $\mathcal C$ is called $\textbf{initial}$ if for any object $c\in \mathcal C$, a morphism $g:U(c)\to U(x)$ is a morphism in $\mathcal{C}$ whenever $f\circ g: U(c)\to U(y)$ is a morphism in $\mathcal C$.
Definition: An... | import Mathlib
open CategoryTheory
namespace CAT_statement_S_0011
universe u uX
variable {X : Type uX} [Category.{vX} X]
namespace AHS
structure ConcreteCat (X : Type v) [Category X] where
C : Type u
[cat : Category C]
U : C β₯€ X
[U_Faithful : U.Faithful]
attribute [instance] ConcreteCat.cat ConcreteCat.U... |
0012 | [
"Category"
] | High | [
"Basic"
] | Exercise 7P, AdΓ‘mek, J., Herrlich, H. and Strecker, G., 1990. Abstract and concrete categories. Wiley-Interscience. | [] | problems/0012.md | CAT_statement/S_0012.lean | Theorem: Let $\mathcal{C}$ be a category and let $f:x\to y$ be a morphism in $\mathcal{C}$.
Then $f$ is a monomorphism in $\mathcal{C}$ if and only if there exists a category $\mathcal{D}$ and a faithful functor $I:\mathcal{C}\to\mathcal{D}$ such that $f$ is a section in $\mathcal{D}$. | import Mathlib
open CategoryTheory Functor
theorem mono_iff_exists_embedding_section
{C : Type u} [Category.{v} C] {X Y : C} (f : X βΆ Y) :
Mono f β β (D : Type (max u v)) (_ : Category.{v} D) (I : C β₯€ D) (_ : Faithful I),
IsSplitMono (I.map f) := by
sorry |
0013 | [
"Category"
] | High | [
"Basic"
] | 3.39 EXAMPLES (6), AdΓ‘mek, J., Herrlich, H. and Strecker, G., 1990. Abstract and concrete categories. Wiley-Interscience. | [] | problems/0013.md | CAT_statement/S_0013.lean | Theorem: The category $\mathcal{T}\mathrm{op}^{CH}$ of compact Hausdorff space is dually equivalent to the category of commutative unital $C^*$-algebras and algebra homomorphisms.\nomenclature{$\mathcal{T}\mathrm{op}^{CH}$}{the category of compact Hausdorff topological spaces} | import Mathlib
open CategoryTheory
universe u
structure CommCStarAlgCat : Type (u + 1) where
of ::
carrier : Type u
[commCStarAlgebra : CommCStarAlgebra carrier]
attribute [instance] CommCStarAlgCat.commCStarAlgebra
namespace CommCStarAlgCat
instance : CoeSort CommCStarAlgCat (Type u) :=
β¨CommCStarAlgCa... |
0014 | [
"Category"
] | High | [
"Basic"
] | 9.3 EXAMPLES (2), AdΓ‘mek, J., Herrlich, H. and Strecker, G., 1990. Abstract and concrete categories. Wiley-Interscience. | [] | problems/0014.md | CAT_statement/S_0014.lean | Definition: Let $(\mathcal C,U)$ be a concrete category over $\mathcal B$.
A morphism $f: x\to y$ in $\mathcal C$ is called $\textbf{initial}$ if for any object $c\in \mathcal C$, a morphism $g:U(c)\to U(x)$ is a morphism in $\mathcal{C}$ whenever $f\circ g: U(c)\to U(y)$ is a morphism in $\mathcal C$.
Definition: An... | import Mathlib
open CategoryTheory
namespace CAT_statement_S_0014
universe u uX
variable {X : Type uX} [Category.{vX} X]
namespace AHS2
structure ConcreteCat (X : Type v) [Category X] where
C : Type u
[cat : Category C]
U : C β₯€ X
[U_Faithful : U.Faithful]
attribute [instance] ConcreteCat.cat ConcreteCat.... |
0015 | [
"Category"
] | High | [
"Basic"
] | 9.3 EXAMPLES (2)(b), AdΓ‘mek, J., Herrlich, H. and Strecker, G., 1990. Abstract and concrete categories. Wiley-Interscience. | [] | problems/0015.md | CAT_statement/S_0015.lean | Definition: Let $(\mathcal C,U)$ be a concrete category over $\mathcal B$.
A morphism $f: x\to y$ in $\mathcal C$ is called $\textbf{initial}$ if for any object $c\in \mathcal C$, a morphism $g:U(c)\to U(x)$ is a morphism in $\mathcal{C}$ whenever $f\circ g: U(c)\to U(y)$ is a morphism in $\mathcal C$.
Definition: An... | import Mathlib
open CategoryTheory
namespace CAT_statement_S_0015
universe u uX
variable {X : Type uX} [Category.{vX} X]
namespace AHS
structure ConcreteCat (X : Type v) [Category X] where
C : Type u
[cat : Category C]
U : C β₯€ X
[U_Faithful : U.Faithful]
attribute [instance] ConcreteCat.cat ConcreteCat.U... |
0016 | [
"Category"
] | High | [
"Basic"
] | 9.3 EXAMPLES (3)(b), AdΓ‘mek, J., Herrlich, H. and Strecker, G., 1990. Abstract and concrete categories. Wiley-Interscience. | [] | problems/0016.md | CAT_statement/S_0016.lean | Definition: Let $(\mathcal C,U)$ be a concrete category over $\mathcal B$.
A morphism $f: x\to y$ in $\mathcal C$ is called $\textbf{initial}$ if for any object $c\in \mathcal C$, a morphism $g:U(c)\to U(x)$ is a morphism in $\mathcal{C}$ whenever $f\circ g: U(c)\to U(y)$ is a morphism in $\mathcal C$.
Definition: An... | import Mathlib
open CategoryTheory
namespace CAT_statement_S_0016
universe u uX
variable {X : Type uX} [Category.{vX} X]
namespace AHS
structure ConcreteCat (X : Type v) [Category X] where
C : Type u
[cat : Category C]
U : C β₯€ X
[U_Faithful : U.Faithful]
attribute [instance] ConcreteCat.cat ConcreteCat.U... |
0017 | [
"Category"
] | High | [
"Basic"
] | 9.3 EXAMPLES (4)(d), AdΓ‘mek, J., Herrlich, H. and Strecker, G., 1990. Abstract and concrete categories. Wiley-Interscience. | [] | problems/0017.md | CAT_statement/S_0017.lean | Definition: Let $(\mathcal C,U)$ be a concrete category over $\mathcal B$.
A morphism $f: x\to y$ in $\mathcal C$ is called $\textbf{initial}$ if for any object $c\in \mathcal C$, a morphism $g:U(c)\to U(x)$ is a morphism in $\mathcal{C}$ whenever $f\circ g: U(c)\to U(y)$ is a morphism in $\mathcal C$.
Definition: An... | import Mathlib
open CategoryTheory Limits TopologicalSpace
namespace CAT_statement_S_0017
universe u uX
variable {X : Type uX} [Category.{vX} X]
namespace AHS
structure ConcreteCat (X : Type v) [Category X] where
C : Type u
[cat : Category C]
U : C β₯€ X
[U_Faithful : U.Faithful]
attribute [instance] Concr... |
0018 | [
"Category"
] | High | [
"Basic"
] | 9.3 EXAMPLES (4)(a), AdΓ‘mek, J., Herrlich, H. and Strecker, G., 1990. Abstract and concrete categories. Wiley-Interscience. | [] | problems/0018.md | CAT_statement/S_0018.lean | Definition: Let $(\mathcal C,U)$ be a concrete category over $\mathcal B$.
A morphism $f: x\to y$ in $\mathcal C$ is called $\textbf{initial}$ if for any object $c\in \mathcal C$, a morphism $g:U(c)\to U(x)$ is a morphism in $\mathcal{C}$ whenever $f\circ g: U(c)\to U(y)$ is a morphism in $\mathcal C$.
Definition: An... | import Mathlib
open CategoryTheory
namespace CAT_statement_S_0018
universe u uX
variable {X : Type uX} [Category.{vX} X]
namespace AHS
structure ConcreteCat (X : Type v) [Category X] where
C : Type u
[cat : Category C]
U : C β₯€ X
[U_Faithful : U.Faithful]
attribute [instance] ConcreteCat.cat ConcreteCat.U... |
0019 | [
"Category"
] | Easy | [
"Adjunction"
] | [] | problems/0019.md | CAT_statement/S_0019.lean | Theorem: A functor $G : \mathcal{D} \to \mathcal{C}$ has a left adjoint if and only if for each $c \in\mathcal{C}$, the comma category $(c \downarrow G)$ has an initial object. | import Mathlib
open CategoryTheory Limits
variable {C : Type*} {D : Type*} [Category.{vβ} C] [Category.{vβ} D]
theorem functor_hasLeftAdjoint_iff_structuredArrow_hasInitial
(G : D β₯€ C) :
G.IsRightAdjoint β β c : C, HasInitial (StructuredArrow c G) := by
sorry | |
0020 | [
"Category"
] | Easy | [
"Adjunction"
] | [] | problems/0020.md | CAT_statement/S_0020.lean | Theorem: Let $\mathcal{C}$ and $\mathcal{D}$ be categories and let $F:\mathcal{C}\to \mathcal{D}$ be a functor that admits a right adjoint $G$.
Then $F$ is fully faithful if and only if $u:\mathrm{Id}_{\mathcal{C}}\to G\circ F$ is isomorphism. | import Mathlib
open CategoryTheory
variable {C D : Type*} [Category C] [Category D] (F : C β₯€ D) (G : D β₯€ C)
theorem fully_faithful_iff_unit_isIso (adj : F β£ G) :
(F.Full β§ F.Faithful) β IsIso adj.unit := by
sorry | |
0021 | [
"Category"
] | Medium | [
"Adjunction"
] | [] | problems/0021.md | CAT_statement/S_0021.lean | Theorem: Let $\mathcal{C}$ and $\mathcal{D}$ be categories and let $F:\mathcal{C}\to \mathcal{D}$ be a functor that admits a right adjoint $G$.
