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category_theory.monad.algebra

Eilenberg-Moore (co)algebras for a (co)monad

This file defines Eilenberg-Moore (co)algebras for a (co)monad, and provides the category instance for them.

Further it defines the adjoint pair of free and forgetful functors, respectively from and to the original category, as well as the adjoint pair of forgetful and cofree functors, respectively from and to the original category.

References

structure category_theory.monad.algebra {C : Type u₁} [category_theory.category C] (T : C C) [category_theory.monad T] :
Type (max u₁ v₁)

An Eilenberg-Moore algebra for a monad T. cf Definition 5.2.3 in [Riehl][riehl2017].

@[ext]

A morphism of Eilenberg–Moore algebras for the monad T.

theorem category_theory.monad.algebra.hom.ext {C : Type u₁} {_inst_1 : category_theory.category C} {T : C C} {_inst_2 : category_theory.monad T} {A B : category_theory.monad.algebra T} (x y : A.hom B) :
x.f = y.fx = y

theorem category_theory.monad.algebra.hom.ext_iff {C : Type u₁} {_inst_1 : category_theory.category C} {T : C C} {_inst_2 : category_theory.monad T} {A B : category_theory.monad.algebra T} (x y : A.hom B) :
x = y x.f = y.f

The identity homomorphism for an Eilenberg–Moore algebra.

Equations

Composition of Eilenberg–Moore algebra homomorphisms.

Equations
@[simp]
theorem category_theory.monad.algebra.hom.comp_f {C : Type u₁} [category_theory.category C] {T : C C} [category_theory.monad T] {P Q R : category_theory.monad.algebra T} (f : P.hom Q) (g : Q.hom R) :
(f.comp g).f = f.f g.f

@[instance]

The category of Eilenberg-Moore algebras for a monad. cf Definition 5.2.4 in [Riehl][riehl2017].

Equations

The forgetful functor from the Eilenberg-Moore category, forgetting the algebraic structure.

Equations
@[simp]

The free functor from the Eilenberg-Moore category, constructing an algebra for any object.

Equations
@[simp]
theorem category_theory.monad.free_obj_a {C : Type u₁} [category_theory.category C] (T : C C) [category_theory.monad T] (X : C) :

@[simp]
theorem category_theory.monad.free_map_f {C : Type u₁} [category_theory.category C] (T : C C) [category_theory.monad T] (X Y : C) (f : X Y) :

The adjunction between the free and forgetful constructions for Eilenberg-Moore algebras for a monad. cf Lemma 5.2.8 of [Riehl][riehl2017].

Equations

Given an algebra morphism whose carrier part is an isomorphism, we get an algebra isomorphism.

Equations
@[nolint]
structure category_theory.comonad.coalgebra {C : Type u₁} [category_theory.category C] (G : C C) [category_theory.comonad G] :
Type (max u₁ v₁)

An Eilenberg-Moore coalgebra for a comonad T.

theorem category_theory.comonad.coalgebra.hom.ext_iff {C : Type u₁} {_inst_1 : category_theory.category C} {G : C C} {_inst_2 : category_theory.comonad G} {A B : category_theory.comonad.coalgebra G} (x y : A.hom B) :
x = y x.f = y.f

@[nolint, ext]

A morphism of Eilenberg-Moore coalgebras for the comonad G.

theorem category_theory.comonad.coalgebra.hom.ext {C : Type u₁} {_inst_1 : category_theory.category C} {G : C C} {_inst_2 : category_theory.comonad G} {A B : category_theory.comonad.coalgebra G} (x y : A.hom B) :
x.f = y.fx = y

The identity homomorphism for an Eilenberg–Moore coalgebra.

Equations
@[simp]
theorem category_theory.comonad.coalgebra.hom.comp_f {C : Type u₁} [category_theory.category C] {G : C C} [category_theory.comonad G] {P Q R : category_theory.comonad.coalgebra G} (f : P.hom Q) (g : Q.hom R) :
(f.comp g).f = f.f g.f

Composition of Eilenberg–Moore coalgebra homomorphisms.

Equations

The forgetful functor from the Eilenberg-Moore category, forgetting the coalgebraic structure.

Equations
@[simp]
theorem category_theory.comonad.cofree_map_f {C : Type u₁} [category_theory.category C] (G : C C) [category_theory.comonad G] (X Y : C) (f : X Y) :

The cofree functor from the Eilenberg-Moore category, constructing a coalgebra for any object.

Equations

The adjunction between the cofree and forgetful constructions for Eilenberg-Moore coalgebras for a comonad.

Equations