Comonade

In category theory, a comonad is a structure dual to that of the monad .

definition

A comonade is a triple consisting of ${\displaystyle (T,\varepsilon ,\psi )}$

• an endofector , ${\displaystyle T\colon C\to C}$
• a natural transformation and ${\displaystyle \varepsilon \colon T\to 1_{C}}$
• a natural transformation , ${\displaystyle \psi \colon T\to T^{2}}$

that meets the following conditions:

• ${\displaystyle T\psi \circ \psi =\psi T\circ \psi }$ and
• ${\displaystyle \varepsilon T\circ \psi =T\varepsilon \circ \psi =1_{T}}$.

Explicitly at the level of morphisms of this means that for each object from true ${\displaystyle C}$ ${\displaystyle X}$ ${\displaystyle C}$

• ${\displaystyle T(\psi _{X})\circ \psi _{X}=\psi _{T(X)}\circ \psi _{X}}$ and
• ${\displaystyle \varepsilon _{T(X)}\circ \psi _{X}=T(\varepsilon _{X})\circ \psi _{X}=1_{T(X)}}$.

Coalgebras

A koalgebra for a comonad on a category is a pair consisting of an object of and a morphism such that and . A homomorphism of coalgebras is a morphism in that satisfies. The koalgebras form a category . ${\displaystyle (T,\varepsilon ,\psi )}$ ${\displaystyle C}$ ${\displaystyle (X,\alpha )}$ ${\displaystyle X}$ ${\displaystyle C}$ ${\displaystyle \alpha \colon X\to TX}$ ${\displaystyle T\alpha \circ \alpha =\psi _{X}\circ \alpha }$ ${\displaystyle \varepsilon _{X}\circ \alpha =1_{X}}$ ${\displaystyle (X,\alpha )\to (Y,\beta )}$ ${\displaystyle f\colon X\to Y}$ ${\displaystyle C}$ ${\displaystyle Tf\circ \alpha =\beta \circ f}$ ${\displaystyle C^{T}}$

There is a canonical functor that is on objects . He is right adjoint to the forget functor . ${\displaystyle A_{T}\colon C\to C^{T}}$ ${\displaystyle X\mapsto (TX,\psi _{X})}$ ${\displaystyle U_{T}\colon C^{T}\to C}$

Comonad to an adjoint functor pair

Let there be categories and , functors, so that it is right adjoint to . One or Koeins of the adjunction are or . Then a comonade is on . ${\displaystyle C,D}$ ${\displaystyle F\colon C\to D}$ ${\displaystyle G\colon D\to C}$ ${\displaystyle F}$ ${\displaystyle G}$ ${\displaystyle \eta \colon 1\to FG}$ ${\displaystyle \varepsilon \colon GF\to 1}$ ${\displaystyle T=(GF,\varepsilon ,G\eta F)}$ ${\displaystyle C}$

One obtains an induced functor such that and holds. The functor is called comonadic if there is an equivalence of categories . The Monadizitätssatz of Jonathan Mock Beck 's criteria for when a functor is komonadisch. ${\displaystyle A\colon D\to C^{T}}$ ${\displaystyle U_{T}\circ A=G}$ ${\displaystyle A\circ F=A_{T}}$ ${\displaystyle G}$ ${\displaystyle A}$

If a comonad is on a category , then the comonad associated with the adjoint functor pair is again . ${\displaystyle T}$ ${\displaystyle C}$ ${\displaystyle (U_{T}\colon C^{T}\to C,A_{T}\colon C\to C^{T})}$ ${\displaystyle T}$

example

In the set category , the endofunctor is that of the formation of -indicated sequences, i.e. H. for every set is , and for sets and as well as mappings is given by . ${\displaystyle T}$ ${\displaystyle \mathbb {N} }$ ${\displaystyle X}$ ${\displaystyle T(X)=X^{\mathbb {N} }}$ ${\displaystyle A}$ ${\displaystyle B}$ ${\displaystyle f\colon A\to B}$ ${\displaystyle T(f)\colon A^{\mathbb {N} }\to B^{\mathbb {N} }}$ ${\displaystyle T(f)(s):=f\circ s}$

Let the natural transformations and be through the families of maps and , ${\displaystyle \varepsilon }$ ${\displaystyle \psi }$ ${\displaystyle \varepsilon _{X}}$ ${\displaystyle \psi _{X}}$

• ${\displaystyle \varepsilon _{X}\colon X^{\mathbb {N} }\to X,\varepsilon _{X}(s):=s(0)}$
• ${\displaystyle \psi _{X}\colon X^{\mathbb {N} }\to (X^{\mathbb {N} })^{\mathbb {N} },\psi _{X}(s)(n)(m):=s(n+m)}$

given for arbitrary quantities . ${\displaystyle X}$

The triple is now a comonade in set . ${\displaystyle (T,\varepsilon ,\psi )}$

The Koalgebras for are figures that satisfy and . With , is , and one can identify the coalgebras with pairs with any map . ${\displaystyle (T,\varepsilon ,\psi )}$ ${\displaystyle \alpha \colon X\to X^{\mathbb {N} }}$ ${\displaystyle \alpha (x)(0)=x}$ ${\displaystyle \alpha (x)(n+m)=\alpha (\alpha (x)(n))(m)}$ ${\displaystyle \alpha _{1}\colon X\to X}$ ${\displaystyle x\mapsto \alpha (x)(1)}$ ${\displaystyle \alpha (x)(n)=\alpha _{1}^{n}(x)}$ ${\displaystyle (X,\alpha _{1})}$ ${\displaystyle \alpha _{1}\colon X\to X}$

Is any amount then correspond Komonadenstrukturen on bijektiv the Monoidstrukturen on . The multiplication on is . For a monoid , the structural mapping of a koalgebra can be identified with other mappings under the power law : ${\displaystyle M}$ ${\displaystyle T(X)=X^{M}}$ ${\displaystyle M}$ ${\displaystyle M}$ ${\displaystyle \psi _{M}(1_{M})\in (M^{M})^{M}}$ ${\displaystyle M}$ ${\displaystyle X\to X^{M}}$ ${\displaystyle (A^{B})^{C}=A^{B\times C}=(A^{C})^{B}}$

• a figure which is an algebra of the Monad is ${\displaystyle X\times M\to X}$ ${\displaystyle T^{*}(X)=X\times M}$
• a monoid homomorphism , d. H. an operation from on . ${\displaystyle M\to X^{X}}$ ${\displaystyle M}$ ${\displaystyle X}$

literature

• Saunders Mac Lane, Categories for the Working Mathematician . Springer-Verlag, Berlin 1971. ISBN 3-540-90035-7