Then $G$ is an equivalence of categories if and only if $F$ is fully faithful and $G$ is conservative. | import Mathlib
open CategoryTheory
variable {C : Type uβ} [Category.{vβ} C] {D : Type uβ} [Category.{vβ} D]
theorem right_adjoint_isEquivalence_iff_left_full_faithful_and_right_conservative
(F : C β₯€ D) (G : D β₯€ C) (adj : F β£ G) :
G.IsEquivalence β (F.Full β§ F.Faithful) β§ G.ReflectsIsomorphisms := by
sorry | |
0022 | [
"Category"
] | Medium | [
"Adjunction"
] | [] | problems/0022.md | CAT_statement/S_0022.lean | Theorem: Let $\mathcal{C}$ and $\mathcal{D}$ be locally small categories and let $F:\mathcal{C}\to \mathcal{D}$ be a functor.
Then $F$ admits a right adjoint if and only if for each $d\in\mathcal{D}$, $\hom_{\mathcal{D}}(F(-),d):\mathcal{C}^{op}\to\mathcal{S}\mathrm{et}$ is representable. | import Mathlib
open CategoryTheory
variable {C : Type uβ} [Category.{v} C] {D : Type uβ} [Category.{v} D]
theorem isLeftAdjoint_iff_yoneda_comp_op_isRepresentable (F : C β₯€ D) :
F.IsLeftAdjoint β β (d : D), (F.op β yoneda.obj d).IsRepresentable := by
sorry | |
0023 | [
"Category"
] | Medium | [
"Adjunction"
] | [] | problems/0023.md | CAT_statement/S_0023.lean | Theorem: Let $A,B$ be rings and $\phi : A \to B$ be a ring homomorphism. The functor $\phi_* : B-\mathcal{M}\mathrm{od} \to A-\mathcal{M}\mathrm{od}$ between their module categories is defined by $(N,l_N) \mapsto (N,l_N \circ (\phi \otimes \mathrm{id}))$. Then the functor $\phi_*$ admits a left adjoint $\phi^* := B \ot... | import Mathlib
open CategoryTheory
theorem ring_hom_induced_functor_has_adjoints
{A B : RingCat} (Ο : A βΆ B) :
β (Ο_pull : ModuleCat B β₯€ ModuleCat A)
(Ο_push : ModuleCat A β₯€ ModuleCat B)
(Ο_coind : ModuleCat A β₯€ ModuleCat B),
Nonempty (Adjunction Ο_push Ο_pull) β§ Nonempty (Adjunction Ο_pull ... | |
0024 | [
"Category"
] | Easy | [
"Adjunction"
] | [] | problems/0024.md | CAT_statement/S_0024.lean | Theorem: The forgetful functor $U : \mathcal{A}\mathrm{b} \to \mathcal{G}\mathrm{rp}$ admits a left adjoint. | import Mathlib
open CategoryTheory
universe u
theorem forget_CommGrp_to_Grp_admits_left_adjoint :
(forgetβ CommGrp.{u} Grp.{u}).IsRightAdjoint := by
sorry | |
0025 | [
"Category"
] | Medium | [
"Adjunction"
] | [] | problems/0025.md | CAT_statement/S_0025.lean | Theorem: Let $\mathcal{C}$, $\mathcal{D}$, $\mathcal{E}$ be categories and $U : \mathcal{D}\to \mathcal{C}$, $V : \mathcal{E}\to \mathcal{C}$, $F : \mathcal{D} \to \mathcal{E}$ be functors such that $V \circ F = U$. Suppose $U, V$ have left adjoints and $\mathcal D$ admits coequalizers. If $V$ reflects split epimorphis... | import Mathlib
open CategoryTheory
variable {C D E : Type*} [Category C] [Category D] [Category E]
namespace CategoryTheory
class Functor.ReflectsSplitEpimorphismsToRegularEpimorphisms (F : Functor C D) : Prop where
reflects : β {X Y} {f : X βΆ Y} [IsSplitEpi (F.map f)], Nonempty (RegularEpi f)
end CategoryTheor... | |
0026 | [
"Category"
] | Medium | [
"Adjunction"
] | Exercise 19I, AdΓ‘mek, J., Herrlich, H. and Strecker, G., 1990. Abstract and concrete categories. Wiley-Interscience. | [] | problems/0026.md | CAT_statement/S_0026.lean | Theorem: Let $F,G,H$ be functors such that $F\dashv G\dashv H$.
Then $F$ is fully faithful if and only if $H$ is fully faithful. | import Mathlib
open CategoryTheory Functor
variable {C : Type uβ} [Category.{vβ} C] {D : Type uβ} [Category.{vβ} D]
variable {F : C β₯€ D} {G : D β₯€ C} {H : C β₯€ D}
theorem fullyFaithful_iff_of_adjoints (hFG : F β£ G) (hGH : G β£ H) :
(F.Full β§ F.Faithful) β (H.Full β§ H.Faithful) := by
sorry |
0027 | [
"Category"
] | Medium | [
"Adjunction"
] | Exercise 18H, AdΓ‘mek, J., Herrlich, H. and Strecker, G., 1990. Abstract and concrete categories. Wiley-Interscience. | [] | problems/0027.md | CAT_statement/S_0027.lean | Theorem: Let $(\mathbb{Z},\leq)$ be a poset, regarded as a category, then $f\in \mathrm{End}(\mathbb{Z})$ has left adjoint if and only if it has a right adjoint. | import Mathlib
open CategoryTheory
theorem int_endofunctor_hasLeftAdjoint_iff_hasRightAdjoint (f : β€ β₯€ β€) :
f.IsRightAdjoint β f.IsLeftAdjoint := by
sorry |
0028 | [
"Category"
] | Medium | [
"Adjunction"
] | Exercise 19F (a), AdΓ‘mek, J., Herrlich, H. and Strecker, G., 1990. Abstract and concrete categories. Wiley-Interscience. | [] | problems/0028.md | CAT_statement/S_0028.lean | Theorem: View the poset $\mathbb{N} = (\mathbb{N},\leq)$ of natural numbers as a category. There is a sequence of distinct functors $G_n:\mathbb{N}\to \mathbb{N}$ such that $G_0(x)=x+1$ and $G_{n+1}\dashv G_n$ for each $n\in \mathbb{N}$. | import Mathlib
open CategoryTheory
theorem exists_sequence_of_distinct_adjoints_nat :
β G : β β (β β₯€ β),
Function.Injective G β§
(β x, (G 0).obj x = x + 1) β§
(β n, Nonempty (G (n + 1) β£ G n)) := by
sorry |
0029 | [
"Category"
] | Medium | [
"Adjunction"
] | [] | problems/0029.md | CAT_statement/S_0029.lean | Theorem: Let $(-)^{\times}: \mathcal{R}\mathrm{ing} \to \mathcal{G}\mathrm{rp}$ mapping a ring to its group of units. Then it admits a left adjoint. | import Mathlib
open CategoryTheory
def RingCat.units : RingCat.{u} β₯€ Grp.{u} where
obj R := .of RΛ£
map f := Grp.ofHom (Units.map f.hom)
theorem exists_leftAdjoint_unitFunctor :
β (left : Grp.{u} β₯€ RingCat.{u}), Nonempty (left β£ RingCat.units.{u}) := by
sorry | |
0030 | [
"Category"
] | High | [
"Reflective"
] | Exercise 4G(d), AdΓ‘mek, J., Herrlich, H. and Strecker, G., 1990. Abstract and concrete categories. Wiley-Interscience. | [] | problems/0030.md | CAT_statement/S_0030.lean | Theorem: There are categories $\mathcal{C}$, $\mathcal{D}$ and $\mathcal{E}$ such that $\mathcal{C}$ is a subcategory of $\mathcal{D}$, $\mathcal{D}$ is a subcategory of $\mathcal{E}$ and $\mathcal{C}$ is reflective in $\mathcal{E}$, but $\mathcal{C}$ is not reflective in $\mathcal{D}$. Here we do not require a reflect... | import Mathlib
open CategoryTheory Functor
universe u v
namespace CategoryTheory
open Category Adjunction
variable {C : Type uβ} {D : Type uβ} {E : Type uβ}
variable [Category.{vβ} C] [Category.{vβ} D] [Category.{vβ} E]
class Reflective2 (R : D β₯€ C) extends R.Faithful where
L : C β₯€ D
adj : L β£ R
end Category... |
0031 | [
"Category"
] | High | [
"Reflective"
] | Exercise 4E(d), AdΓ‘mek, J., Herrlich, H. and Strecker, G., 1990. Abstract and concrete categories. Wiley-Interscience. | [] | problems/0031.md | CAT_statement/S_0031.lean | Theorem: Neither the category $\mathcal{S}\mathrm{et}$ of sets nor the category $\mathcal{T}\mathrm{op}$ of topological spaces has a proper isomorphism-closed full subcategory that is both reflective and coreflective. | import Mathlib
open CategoryTheory
theorem not_reflective_and_coreflective (P : ObjectProperty (Type u))
(h : P.IsClosedUnderIsomorphisms) (hproper : β X : Type u, Β¬ P X) :
IsEmpty (Reflective P.ΞΉ) β¨ IsEmpty (Coreflective P.ΞΉ) := by
sorry |
0032 | [
"Category"
] | High | [
"Reflective"
] | Exercise 4D(a), AdΓ‘mek, J., Herrlich, H. and Strecker, G., 1990. Abstract and concrete categories. Wiley-Interscience. | [] | problems/0032.md | CAT_statement/S_0032.lean | Theorem: $\mathcal{S}\mathrm{et}$ has precisely three full, isomorphism-closed, reflective subcategories. | import Mathlib
open CategoryTheory Functor Limits
namespace CAT_statement_S_0032
def IsIsoClosed (P : Type u β Prop) : Prop :=
β {X Y : Type u}, Nonempty (X β
Y) β P X β P Y
def SubcategoryEquiv (P Q : Type u β Prop) : Prop :=
β X, P X β Q X
def IsReflectiveSubcategory (P : Type u β Prop) : Prop :=
Nonempty ... |
0033 | [
"Category"
] | High | [
"Reflective"
] | Exercise 4D(b), AdΓ‘mek, J., Herrlich, H. and Strecker, G., 1990. Abstract and concrete categories. Wiley-Interscience. | [] | problems/0033.md | CAT_statement/S_0033.lean | Theorem: $\mathcal{T}\mathrm{op}^{CH}$ has precisely two full, isomorphism-closed, coreflective subcategories. | import Mathlib
open CategoryTheory Topology
namespace CAT_statement_S_0033
structure FullCoreflectiveSubcategory (C : Type u) [Category.{v} C] where
obj : ObjectProperty C
iso_closed : obj.IsClosedUnderIsomorphisms
coreflective : Coreflective obj.ΞΉ
theorem CompHaus_has_precisely_two_coreflective_subcategories... |
0034 | [
"Category"
] | Medium | [
"Concrete"
] | Exercise 10R, AdΓ‘mek, J., Herrlich, H. and Strecker, G., 1990. Abstract and concrete categories. Wiley-Interscience. | [] | problems/0034.md | CAT_statement/S_0034.lean | Definition: Let $(\mathcal C, U)$ be a concrete category over $\mathcal B$.
