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 #
- [Riehl, Category theory in context, Section 5.2.4][riehl2017]
- A : C
The underlying object associated to an algebra.
- a : T.obj s.A ⟶ s.A
The structure morphism associated to an algebra.
- unit : CategoryTheory.CategoryStruct.comp ((CategoryTheory.Monad.η T).app s.A) s.a = CategoryTheory.CategoryStruct.id s.A
The unit axiom associated to an algebra.
- assoc : CategoryTheory.CategoryStruct.comp ((CategoryTheory.Monad.μ T).app s.A) s.a = CategoryTheory.CategoryStruct.comp (T.map s.a) s.a
The associativity axiom associated to an algebra.
An Eilenberg-Moore algebra for a monad T
.
cf Definition 5.2.3 in [Riehl][riehl2017].
Instances For
- f : A.A ⟶ B.A
The underlying morphism associated to a morphism of algebras.
- h : CategoryTheory.CategoryStruct.comp (T.map s.f) B.a = CategoryTheory.CategoryStruct.comp A.a s.f
Compatibility with the structure morphism, for a morphism of algebras.
A morphism of Eilenberg–Moore algebras for the monad T
.
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The identity homomorphism for an Eilenberg–Moore algebra.
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Composition of Eilenberg–Moore algebra homomorphisms.
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The category of Eilenberg-Moore algebras for a monad. cf Definition 5.2.4 in [Riehl][riehl2017].
To construct an isomorphism of algebras, it suffices to give an isomorphism of the carriers which commutes with the structure morphisms.
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The forgetful functor from the Eilenberg-Moore category, forgetting the algebraic structure.
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The free functor from the Eilenberg-Moore category, constructing an algebra for any object.
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The adjunction between the free and forgetful constructions for Eilenberg-Moore algebras for a monad. cf Lemma 5.2.8 of [Riehl][riehl2017].
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Given an algebra morphism whose carrier part is an isomorphism, we get an algebra isomorphism.
Given an algebra morphism whose carrier part is an epimorphism, we get an algebra epimorphism.
Given an algebra morphism whose carrier part is a monomorphism, we get an algebra monomorphism.
Given a monad morphism from T₂
to T₁
, we get a functor from the algebras of T₁
to algebras of
T₂
.
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The identity monad morphism induces the identity functor from the category of algebras to itself.
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A composition of monad morphisms gives the composition of corresponding functors.
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If f
and g
are two equal morphisms of monads, then the functors of algebras induced by them
are isomorphic.
We define it like this as opposed to using eqToIso
so that the components are nicer to prove
lemmas about.
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Isomorphic monads give equivalent categories of algebras. Furthermore, they are equivalent as
categories over C
, that is, we have algebraEquivOfIsoMonads h ⋙ forget = forget
.
Instances For
- A : C
The underlying object associated to a coalgebra.
- a : s.A ⟶ G.obj s.A
The structure morphism associated to a coalgebra.
- counit : CategoryTheory.CategoryStruct.comp s.a ((CategoryTheory.Comonad.ε G).app s.A) = CategoryTheory.CategoryStruct.id s.A
The counit axiom associated to a coalgebra.
- coassoc : CategoryTheory.CategoryStruct.comp s.a ((CategoryTheory.Comonad.δ G).app s.A) = CategoryTheory.CategoryStruct.comp s.a (G.map s.a)
The coassociativity axiom associated to a coalgebra.
An Eilenberg-Moore coalgebra for a comonad T
.
Instances For
- f : A.A ⟶ B.A
The underlying morphism associated to a morphism of coalgebras.
- h : CategoryTheory.CategoryStruct.comp A.a (G.map s.f) = CategoryTheory.CategoryStruct.comp s.f B.a
Compatibility with the structure morphism, for a morphism of coalgebras.
A morphism of Eilenberg-Moore coalgebras for the comonad G
.
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The identity homomorphism for an Eilenberg–Moore coalgebra.
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Composition of Eilenberg–Moore coalgebra homomorphisms.
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The category of Eilenberg-Moore coalgebras for a comonad.
The category of Eilenberg-Moore coalgebras for a comonad.
To construct an isomorphism of coalgebras, it suffices to give an isomorphism of the carriers which commutes with the structure morphisms.
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The forgetful functor from the Eilenberg-Moore category, forgetting the coalgebraic structure.
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The cofree functor from the Eilenberg-Moore category, constructing a coalgebra for any object.
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The adjunction between the cofree and forgetful constructions for Eilenberg-Moore coalgebras for a comonad.
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Given a coalgebra morphism whose carrier part is an isomorphism, we get a coalgebra isomorphism.
Given a coalgebra morphism whose carrier part is an epimorphism, we get an algebra epimorphism.
Given a coalgebra morphism whose carrier part is a monomorphism, we get an algebra monomorphism.