Skip to content

Commit

Permalink
bump to nightly-2023-07-06-07
Browse files Browse the repository at this point in the history
  • Loading branch information
leanprover-community-bot committed Jul 6, 2023
1 parent c41d1b1 commit eafde23
Show file tree
Hide file tree
Showing 5 changed files with 177 additions and 43 deletions.
12 changes: 12 additions & 0 deletions Mathbin/CategoryTheory/Monoidal/Internal/Module.lean
Expand Up @@ -74,11 +74,14 @@ instance (A : Mon_ (ModuleCat.{u} R)) : Algebra R A.pt :=
exact h₁.trans h₂.symm
smul_def' := fun r a => (LinearMap.congr_fun A.one_mul (r ⊗ₜ a)).symm }

#print ModuleCat.MonModuleEquivalenceAlgebra.algebraMap /-
@[simp]
theorem algebraMap (A : Mon_ (ModuleCat.{u} R)) (r : R) : algebraMap R A.pt r = A.one r :=
rfl
#align Module.Mon_Module_equivalence_Algebra.algebra_map ModuleCat.MonModuleEquivalenceAlgebra.algebraMap
-/

#print ModuleCat.MonModuleEquivalenceAlgebra.functor /-
/-- Converting a monoid object in `Module R` to a bundled algebra.
-/
@[simps]
Expand All @@ -92,7 +95,9 @@ def functor : Mon_ (ModuleCat.{u} R) ⥤ AlgebraCat R
map_mul' := fun x y => LinearMap.congr_fun f.MulHom (x ⊗ₜ y)
commutes' := fun r => LinearMap.congr_fun f.OneHom r }
#align Module.Mon_Module_equivalence_Algebra.functor ModuleCat.MonModuleEquivalenceAlgebra.functor
-/

#print ModuleCat.MonModuleEquivalenceAlgebra.inverseObj /-
/-- Converting a bundled algebra to a monoid object in `Module R`.
-/
@[simps]
Expand Down Expand Up @@ -122,7 +127,9 @@ def inverseObj (A : AlgebraCat.{u} R) : Mon_ (ModuleCat.{u} R)
monoidal_category.associator_hom_apply]
simp only [LinearMap.mul'_apply, mul_assoc]
#align Module.Mon_Module_equivalence_Algebra.inverse_obj ModuleCat.MonModuleEquivalenceAlgebra.inverseObj
-/

#print ModuleCat.MonModuleEquivalenceAlgebra.inverse /-
/-- Converting a bundled algebra to a monoid object in `Module R`.
-/
@[simps]
Expand All @@ -134,11 +141,13 @@ def inverse : AlgebraCat.{u} R ⥤ Mon_ (ModuleCat.{u} R)
one_hom' := LinearMap.ext f.commutes
mul_hom' := TensorProduct.ext <| LinearMap.ext₂ <| map_mul f }
#align Module.Mon_Module_equivalence_Algebra.inverse ModuleCat.MonModuleEquivalenceAlgebra.inverse
-/

end MonModuleEquivalenceAlgebra

open MonModuleEquivalenceAlgebra

#print ModuleCat.monModuleEquivalenceAlgebra /-
/-- The category of internal monoid objects in `Module R`
is equivalent to the category of "native" bundled `R`-algebras.
-/
Expand Down Expand Up @@ -181,7 +190,9 @@ def monModuleEquivalenceAlgebra : Mon_ (ModuleCat.{u} R) ≌ AlgebraCat R
commutes' := fun r => rfl } })
(by intros; rfl)
#align Module.Mon_Module_equivalence_Algebra ModuleCat.monModuleEquivalenceAlgebra
-/

#print ModuleCat.monModuleEquivalenceAlgebraForget /-
/-- The equivalence `Mon_ (Module R) ≌ Algebra R`
is naturally compatible with the forgetful functors to `Module R`.
-/
Expand All @@ -200,6 +211,7 @@ def monModuleEquivalenceAlgebraForget :
map_smul' := fun c x => rfl } })
(by tidy)
#align Module.Mon_Module_equivalence_Algebra_forget ModuleCat.monModuleEquivalenceAlgebraForget
-/

end ModuleCat

0 comments on commit eafde23

Please sign in to comment.