A $\textbf{universal arrow}$ over $x\in \mathcal B$ is a structured arrow $u:x\to U(c)$ with domain $x$ that has the following universal property: for each structured arrow $f: x\to U(b)$ with domain $x$ there exists a unique morphism $\underl... | import Mathlib
open CategoryTheory Limits Functor Opposite
namespace CAT_statement_S_0034
variable {C : Type u} [Category.{v} C]
def IsFreeObject (U : C β₯€ Type v) (d : C) (I : Type v) : Prop :=
β (Ξ· : I βΆ U.obj d), β {y : C} (f : I βΆ U.obj y), β! (g : d βΆ y), U.map g β Ξ· = f
def IsCopower (x d : C) (I : Type v) ... |
0035 | [
"Category"
] | High | [
"Concrete"
] | Exercise 5O(k), AdΓ‘mek, J., Herrlich, H. and Strecker, G., 1990. Abstract and concrete categories. Wiley-Interscience. | [] | problems/0035.md | CAT_statement/S_0035.lean | Definition: A full concrete embedding is called a realization.
Theorem: There is a construct $(\mathcal{C},U)$ such that every construct has a realization to $(\mathcal{C},U)$. | import Mathlib
open CategoryTheory
namespace CAT_statement_S_0035
structure Construct where
C : Type u
[str : Category.{v} C]
U : C β₯€ Type u
[faithful : Functor.Faithful U]
attribute [instance] Construct.str Construct.faithful
def IsRealization (S T : Construct.{u, v}) (F : S.C β₯€ T.C) : Prop :=
F β T.U =... |
0036 | [
"Category"
] | High | [
"Concrete"
] | Exercise 5J(d), AdΓ‘mek, J., Herrlich, H. and Strecker, G., 1990. Abstract and concrete categories. Wiley-Interscience. | [] | problems/0036.md | CAT_statement/S_0036.lean | Definition: A category $\mathcal C$ is called $\textbf{concretizable}$ over a category $\mathcal B$ if there exists a faithful functor $U:\mathcal C\to \mathcal B$.
Theorem: There exist categories that are not concretizable over $\mathcal{S}\mathrm{et}$. | import Mathlib
open CategoryTheory
theorem exists_category_not_concretizable_over_Type :
β (C : Type u) (_ : Category.{v} C), Β¬ β (F : C β₯€ Type v), F.Faithful := by
sorry |
0037 | [
"Category"
] | High | [
"Concrete"
] | Exercise 5D(d), AdΓ‘mek, J., Herrlich, H. and Strecker, G., 1990. Abstract and concrete categories. Wiley-Interscience. | [] | problems/0037.md | CAT_statement/S_0037.lean | Theorem: There are precisely two concrete functors from the category $\mathcal{S}\mathrm{et}$ of sets to the category $\mathcal{T}\mathrm{op}$ of topological spaces, but a proper class of concrete functors from $\mathcal{T}\mathrm{op}$ into itself. | import Mathlib
open CategoryTheory
namespace CAT_statement_S_0037
universe u v w
variable {X : Type uX} [Category.{vX} X]
structure ConcreteCat (X : Type v) [Category X] where
C : Type u
[cat : Category C]
U : C β₯€ X
[U_Faithful : U.Faithful]
attribute [instance] ConcreteCat.cat ConcreteCat.U_Faithful
de... |
0038 | [
"Category"
] | High | [
"Concrete"
] | Exercise 10L, AdΓ‘mek, J., Herrlich, H. and Strecker, G., 1990. Abstract and concrete categories. Wiley-Interscience. | [] | problems/0038.md | CAT_statement/S_0038.lean | Definition: Let $\mathcal C$ be a category and let $c\in \mathcal C$ be an object.
A $\textbf{regular subobject}$ of $c$ is a pair $(x,i)$ where $i$ is a regular monomorphism.
Definition: Let $\mathcal C$ be a category.
$\mathcal C$ is called $\textbf{regular wellpowered}$ if no object in $\mathcal C$ has a proper cla... | import Mathlib
namespace CAT_statement_S_0038
open CategoryTheory Limits
universe u v w
variable {C : Type u} [Category C] [HasFiniteLimits C]
def IsConcretizable (X : Type v) [Category X] (D: Type u) [Category D] : Prop :=
β (U : D β₯€ X), U.Faithful
variable (C)
class RegularWellPowered : Prop where
regularS... |
0039 | [
"Category"
] | High | [
"Concrete"
] | Exercise 5L, AdΓ‘mek, J., Herrlich, H. and Strecker, G., 1990. Abstract and concrete categories. Wiley-Interscience. | [] | problems/0039.md | CAT_statement/S_0039.lean | Definition: A $\textbf{construct}$ is a concrete category over the category $\mathcal{S}\mathrm{et}$ of sets.
Theorem: Let $\mathcal{F}\mathrm{rm}$ be the construct of frames (i.e., distributive suplattices) and frame homomorphisms. Then there is a unique concrete functor $T: \mathcal{T}\mathrm{op}^{op}_0 \to \mathcal... | import Mathlib
open CategoryTheory Topology
universe u v w
variable {X : Type uX} [Category.{vX} X]
namespace CAT_statement_S_0039
structure ConcreteCat (X : Type v) [Category X] where
C : Type u
[cat : Category C]
U : C β₯€ X
[U_Faithful : U.Faithful]
attribute [instance] ConcreteCat.cat ConcreteCat.U_Fait... |
0040 | [
"Category"
] | High | [
"Concrete"
] | Exercise 8G(a), AdΓ‘mek, J., Herrlich, H. and Strecker, G., 1990. Abstract and concrete categories. Wiley-Interscience. | [] | problems/0040.md | CAT_statement/S_0040.lean | Definition: Let $(\mathcal C, U)$ be a concrete category over $\mathcal B$.
A $\textbf{universal arrow}$ over $x\in \mathcal B$ is a structured arrow $u:x\to U(c)$ with domain $x$ that has the following universal property: for each structured arrow $f: x\to U(b)$ with domain $x$ there exists a unique morphism $\underl... | import Mathlib
open CategoryTheory
namespace CAT_statement_S_0040
universe u v w
variable {X : Type uX} [Category.{vX} X]
structure ConcreteCat (X : Type v) [Category X] where
C : Type u
[cat : Category C]
U : C β₯€ X
[U_Faithful : U.Faithful]
attribute [instance] ConcreteCat.cat ConcreteCat.U_Faithful
ab... |
0041 | [
"Category"
] | High | [
"Concrete"
] | Exercise 8G(b), AdΓ‘mek, J., Herrlich, H. and Strecker, G., 1990. Abstract and concrete categories. Wiley-Interscience. | [] | problems/0041.md | CAT_statement/S_0041.lean | Definition: Let $(\mathcal C, U)$ be a concrete category over $\mathcal B$. A $\textbf{universal arrow}$ over $x \in \mathcal B$ is a morphism $u:x\to U(c)$ that has the following universal property: for each morphism $f: x\to U(b)$ there exists a unique morphism $f' : c\to b$ such that $U(f') \circ u = f$.
Definition... | import Mathlib
open CategoryTheory
universe u v w
namespace CAT_statement_S_0041
structure FreeObject {C : Type u} [Category.{v} C] [HasForget.{w} C] (x : Type w) where
(obj : C)
(emb : x βΆ (forget C).obj obj)
(uniq : β (Y : C) (f : x βΆ (forget C).obj Y), β! (g : obj βΆ Y), emb β« (forget C).map g = f)
theore... |
0042 | [
"Category"
] | Easy | [
"Limit"
] | [] | problems/0042.md | CAT_statement/S_0042.lean | Theorem: Let $\mathcal{C}$ and $\mathcal{D}$ be categories and let $F:\mathcal{C}\to \mathcal{D}$ be a fully faithful functor.
Then $F$ reflects any limits and colimits admitted in the codomain category. | import Mathlib
open CategoryTheory Limits Functor
variable {C : Type uβ} [Category.{vβ} C] {D : Type uβ} [Category.{vβ} D]
theorem fully_faithful_reflects_limits_and_colimits (F : C β₯€ D) [Full F] [Faithful F] :
ReflectsLimits F β§ ReflectsColimits F := by
sorry | |
0043 | [
"Category"
] | Easy | [
"Limit"
] | [] | problems/0043.md | CAT_statement/S_0043.lean | Theorem: The one point set $\{*\}$ form a separator in Set, and the two point set $\{a,b\}$ form a coseparating set in Set. | import Mathlib
open CategoryTheory Function Classical
theorem PUnit_isSeparator : IsSeparator (PUnit : Type u) := by
sorry
theorem ULiftBool_isCoseparator : IsCoseparator (ULift.{u} Bool) := by
sorry | |
0044 | [
"Category"
] | Easy | [
"Limit"
] | [] | problems/0044.md | CAT_statement/S_0044.lean | Theorem: Filtered colimits commute with finite limits in $\mathcal{S}\mathrm{et}$. | import Mathlib
open CategoryTheory Limits
variable {J : Type u} [SmallCategory J] [FinCategory J]
variable {K : Type u} [SmallCategory K] [IsFiltered K]
variable (F : J β₯€ K β₯€ Type u)
theorem filteredColimitsCommuteWithFiniteLimits :
Nonempty (colimit (limit F) β
limit (colimit F.flip)) := by
sorry | |
0045 | [
"Category"
] | Easy | [
"Limit"
] | [] | problems/0045.md | CAT_statement/S_0045.lean | Theorem: Let $\omega$ be the ordinal of natural numbers.
Consider $F:\omega^{op}\to \mathcal{R}\mathrm{ing}$ with $F_n:=\mathbb{Z}/p^n\mathbb{Z}$ and $f_n:F_{n+1}\to F_n$.
Then the limit exists. | import Mathlib
open CategoryTheory Limits Opposite
variable (p : β)
noncomputable def pAdicFunctor : βα΅α΅ β₯€ RingCat where
obj n := RingCat.of (ZMod (p ^ (unop n)))
map {m n} f := RingCat.ofHom <|
ZMod.castHom (pow_dvd_pow p (leOfHom f.unop)) (ZMod (p ^ (unop n)))
map_id := by
intro n
ext x
simp... | |
0046 | [
"Category"
] | Easy | [
"Limit"
] | [] | problems/0046.md | CAT_statement/S_0046.lean | Theorem: Let $\mathcal{C}$ and $\mathcal{D}$ be a small category and let $F,G:\mathcal{C}\to\mathcal{D}$ be two functors.
Then we have $\mathrm{Nat}(F,G)\cong \int_{c\in\mathcal{C}}\hom_{\mathcal{D}}(F(c),G(c))$. | import Mathlib
open CategoryTheory Limits
variable {C : Type u} [SmallCategory C]
variable {D : Type u} [SmallCategory D]
variable (F G : C β₯€ D)
def homIntegrandBifunctor : Cα΅α΅ Γ C β₯€ Type u :=
(Functor.prod F.op G) β (Functor.hom D)
theorem natTransIsoEnd :
Nonempty (NatTrans F G β
end_ (curryObj (homIntegra... | |
0047 | [
"Category"
] | Easy | [
"Limit"
] | [] | problems/0047.md | CAT_statement/S_0047.lean | Theorem: There is no equivalence of categories between $\mathcal{S}\mathrm{et}$ and $\mathcal{S}\mathrm{et}^{op}$. | import Mathlib
open CategoryTheory
theorem no_equiv_between_Set_and_op : Β¬ Nonempty (Equivalence (Type u) (Type u)α΅α΅) := by
sorry | |
0048 | [
"Category"
] | Easy | [
"Limit"
] | [] | problems/0048.md | CAT_statement/S_0048.lean | Theorem: A reflective subcategory $\mathcal{C}$ of a cocomplete category $\mathcal{D}$ is also cocomplete. | import Mathlib
open CategoryTheory Limits
variable {C : Type u} [Category.{v} C] {D : Type u} [Category.{v} D]
theorem hasColimits_of_reflective (i : C β₯€ D) [Reflective i] [HasColimits D] :
HasColimits C := by
sorry | |
0049 | [
"Category"
] | Easy | [
"Limit"
] | [] | problems/0049.md | CAT_statement/S_0049.lean | Theorem: Let $\mathcal{C}$ and $\mathcal{E}$ be two categories and let $F:\mathcal{C}\to \mathcal{E}$ be a functor.
Let $\bullet $ be the terminal category consisting of a unique object $\bullet$ and a unique morphism.
Then a colimit of $F$ is a left Kan extension of $F$ along $K:\mathcal{C}\to \bullet$, i.e. $... | import Mathlib
open CategoryTheory Limits
universe uβ vβ uβ vβ
variable {C : Type uβ} [Category.{vβ} C]
variable {E : Type uβ} [Category.{vβ} E]
theorem colimit_is_leftKanExtension_along_to_terminal
(F : C β₯€ E) (K : C β₯€ PUnit) [HasColimit F] [K.HasLeftKanExtension F] :
Nonempty ((K.leftKanExtension F).obj ... | |
0050 | [
"Category"
] | Medium | [
"Limit"
] | [] | problems/0050.md | CAT_statement/S_0050.lean | Definition: Let $\mathcal C$ be a locally small category.
An object $c \in\mathcal C$ is called $\textbf{compact}$ if $\mathrm{hom}_{\mathcal C} (c,-)$ preserves filtered colimits.
Theorem: For $\mathcal{S}\mathrm{et}$, an object is compact if and only if it is a finite set. | import Mathlib
open CategoryTheory Limits
theorem isCompactObject_iff_finite_type (X : Type u) :
PreservesFilteredColimits (coyoneda.obj (Opposite.op X)) β Finite X := by
sorry | |
0051 | [
"Category"
] | Medium | [
"Limit"
] | [] | problems/0051.md | CAT_statement/S_0051.lean | Theorem: Let $\mathcal{C}$ be a category.
Then $\mathcal{C}$ admits all small limits if and only if $\mathcal{C}$ admits all small products and pullbacks. | import Mathlib
open CategoryTheory Limits
variable {C : Type u} [Category.{v} C]
theorem has_limits_iff_has_products_and_pullbacks :
HasLimitsOfSize.{v, v} C β (β (J : Type v), HasLimitsOfShape (Discrete J) C) β§ HasLimitsOfShape WalkingCospan C := by
sorry | |
0052 | [
"Category"
] | Medium | [
"Limit"
] | [] | problems/0052.md | CAT_statement/S_0052.lean | Theorem: Let $X,Y,Z$ be objects in $\mathcal{S}\mathrm{et}$ with morphisms $f:X\to Z$ and $g:Y\to Z$.
Then $\{(x,y)\in X\times Y\mid f(x)=g(y)\}$ is the pullback $X\times_Z Y$ of $X$ and $Y$ over $Z$. | import Mathlib
open CategoryTheory Limits Functor Types Function Pullback
theorem Function.isPullback_pulllback {X Y Z : Type u} (f : X β Z) (g : Y β Z) :
IsPullback (C := Type u) (fst (f := f) (g := g)) snd f g := by
sorry | |
0053 | [
"Category"
] | Medium | [
"Limit"
] | [] | problems/0053.md | CAT_statement/S_0053.lean | Theorem: Let $\mathcal{D}$ be a small-complete locally small category, a functor $ G : \mathcal{D}\to \mathcal{C}$ has a left adjoint if and only if $G$ is continuous and for each $c\in \mathcal{C}$, the comma category $(c\downarrow G)$ admits an initial object. | import Mathlib
open CategoryTheory Limits
variable {D : Type u} [Category.{v} D] [HasLimits D] [LocallySmall.{v} D]
variable {C : Type u} [Category.{v} C]
variable (G : D β₯€ C)
theorem has_left_adjoint_iff_continuous_and_initials :
G.IsRightAdjoint β PreservesLimits G β§ β (c : C), HasInitial (StructuredArrow c G)... | |
0054 | [
"Category"
] | Medium | [
"Limit"
] | [] | problems/0054.md | CAT_statement/S_0054.lean | Theorem: Let $\mathcal{B}$ be a complete category.
Then $\mathcal{B}$ has an initial object if and only if there exists a small set $I$ and an $I$-indexed family of objects $x_i$ such that, for every $s\in \mathcal{B}$, there is an $i\in I$ and an arrow $x_i \to s$. | import Mathlib
open CategoryTheory Limits
variable {B : Type u} [Category.{v} B]
theorem hasInitial_iff_exists_weakly_initial [HasLimits B] :
HasInitial B β β (I : Type v) (x : I β B), β (s : B), β (i : I), Nonempty (x i βΆ s) := by
sorry | |
0055 | [
"Category"
] | Medium | [
"Limit"
] | [] | problems/0055.md | CAT_statement/S_0055.lean | Theorem: The forgetful functor $U$ : $\mathcal{G}\mathrm{rp}$, $\mathcal{A}\mathrm{b}$, $\mathcal{R}\mathrm{ing}\to \mathcal{S}\mathrm{et}$ creates limits, but they do not preserve coproducts. | import Mathlib
open CategoryTheory Limits
theorem forget_Grp_createsLimits_but_not_coproducts :
Nonempty (CreatesLimits (forget Grp.{u})) β§
Β¬ Nonempty (PreservesColimitsOfShape (Discrete Bool) (forget Grp.{u})) := by
sorry
theorem forget_Ab_createsLimits_but_not_coproducts :
Nonempty (CreatesLimits ... | |
0056 | [
"Category"
] | High | [
"Limit"
] | [] | problems/0056.md | CAT_statement/S_0056.lean | Theorem: Let $\mathcal{C}$ and $\mathcal{D}$ be categories and let $G:\mathcal{D}\to \mathcal{C}$ be a functor.
Then $G : \mathcal{D} \to \mathcal{C}$ has a left adjoint if and only if the right Kan extension $\mathrm{Ran}_G \mathrm{Id}_{\mathcal{D}} : C \to D$ exists and is preserved by $G$ (i.e.$G \circ \mathrm{R... | import Mathlib
open CategoryTheory Functor
variable {C : Type uβ} [Category.{vβ} C] {D : Type uβ} [Category.{vβ} D]
theorem hasLeftAdjoint_iff_ran_id_preserved (G : D β₯€ C) :
G.IsRightAdjoint β
β (R : C β₯€ D) (Ξ± : G β R βΆ π D),
R.IsRightKanExtension Ξ± β§
(R β G).IsRightKanExtension ((associator ... | |
0057 | [
"Category"
] | Medium | [
"Limit"
] | Exercise 13F, AdΓ‘mek, J., Herrlich, H. and Strecker, G., 1990. Abstract and concrete categories. Wiley-Interscience. | [] | problems/0057.md | CAT_statement/S_0057.lean | Theorem: A functor that reflects equalizers (or finite products) reflects isomorphisms. | import Mathlib
open CategoryTheory Limits
variable {C : Type u} [Category.{v} C] {D : Type u'} [Category.{v'} D]
theorem reflectsIsomorphisms_of_reflects_equalizers (F : C β₯€ D)
[ReflectsLimitsOfShape WalkingParallelPair F] : F.ReflectsIsomorphisms := by
sorry
theorem reflectsIsomorphisms_of_reflects_finite_p... |
0058 | [
"Category"
] | High | [
"Limit"
] | [] | problems/0058.md | CAT_statement/S_0058.lean | Definition: Let $\mathcal C$ be a locally small category.
An object $c \in\mathcal C$ is called $\textbf{compact}$ if $\mathrm{hom}_{\mathcal C} (c,-)$ preserves filtered colimits.
Theorem: A topological space $X$ is compact if and only if it is a compact object in the category $\mathcal{O}\mathrm{p}(X)$, the catego... | import Mathlib
open CategoryTheory
namespace CAT_statement_S_0058
universe u
variable (X : Type u) [TopologicalSpace X]
abbrev Op (X : Type u) [TopologicalSpace X] := TopologicalSpace.Opens X
theorem compactSpace_iff_finitelyPresented_top :
CompactSpace X β IsFinitelyPresentable (C := Op X) (β€ : Op X) := by
... | |
0059 | [
"Category"
] | High | [
"Limit"
] | Exercise 13D(b), AdΓ‘mek, J., Herrlich, H. and Strecker, G., 1990. Abstract and concrete categories. Wiley-Interscience. | [] | problems/0059.md | CAT_statement/S_0059.lean | Definition: A functor $F : \mathcal C \to \mathcal D$ is said to $\textbf{lift limits}$ if for every diagram $D: \mathcal I\to \mathcal C$ and every limit $L$ of $F\circ D$, there exists a limit $L'\in\mathcal D$ such that $F(L')\cong L$.
Theorem: There is a functor that lifts limits but is not faithful. | import Mathlib
open CategoryTheory Limits
namespace CAT_statement_S_0059
universe w' w'β w wβ vβ vβ vβ uβ uβ uβ
variable {C : Type uβ} [Category.{vβ} C]
variable {D : Type uβ} [Category.{vβ} D]
variable {J : Type w} [Category.{w'} J] {K : J β₯€ C}
structure LiftableConeβ (K : J β₯€ C) (F : C β₯€ D) (c : Cone (K β F)) wh... |
0060 | [
"Category"
] | High | [
"Limit"
] | [] | problems/0060.md | CAT_statement/S_0060.lean | Theorem: Suppose $\mathcal{B}$ is locally small, complete, has a small coseparating set $S$, and has the property that every family of subobjects has an intersection. Then $\mathcal{B}$ has an initial object. | import Mathlib
open CategoryTheory Limits
theorem has_initial_of_locally_small_complete_coseparating {β¬ : Type u} [Category.{v} β¬]
[LocallySmall.{w} β¬] [HasLimitsOfSize.{w, w} β¬] {S : Set β¬} [Small.{w} S]
(hS : IsCoseparating S) (h : β (A : β¬), β (s : Set (Subobject A)), β (f : Subobject A),
IsGLB s f)... | |
0061 | [
"Category"
] | High | [
"Limit"
] | [] | problems/0061.md | CAT_statement/S_0061.lean | Theorem: Let $\mathcal{C} = \mathrm{Vec}_{\mathbb{k}}$ the category of finite dimensional vector spaces over a field $\mathbb{k}$. Then the coend of the hom functor $\mathrm{Hom} \colon \mathcal{C}^{\mathrm{op}} \times \mathcal{C} \to \mathcal{C}$ is $\mathbb{k}$ equipped with the usual trace maps $\mathrm{Hom}(V,V) \t... | import Mathlib
open CategoryTheory Limits
theorem coend_hom_is_trace_of_matrices
(π : Type u) [Field π] :
β (F : (ModuleCat π)α΅α΅ β₯€ ModuleCat π β₯€ ModuleCat π),
(β X Y, (F.obj (Opposite.op X)).obj Y β
ModuleCat.of π (X ββ[π] Y)) β
β (T : ModuleCat π),
(β (tr : β X, (F.obj (Opposite.o... | |
0062 | [
"Category"
] | High | [
"Limit"
] | Exercise 12N(b), AdΓ‘mek, J., Herrlich, H. and Strecker, G., 1990. Abstract and concrete categories. Wiley-Interscience. | [] | problems/0062.md | CAT_statement/S_0062.lean | Definition: Let $\mathcal C$ be a category. Let $S$ be a family of subobjects $(s_n,i_n)$ of an object $c\in \mathcal C$, indexed by a class $I$.
A subobject $(x,i:x\to c)$ of $c$ is called an $\textbf{intersection}$ of $S$ provided that the following two conditions are satisfied:
(1) $i$ factors through each $i_n$ i... | import Mathlib
open CategoryTheory Limits
namespace CAT_statement_S_0062
universe u v
variable {C : Type u} [Category.{v} C]
def IsIntersectionOf {B : C} (A : Subobject B) (S : Set (Subobject B)) : Prop :=
(β Ai, Ai β S β A β€ Ai) β§
(β A' : Subobject B, (β Ai, Ai β S β A' β€ Ai) β A' β€ A)
def HasIntersections (... |
0063 | [
"Category"
] | High | [
"Limit"
] | [] | problems/0063.md | CAT_statement/S_0063.lean | Definition: Let $\mathcal C$ be a locally small category.
An object $c \in\mathcal C$ is called $\textbf{compact}$ if $\mathrm{hom}_{\mathcal C} (x,-)$ preserves filtered colimits.
Theorem: For $\mathcal{G}\mathrm{rp}$, an object is compact if and only if it is finitely presented as a group. Every group can be reali... | import Mathlib
open CategoryTheory
namespace CAT_statement_S_0063
universe u
def IsFinitelyPresentedGrp (X : Type u) [Group X] : Prop :=
β (Ξ± : Type u) (rels : Set (FreeGroup Ξ±)), Finite Ξ± β§ rels.Finite β§ Nonempty (X β* PresentedGroup rels)
theorem isCompactObject_Grp_iff_finite_presented (X : Type u) [Group X... | |
0064 | [
"Category"
] | High | [
"Limit"
] | [] | problems/0064.md | CAT_statement/S_0064.lean | Definition: Let $\mathcal C$ be a locally small category.
An object $c \in\mathcal C$ is called $\textbf{compact}$ if $\mathrm{hom}_{\mathcal C} (x,-)$ preserves filtered colimits.
Theorem: Let $A$ be a ring.
For the category of right $A$-modules $\mathcal{A}\mathrm{b}_A$, an object is compact if and only if it... | import Mathlib
open CategoryTheory
universe u v w
theorem isCompactObject_Grp_iff_finite_presented {A : Type u} [Ring A] (X : Type v) [Group X] [AddCommGroup X] [Module A X] : CategoryTheory.IsFinitelyPresentable (ModuleCat.of A X) β Module.FinitePresentation A X := by
sorry
theorem module_realized_as_direct_l... | |
0065 | [
"Category"
] | High | [
"Limit"
] | [] | problems/0065.md | CAT_statement/S_0065.lean | Theorem: Let $\mathcal{C}$ be a complete, wellpowered, cowellpowered and have a separator $s$.
Then $\mathcal{C}$ is cocomplete if and only if for each set $I$, there exists an $I$-th copower of $S$ in $\mathcal{C}$. | import Mathlib
open CategoryTheory Limits
variable {C : Type u} [Category.{v} C]
theorem hasColimits_iff_hasCoprod_of_separator
[HasLimits C]
[WellPowered C]
[WellPowered Cα΅α΅]
(S : C) (hS : IsSeparator S) :
HasColimits C β β (I : Type v), HasColimit (Discrete.functor (fun (_ : I) => S)) := by
s... | |
0066 | [
"Category"
] | High | [
"Limit"
] | Exercise 12N(c), AdΓ‘mek, J., Herrlich, H. and Strecker, G., 1990. Abstract and concrete categories. Wiley-Interscience. | [] | problems/0066.md | CAT_statement/S_0066.lean | Definition: Let $\mathcal C$ be a category. Let $S$ be a family of subobjects $(s_n,i_n)$ of an object $c\in \mathcal C$, indexed by a class $I$.
A subobject $(x,i:x\to c)$ of $c$ is called an $\textbf{intersection}$ of $S$ provided that the following two conditions are satisfied:
(1) $i$ factors through each $i_n$ i... | import Mathlib
open CategoryTheory Limits
namespace CAT_statement_S_0066
universe u v
variable {C : Type u} [Category.{v} C]
def IsIntersectionOf {B : C} (A : Subobject B) (S : Set (Subobject B)) : Prop :=
(β Ai, Ai β S β A β€ Ai) β§
(β A' : Subobject B, (β Ai, Ai β S β A' β€ Ai) β A' β€ A)
def HasIntersections (... |
0067 | [
"Category"
] | Medium | [
"Limit"
] | [] | problems/0067.md | CAT_statement/S_0067.lean | Theorem: Let $\omega$ be the ordinal of natural numbers.
Consider $F:\omega^{op}\to \mathcal{R}\mathrm{ing}$ with $F_n:=k[x]/(x^n)$ and $f_n:k[x]/(x^{n+1})\to k[x]/(x^n)$.
Then the limit exists and is isomorphic to $k[[x]]$. | import Mathlib
open CategoryTheory Polynomial Limits
universe u
namespace CAT_statement_S_0067
variable (k : Type u) [Field k]
noncomputable def F : Natα΅α΅ β₯€ RingCat :=
{
obj := fun β¨nβ© => RingCat.of ((k[X] β§Έ Ideal.span {(X ^ n : k[X])}))
map := fun {A B} f => match A, B with
| β¨nβ©, β¨mβ© => match f wi... | |
0068 | [
"Category"
] | High | [
"Limit"
] | [] | problems/0068.md | CAT_statement/S_0068.lean | Theorem: There is a category $\mathcal{C}$ such that there exists two regular epimorphisms $f:c\to d$ and $g:c'\to d'$ in which the product of $f$ and $g$ is not regularly epic. | import Mathlib
open CategoryTheory Limits
universe u
theorem regular_epimorphism_not_product_regular_epimorphism : β (C : Type (u+1)) (inst : Category C) (c d c' d' : C) (f : c βΆ d) (g : c' βΆ d') (instβ : RegularEpi f) (instβ : RegularEpi g) (hasProdβ : HasBinaryProduct c c') (hasProdβ : HasBinaryProduct d d'), IsE... | |
0069 | [
"Category"
] | High | [
"Limit"
] | Exercise 11O, AdΓ‘mek, J., Herrlich, H. and Strecker, G., 1990. Abstract and concrete categories. Wiley-Interscience. | [] | problems/0069.md | CAT_statement/S_0069.lean | Theorem: An abelian group is torsion free if and only if it is a directed colimit in $\mathcal{A}\mathrm{b}$ of free abelian groups. | import Mathlib
open CategoryTheory Limits
theorem torsionFree_iff_isFilteredColimit_free
(A : ModuleCat β€) :
NoZeroSMulDivisors β€ A β
β (J : Type) (_ : SmallCategory J) (_ : IsFiltered J)
(F : J β₯€ ModuleCat β€),
(β j : J, Module.Free β€ (F.obj j)) β§
Nonempty (A β
colimit F) := by
... |
0070 | [
"Category"
] | High | [
"Limit"
] | Exercise 13C, AdΓ‘mek, J., Herrlich, H. and Strecker, G., 1990. Abstract and concrete categories. Wiley-Interscience. | [] | problems/0070.md | CAT_statement/S_0070.lean | Definition: A concrete category $(\mathcal C, U)$ over $\mathcal B$ is said to $\textbf{have (small) concrete limits}$ if $\mathcal C$ has all small limits and $U$ preserves them.
Theorem: Let $(\mathcal{C},U)$ have small concrete limits.
Then $U$ reflects small limits if and only if $U$ reflects isomorphisms. | import Mathlib
open CategoryTheory Limits
variable {C : Type u} [Category.{v} C]
variable {D : Type u'} [Category.{v'} D]
variable (U : C β₯€ D)
theorem reflects_limits_iff_reflects_isomorphisms_preserves_limits
[HasLimitsOfSize.{v, v} C]
[PreservesLimitsOfSize.{v, v} U]
[CategoryTheory.Functor.Faithful U... |
0071 | [
"Category"
] | Medium | [
"Limit"
] | Exercise 13D(a), AdΓ‘mek, J., Herrlich, H. and Strecker, G., 1990. Abstract and concrete categories. Wiley-Interscience. | [] | problems/0071.md | CAT_statement/S_0071.lean | Definition: A functor $F : \mathcal C \to \mathcal D$ is said to $\textbf{lift limits}$ if for every diagram $D: \mathcal I\to \mathcal C$ and every limit $L$ of $F\circ D$, there exists a limit $L'\in\mathcal D$ such that $F(L')\cong L$.
Theorem: A functor that lifts equalizers is faithful if and only if it reflects... | import Mathlib
open CategoryTheory Limits
namespace CAT_statement_S_0071
universe uC vC uD vD w w'
variable {C : Type uC} [Category.{vC} C]
variable {D : Type uD} [Category.{vD} D]
variable (F : C β₯€ D)
variable {J : Type w} [Category.{w'} J]
class LiftsLimit (K : J β₯€ C) (F : C β₯€ D): Prop where
lifts {c : Cone... |
0072 | [
"Category"
] | High | [
"Limit"
] | Exercise 12K, AdΓ‘mek, J., Herrlich, H. and Strecker, G., 1990. Abstract and concrete categories. Wiley-Interscience. | [] | problems/0072.md | CAT_statement/S_0072.lean | Theorem: A full subcategory of the category $\mathcal{T}\mathrm{op}^{CH}$ of compact Hausdorff spaces is reflective if and only if it is cocomplete and contains at least one non-empty space. | import Mathlib
open CategoryTheory Limits Topology
universe u
variable {D : Type (u+1)} [Category.{u} D]
variable (i : D β₯€ CompHaus.{u})
variable [CategoryTheory.Functor.Full i] [CategoryTheory.Functor.Faithful i]
theorem reflective_iff_cocomplete_and_contains_nonempty_of_full_subcategory_CompHaus :
Nonempty (C... |
0073 | [
"Category"
] | Medium | [
"Limit"
] | [] | problems/0073.md | CAT_statement/S_0073.lean | Definition: A functor $F : \mathcal C \to \mathcal D$ is said to $\textbf{lift limits}$ if for every diagram $D: \mathcal I\to \mathcal C$ and every limit $L$ of $F\circ D$, there exists a limit $L'\in\mathcal D$ such that $F(L')\cong L$.
Theorem: The forgetful functor $U:\mathcal{T}\mathrm{op}\to \mathcal{S}\mathrm{... | import Mathlib
open CategoryTheory Limits
namespace CAT_statement_S_0073
universe w' wβ' w wβ vβ vβ vβ uβ uβ uβ
variable {C : Type uβ} [Category.{vβ} C]
variable {D : Type uβ} [Category.{vβ} D]
variable {J : Type w} [Category.{w'} J] {K : J β₯€ C}
class LiftsLimit (K : J β₯€ C) (F : C β₯€ D): Prop where
lifts {c : ... | |
0074 | [
"Category"
] | High | [
"Cocompletion"
] | Corollary A.5, JiΕΓ AdΓ‘mek, Liang-Ting Chen, Stefan Milius, and Henning Urbat, Reiterman's Theorem on Finite Algebras for a Monad, arXiv:2101.00942 | [] | problems/0074.md | CAT_statement/S_0074.lean | Theorem: Let $\mathcal C$ be a small category. A category $\mathcal L$ containing $\mathcal C$ as a full subcategory is an
pro-completion of $\mathcal C$ if and only if the following conditions hold:
(1) $\mathcal L$ has cofiltered colimits,
(2) every object of $\mathcal L$ is the colimit of a cofiltered diagram in $... | import Mathlib
open CategoryTheory Limits
universe u v w uβ vβ
namespace CAT_statement_S_0074
noncomputable section
abbrev Pro (C : Type u) [Category.{v} C] : Type (max u (v + 1)) := (Ind (Cα΅α΅))α΅α΅
abbrev proYoneda (C : Type u) [SmallCategory C] : C β₯€ Pro C :=
CategoryTheory.opOp C β (CategoryTheory.Ind.yoneda... |
0075 | [
"Category"
] | High | [
"Cocompletion"
] | Proposition 3.2, Chen Ruiyuan 2021, On sifted colimits in the presence of pullbacks, arXiv:2109.12708 | [] | problems/0075.md | CAT_statement/S_0075.lean | Definition: A category is called sifted if the category of cocones over any finite discrete family of objects in it is connected.
Notation: $\mathrm{Rec}(\mathcal{C})$ := free cocompletion of $\mathcal{C}$ under reflexive coequalizers.
Theorem: For a sifted category with pullbacks $\mathcal C$, $\mathrm{Rec}(\mathc... | import Mathlib
namespace CAT_statement_S_0075
open CategoryTheory Limits
universe u v
namespace CategoryTheory.Limits
open Limits Functor
variable {C : Type u} [Category.{v} C]
variable (C) in
abbrev Psh (C : Type u) [Category.{v} C] : Type (max u (v + 1)) :=
Cα΅α΅ β₯€ Type v
inductive RecObjectPresentation : Ps... |
0076 | [
"Category"
] | High | [
"Cocompletion"
] | JiΕΓ AdΓ‘mek, Liang-Ting Chen, Stefan Milius, and Henning Urbat, Reiterman's Theorem on Finite Algebras for a Monad, arXiv:2101.00942 | [] | problems/0076.md | CAT_statement/S_0076.lean | Theorem: Let $\mathcal Set^{fin}$ be the category of finite sets and functions. Its pro-completion is the category
$$ \mathrm{Pro}(\mathcal Set^{fin}) = \mathcal Stone$$
of Stone spaces, i.e. compact topological spaces in which distinct elements can be separated by
clopen subsets. Morphisms are the continuous functions... | import Mathlib
open CategoryTheory
universe v u
abbrev Pro (C : Type u) [Category.{v} C] : Type (max u (v + 1)) := (Ind (Cα΅α΅))α΅α΅
theorem pro_fintypecat_equiv_profinite : Nonempty ((Pro (FintypeCat)) β Profinite) := by
sorry |
0077 | [
"Category"
] | High | [
"Cocompletion"
] | Theorem 5.1, Chen Ruiyuan 2021, On sifted colimits in the presence of pullbacks, arXiv:2109.12708 | [] | problems/0077.md | CAT_statement/S_0077.lean | Notation:
$\mathrm{Sind}(\mathcal{C})$ := free cocompletion of $\mathcal{C}$ under small sifted colimits;
$\mathrm{Ind}(\mathcal{C})$ := free cocompletion of $\mathcal{C}$ under small filtered colimits;
$\mathrm{Rec}(\mathcal{C})$ := free cocompletion of $\mathcal{C}$ under reflexive coequalizers.
Theorem:
Let $\ma... | import Mathlib
namespace CAT_statement_S_0077
open CategoryTheory Limits
universe u v
namespace CategoryTheory.Limits
open Limits Functor
variable {C : Type u} [Category.{v} C]
abbrev Psh (C : Type u) [Category.{v} C] : Type (max u (v + 1)) :=
Cα΅α΅ β₯€ Type v
inductive RecObjectPresentation : Psh C β Type (max u... |
0078 | [
"Category"
] | High | [
"Cocompletion"
] | Lemma 6.2, Chen Ruiyuan 2021, On sifted colimits in the presence of pullbacks, arXiv:2109.12708 | [] | problems/0078.md | CAT_statement/S_0078.lean | Definition: For $F : \mathcal{C} \to \mathcal{D}$, we define the induced cocontinuous functor $\mathrm{Lan}_{F^{op}} : \mathcal{P}sh(\mathcal{C}) \to \mathcal{P}sh(\mathcal{D})$, by $\phi\mapsto \phi \star yF$, where $\phi\star yF$ is the $\phi$-weighted colimit of the diagram $yF$ and $y$ is the Yoneda embedding.
No... | import Mathlib
namespace CAT_statement_S_0078
open CategoryTheory Limits Functor
universe u v
namespace CategoryTheory
namespace Limits
variable {C : Type u} [Category.{v} C]
structure SindObjectPresentation (A : Cα΅α΅ β₯€ Type v) where
I : Type v
[β : SmallCategory I]
[hI : IsSifted I]
F : I β₯€ C
ΞΉ : F β ... |
0079 | [
"Category"
] | Easy | [
"Abelian"
] | [] | problems/0079.md | CAT_statement/S_0079.lean | Theorem: Let $\mathcal{A}$ be an additive category.
Let $x,y,z$ be objects in $\mathcal{A}$.
Then the composition $\hom_{\mathcal{A}}(y,z)\times \hom_{\mathcal{A}}(x,y)\to \hom_{\mathcal{A}}(x,z)$ is bilinear map. | import Mathlib
open CategoryTheory
variable {C : Type u} [Category.{v} C] [Preadditive C]
structure IsBilinear {X Y Z : C} (f : (Y βΆ Z) β ((X βΆ Y) β (X βΆ Z))) : Prop where
map_add_left : β (a b : Y βΆ Z) (g : X βΆ Y),
f (a + b) g = f a g + f b g
map_add_right : β (a : Y βΆ Z) (g h : X βΆ Y), f a (g + h) = f a g... | |
0080 | [
"Category"
] | Easy | [
"Abelian"
] | [] | problems/0080.md | CAT_statement/S_0080.lean | Theorem: Let $\mathcal{A}$ be an abelian category and let $f$ be a morphism in $\mathcal{A}$.
Then $f$ is an isomorphism if and only if $f$ is monic and epic. | import Mathlib
open CategoryTheory
variable {C : Type*} [Category C] [Abelian C]
theorem isIso_iff_mono_and_epi {X Y : C} (f : X βΆ Y) :
IsIso f β (Mono f β§ Epi f) := by
sorry | |
0081 | [
"Category"
] | Easy | [
"Abelian"
] | [] | problems/0081.md | CAT_statement/S_0081.lean | Theorem: Let $\mathcal{A}$ be an abelian category and let $f$ be a morphism in $\mathcal{A}$.
Then $f$ is monic if and only if $\ker(f)=0$. | import Mathlib
open CategoryTheory Limits Category
variable {C : Type*} [Category C] [Abelian C]
theorem mono_iff_isZero_kernel {X Y : C} (f : X βΆ Y) :
Mono f β IsZero (kernel f) := by
sorry | |
0082 | [
"Category"
] | Easy | [
"Abelian"
] | [] | problems/0082.md | CAT_statement/S_0082.lean | Theorem: $\mathbb{k}$ is the unique (up to isomorphism) simple object in $\mathrm{Vect}_{\mathbb{k}}$. | import Mathlib
open Module
variable (π : Type u) [Field π]
instance isSimpleModule_self : IsSimpleModule π π := by
constructor
intro N
have : IsSimpleOrder (Submodule π π) := by infer_instance
exact eq_bot_or_eq_top N
theorem unique_simple_object (M : Type v) [AddCommGroup M] [Module π M] [IsSimpleMo... | |
0083 | [
"Category"
] | Medium | [
"Abelian"
] | [] | problems/0083.md | CAT_statement/S_0083.lean | Theorem: $\mathbb{Z}_p$ is simple object in $\mathcal{A}\mathrm{b}$ when $p$ is prime number. | import Mathlib
open CategoryTheory
variable (p : β) [Fact p.Prime]
theorem ZMod_simple : CategoryTheory.Simple (ModuleCat.of β€ (ZMod p)) := by
sorry | |
0084 | [
"Category"
] | Medium | [
"Abelian"
] | [] | problems/0084.md | CAT_statement/S_0084.lean | Theorem: $\mathcal{G}\mathrm{rp}$ is not an additive category. | import Mathlib
open CategoryTheory Limits
def IsAdditiveCategory (C : Type u) [Category.{v} C] : Prop :=
β (_ : Preadditive C), HasZeroObject C β§ HasFiniteBiproducts C
theorem Grp_not_is_additive : IsEmpty (IsAdditiveCategory Grp.{u}) := by
sorry | |
0085 | [
"Category"
] | Medium | [
"Abelian"
] | [] | problems/0085.md | CAT_statement/S_0085.lean | Definition: A functor is called $\textbf{left exact}$ if it preserves all finite limits.
Theorem: Let $\mathcal{A}$ and $\mathcal{B}$ be abelian categories and $F:\mathcal{A}\to\mathcal{B}$ be a functor. Then $F$ is left exact if and only if $F$ is additive and $F$ maps each exact sequence $0\to x\to y\to z$ to an ex... | import Mathlib
open CategoryTheory Functor Limits ShortComplex
variable {C D : Type*} [Category C] [Category D]
variable [Abelian C] [Abelian D]
theorem preservesFiniteLimits_tfae
(F : C β₯€ D) [F.Additive] : List.TFAE
[
β (S : ShortComplex C), S.ShortExact β (S.map F).Exact β§ Mono (F.map S.f),
β (... | |
0086 | [
"Category"
] | Medium | [
"Abelian"
] | [] | problems/0086.md | CAT_statement/S_0086.lean | Theorem: Let $\mathcal{A}$ be an abelian category and let $P\in\mathcal{A}$.
Then $\hom_{\mathcal{A}}(P,-):\mathcal{A}\to \mathcal{A}\mathrm{b}$ is right exact if and only if $\hom_{\mathcal{A}}(P,-):\mathcal{A}\to \mathcal{A}\mathrm{b}$ preserves epimorphism. | import Mathlib
open CategoryTheory Limits Opposite
variable {A : Type u} [Category.{v} A] [Abelian A]
theorem hom_rightExact_iff_preserves_epi (P : A) :
PreservesFiniteColimits (preadditiveCoyoneda.obj (op P)) β
Functor.PreservesEpimorphisms (preadditiveCoyoneda.obj (op P)) := by
sorry | |
0087 | [
"Category"
] | High | [
"Abelian"
] | [] | problems/0087.md | CAT_statement/S_0087.lean | Definition: An Abelian category $\mathcal A$ is called $\textbf{semisimple}$ if any short exact sequence in $\mathcal A$ is splittable.
Theorem: Let $\mathcal{A}$ be an abelian category.
Then the followings are equivalent:
\begin{enumerate}
\item $\mathcal{A}$ is semisimple;
\item any obejct i... | import Mathlib
open CategoryTheory Limits
variable {A : Type u} [Category.{v} A] [Abelian A]
def IsSemisimple (A : Type u) [Category.{v} A] [Abelian A] : Prop :=
β (S : ShortComplex A), S.ShortExact β Nonempty S.Splitting
theorem isSemisimple_iff_injective_iff_projective :
(IsSemisimple A β β (X : A), Injecti... | |
0088 | [
"Category"
] | Medium | [
"Abelian"
] | [] | problems/0088.md | CAT_statement/S_0088.lean | Theorem: Let $\mathcal{A}$ be an abelian category.
If $x,y$ are simple objects in $\mathcal{A}$. Then each non-zero $f:x\to y$ are isomorphism.
In particular, if $x$ is simple, then $\hom_{\mathcal{A}}(x,x)$ is a division ring; if $x\neq y$ ,then $\hom_{\mathcal{A}}(x,y)=0$. | import Mathlib
open CategoryTheory
variable {π : Type*} [Category π] [Abelian π]
theorem simple_objects_nonzero_morphisms_iso
{x y : π} [Simple x] [Simple y] (f : x βΆ y) (h : f β 0) :
IsIso f := by
sorry
theorem simple_object_end_is_division_ring
(x : π) [Simple x] :
Nonempty (DivisionRing... | |
0089 | [
"Category"
] | Medium | [
"Abelian"
] | [] | problems/0089.md | CAT_statement/S_0089.lean | Definition: Let $\mathcal C$ be a category with zero morphisms. An object $c\in\mathcal C$ is called $\textbf{mono-simple}$ if it has no proper subobjects. An object $c\in\mathcal C$ is called $\textbf{epi-simple}$ if it has no proper quotient objects.
Theorem: Let $\mathcal{A}$ be a pre-additive category in which all... | import Mathlib
open CategoryTheory
class IsSplitMonoCategory (A : Type*) [Category A] where
splitMonoOfMono {X Y : A} (f : X βΆ Y) [Mono f] : Nonempty (SplitMono f)
class IsSplitEpiCategory (A : Type*) [Category A] where
splitEpiOfEpi {X Y : A} (f : X βΆ Y) [Epi f] : Nonempty (SplitEpi f)
variable {A : Type*} [Ca... | |
0090 | [
"Category"
] | Medium | [
"Abelian"
] | [] | problems/0090.md | CAT_statement/S_0090.lean | Definition: A category is called $\textbf{normal}$ if each monomorphism is a kernel.
Definition: A category is called $\textbf{conormal}$ if each epimorphism is a cokernel.
Definition: A category is called $\textbf{binormal}$ if it is both normal and conormal.
Definition: Let $\mathcal C$ be a category.
An object $c... | import Mathlib
open CategoryTheory
variable {A : Type*} [Category A] [Limits.HasZeroMorphisms A]
[IsNormalMonoCategory A]
[IsNormalEpiCategory A]
[Limits.HasKernels A]
[Limits.HasCokernels A]
theorem binormal_mono_simple_iff_epi_simple (x : A) :
(β (y : A) (f : y βΆ x) [Mono f], f = 0 β¨ IsIso f) β
(β... | |
0091 | [
"Category"
] | Easy | [
"Monad"
] | [] | problems/0091.md | CAT_statement/S_0091.lean | Theorem: For any monad $(T,\mu,\eta)$ on a category $\mathcal{C}$ and let $\mathcal{C}^T$ be its Elienberg-Moore category.
Let $U:\mathcal{C}^T\to\mathcal{C}$ be the forgetful functor, then it admits a left adjoint. | import Mathlib
open CategoryTheory
variable {C : Type uβ} [Category.{vβ} C]
theorem monad_forget_has_left_adjoint (T : Monad C) :
T.forget.IsRightAdjoint := by
sorry | |
0092 | [
"Category"
] | Medium | [
"Monad"
] | [] | problems/0092.md | CAT_statement/S_0092.lean | Theorem: Let $R$ be a ring. The forgetful functor from the category $\mathcal{M}\mathrm{od}_R$ of $R$-modules to the category $\mathcal{A}\mathrm{b}$ of abelian groups creates all colimits that $\mathcal{A}\mathrm{b}$ admits. | import Mathlib
open CategoryTheory Limits
variable {R : Type u} [CommRing R]
theorem ModuleCat.forgetCreatesColimits :
Nonempty (CreatesColimits (forgetβ (ModuleCat R) AddCommGrp)) := by
sorry | |
0093 | [
"Category"
] | Medium | [
"Monad"
] | [] | problems/0093.md | CAT_statement/S_0093.lean | Theorem: Suppose $\mathcal{C}$ is cocomplete and $G :\mathcal{D} \to \mathcal{C}$ is monadic.
Then $\mathcal{D}$ is cocomplete if and only if $\mathcal{D}$ has coequalizers. | import Mathlib
open CategoryTheory Limits
universe uC uD vC vD w w'
variable {C : Type uC} [Category.{vC} C]
variable {D : Type uD} [Category.{vD} D]
variable (G : D β₯€ C)
theorem cocomplete_iff_hasCoequalizers_of_monadic
[HasColimitsOfSize.{w, w'} C] [MonadicRightAdjoint G] :
HasColimitsOfSize.{w, w'} D β Has... | |
0094 | [
"Category"
] | Easy | [
"Monad"
] | Exercise 20F(b), AdΓ‘mek, J., Herrlich, H. and Strecker, G., 1990. Abstract and concrete categories. Wiley-Interscience. | [] | problems/0094.md | CAT_statement/S_0094.lean | Theorem: Let $\mathcal{B}$ be a category and $\mathcal{C}$ be a full reflective subcategory of $\mathcal{B}$. Then the monad associated to the inclusion functor $U : \mathcal{C} \to \mathcal{B}$ is idempotent. | import Mathlib
open CategoryTheory Functor
namespace CAT_statement_S_0094
variable {C : Type*} [Category C]
variable {B : Type*} [Category B]
noncomputable def monadOfRightAdjoint (U : Functor C B) [IsRightAdjoint U] : Monad B :=
(Adjunction.ofIsRightAdjoint U).toMonad
def IsIsoClosed (U : Functor C B) := β (x :... |
0095 | [
"Category"
] | Medium | [
"Monad"
] | [] | problems/0095.md | CAT_statement/S_0095.lean | Theorem: If $\mathcal{D}$ admits coequalizers, a functor $G : \mathcal{D} \to \mathcal{C}$ is monadic if $G$ has a left adjoint, conservative and preserves coequalizers. | import Mathlib
open CategoryTheory Limits
universe uβ uβ vβ
variable {C : Type uβ} {D : Type uβ} [Category.{vβ} C] [Category.{vβ} D]
variable {G : D β₯€ C} {F : C β₯€ D} (adjFG : F β£ G)
variable [HasCoequalizers D]
variable [G.ReflectsIsomorphisms]
variable [PreservesColimitsOfShape WalkingParallelPair G]
theorem mona... | |
0096 | [
"Category"
] | Medium | [
"Monad"
] | [] | problems/0096.md | CAT_statement/S_0096.lean | Theorem: Consider the adjunction $-\otimes_{\mathbb{Z}}R:\mathcal{A}\mathrm{b}\to \mathcal{A}\mathrm{b}_R$ and $U:\mathcal{A}\mathrm{b}_R\to \mathcal{A}\mathrm{b}$. We obtain a monad $T$.
The $T$-modules are right $R$-modules. | import Mathlib
open CategoryTheory
namespace CAT_statement_S_0096
universe u v
variable (R : Type u) [CommRing R]
abbrev intToR : β€ β+* R := Int.castRingHom R
noncomputable abbrev U : ModuleCat.{max u v} R β₯€ ModuleCat.{max u v} β€ :=
ModuleCat.restrictScalars (intToR R)
noncomputable abbrev F : ModuleCat.{max... | |
0097 | [
"Category"
] | Medium | [
"Monad"
] | [] | problems/0097.md | CAT_statement/S_0097.lean | Theorem: Let $\mathcal{C}$, $\mathcal{D}$ be categories and $F : \mathcal{C}\to \mathcal{D}$ be a left adjoint functor to $G: \mathcal{D} \to \mathcal{C}$.
Denote the induced monad of the adjunction $F \dashv G$ by $T := GF$.
Let $K : \mathcal{D} \to \mathcal{C}^T$ be the comparison functor.
If $\mathcal{... | import Mathlib
open CategoryTheory Monad
universe uβ uβ vβ
variable {C : Type uβ} [Category.{vβ} C] {D : Type uβ} [Category.{vβ} D]
variable (F : C β₯€ D) (G : D β₯€ C) (adj : F β£ G)
theorem comparison_adjunction
[β (A : adj.toMonad.Algebra), Limits.HasCoequalizer (F.map A.a) (adj.counit.app (F.obj A.A))] :
β K... | |
0098 | [
"Category"
] | Medium | [
"Monad"
] | [] | problems/0098.md | CAT_statement/S_0098.lean | Definition: For any monad $T$ on $\mathcal C$, we define a category $\mathrm{Adj}_T$ whose objects are adjunctions $(F:\mathcal C\to \mathcal D,G,\eta,\epsilon)$ which induce the same monad $T$, and a morphism between $(F:\mathcal C\to \mathcal D,G,\eta,\epsilon)$ and $(F':\mathcal C\to \mathcal D',G',\eta',\epsilon')$... | import Mathlib
open CategoryTheory Monad
namespace CAT_statement_S_0098
variable {C : Type*} [Category C]
structure AdjCat (T : Monad C) where
D : Type*
[category : Category D]
F : Functor C D
U : Functor D C
adj : F β£ U
monad_eq : T β
Adjunction.toMonad adj
namespace AdjCat
variable {T : Monad C}
in... | |
0099 | [
"Category"
] | Medium | [
"Monad"
] | Exercise 20F(c), AdΓ‘mek, J., Herrlich, H. and Strecker, G., 1990. Abstract and concrete categories. Wiley-Interscience. | [] | problems/0099.md | CAT_statement/S_0099.lean | Theorem: The monad associated with the forgetful functor $\mathcal{T}\mathrm{op}\to \mathcal{S}\mathrm{et}$ is idempotent. | import Mathlib
open CategoryTheory
theorem monad_Top_idempotent : IsIso TopCat.adjβ.toMonad.ΞΌ := by
sorry |
0100 | [
"Category"
] | High | [
"Monad"
] | [] | problems/0100.md | CAT_statement/S_0100.lean | Theorem: Let $\mathcal{C}$, $\mathcal{D}$ be categories and $F : \mathcal{C} \to \mathcal{D}$ be a left adjoint functor to $G: \mathcal{D}\to \mathcal{C}$.
Denote the induced monad of the adjunction $F \dashv G$ by $T$. The following statements are equivalent:
\begin{enumerate}
\item The comparison fun... | import Mathlib
open CategoryTheory Limits
universe v u u'
namespace CAT_statement_S0100
variable {C : Type u} [Category.{v} C]
variable {D : Type u'} [Category.{v} D]
variable (F : C β₯€ D) (G : D β₯€ C)
variable (adj : F β£ G)
abbrev FG : D β₯€ D := G β F
abbrev K : D β₯€ (adj.toMonad).Algebra :=
Monad.comparison adj
... |
LeanCat: A Lean Dataset for Evaluating Library-Grounded Category-Theoretic Reasoning
This is an anonymized review artifact.
LeanCat is a dataset of 100 statement-level problems in Lean 4 (mathlib), designed to stress-test abstraction-heavy, library-grounded reasoning in formal mathematics. This repository contains Part I: 1-Category Theory.
Overview
LeanCat addresses a critical gap in automated theorem proving evaluation datasets by focusing on category theory - the unifying language of modern mathematics that requires sophisticated abstraction and library navigation skills. While existing benchmarks target olympiad-style problems or undergraduate mathematics, LeanCat challenges AI systems with research-level categorical reasoning.
Repository Structure
LeanCat/
βββ CAT_statement/ # Formal Lean 4 statements of dataset problems
βββ problems/ # Natural language problem descriptions (Markdown)
βββ data/ # JSONL records for Hugging Face dataset viewing
β βββ leancat_records.jsonl # One record per dataset problem
βββ configs/ # Evaluation protocol configuration
βββ prompts/ # Prompt templates used by baseline protocols
βββ scripts/ # Dataset validation utilities
βββ .github/
βββ CAT_statement.lean # Main Lean 4 file containing all statement imports
βββ DATASET_CARD.md # Dataset documentation and responsible-use notes
βββ EVALUATION.md # Proof validation and reporting protocol
βββ lakefile.lean
βββ lean-toolchain # Use Lean version 4.19.0
βββ metadata.json # Problem metadata (difficulty, tags, refs)
βββ lake-manifest.json
βββ .gitignore
βββ LICENSE
Quick Start
- Install Lean via
elan: https://leanprover-community.github.io/get_started.html - Build the project:
# Build with lake
lake build
The dataset artifact builds with Lean 4.19.0 and mathlib 4.19.0.
To validate the artifact structure, run:
python scripts/validate_dataset.py
For Hugging Face hosting and programmatic loading, the derived JSONL index is
provided at data/leancat_records.jsonl. It contains one record per problem,
including the problem id, metadata fields, source file paths, natural-language
statement, and Lean formal statement. Regenerate it after metadata or statement
edits with:
python scripts/generate_hf_records.py
The NeurIPS Croissant metadata file with Responsible AI fields is provided as
croissant.json. This file is the validated submission version for
OpenReview; the Hugging Face Croissant API may still expose the platform's
automatically generated core-metadata version.
To run the provided evaluation drivers with an OpenAI-compatible chat API:
$env:OPENAI_API_KEY="your_key"
$env:OPENAI_BASE_URL="https://api.openai.com/v1"
python scripts/passk.py --start 1 --end 1 --model gpt-5.2 -k 4
python scripts/leanbridge.py --start 1 --end 1 --model gpt-5.2 --max-iterations 4
On Linux/macOS:
export OPENAI_API_KEY="your_key"
export OPENAI_BASE_URL="https://api.openai.com/v1"
python scripts/passk.py --start 1 --end 1 --model gpt-5.2 -k 4
python scripts/leanbridge.py --start 1 --end 1 --model gpt-5.2 --max-iterations 4
leanbridge.py uses the local LeanExplore backend by default. Install and
prepare it with pip install lean-explore[local] and lean-explore data fetch.
The local LeanExplore service is initialized once per process and reused across
queries. Use --search-backend none for a no-search refinement loop. Outputs
are written under results/, which is ignored by git.
Evaluation Protocol
Each Lean file contains a dataset statement with one or more sorry placeholders. A problem is solved when the placeholders are replaced by a proof and the file is accepted by Lean under the pinned toolchain and dependencies in this artifact. The aggregate project can be checked with lake build.
See EVALUATION.md for the full proof validity and reporting protocol. See
DATASET_CARD.md for dataset documentation and responsible-use notes.
Benchmark Content
Problem Categories (100 problems total for 1-Category Theory)
Basic Category Properties (Problems 1-18): Fundamental results about categories, morphisms, monomorphisms, epimorphisms, initial/terminal objects
Adjunctions (Problems 19β29): Adjoint functors, universal properties, comma categories
Reflective and Coreflective Subcategories (Problems 30-33): Subcategory properties and classifications
Concrete Categories (Problems 34-41): Categories with faithful forgetful functors to Set
Limits and Colimits (Problems 42-73): The largest cluster covering limits, colimits, and related constructions
Cocompletions (Problems 74-78): Recent work on cocompletions requiring new definitions
Abelian Categories (Problems 79-90): Homological algebra concepts, kernels, cokernels, exact sequences
Monads (Problems 91-100): Monads, Kleisli and Eilenberg-Moore categories
Difficulty Distribution
- Easy: 20 problems (β€5.54/10 difficulty score)
- Medium: 40 problems (5.54-7.8/10 difficulty score)
- High: 40 problems (β₯7.8/10 difficulty score)
License
Dataset contents, including LeanCat problem statements, natural-language descriptions, and metadata, are released under CC BY 4.0. Evaluation scripts and software code are released under the repository MIT license.
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