# nLab algebra for a profunctor

category theory

## Applications

#### Algebra

higher algebra

universal algebra

# Contents

## Idea

The notion of algebra over an endo-profunctor ($C$-$C$-bimodule) is a joint generalization of the notions algebra for an endofunctor and coalgebra for an endofunctor.

## Definition

For a category $C$ and a $C$-$C$ bimodule $H:{C}^{\mathrm{op}}×C\to \mathrm{Set}$, an algebra for $H$ is given by a functor $X:D\to C$ and an extranatural transformation $*\to H\left(X,X\right)$, where $*:1\to \mathrm{Set}$ is constant at the point. $X$ is called the carrier of the algebra. A morphism $\left(X,\alpha \right)\to \left(Y,\beta \right)$ of $H$-algebras is given by a natural transformation $\varphi :X⇒Y$ such that $H\left(X,\varphi \right)\circ \alpha =H\left(\varphi ,Y\right)\circ \beta$.

If $D$ is the one-object category, an algebra $\left(X,\alpha \right)$ is given by an object $X$ in $C$ and an element $\alpha \in H\left(X,X\right)$. A morphism between two algebras $\left(X,\alpha \right)$ and $\left(Y,\beta \right)$ is then a morphism $m:X\to Y$ in $C$ such that $H\left(X,m\right)\left(\alpha \right)=H\left(m,Y\right)\left(\beta \right)$, these both being elements of $H\left(X,Y\right)$.

There is an an obvious forgetful functor into $C$ from the category of algebras for $H$, which sends each algebra to its carrier and each algebra morphism to its underlying morphism in $C$; among other properties, this functor is always faithful and conservative.

In fact, the category $\mathrm{Alg}\left(H\right)$, together with its forgetful functor $U:\mathrm{Alg}\left(H\right)\to C$, has the universal property of an Eilenberg-Moore object, namely that of being the universal $H$-algebra. Specifically, it is a terminal object in the category whose objects are functors $G:D\to C$ equipped with an extranatural transformation $*\to H\left(G-,G?\right)$. For such an extranatural transformation consists of, for every $d\in D$, an element ${\xi }_{d}\in H\left(Gd,Gd\right)$, such that for every morphism $v:d\to e$ in $D$, we have $H\left({\mathrm{id}}_{d},v\right)\left({\xi }_{d}\right)=H\left(v,{\mathrm{id}}_{e}\right)\left({\xi }_{e}\right)$. This is precisely the data of a functor $D\to \mathrm{Alg}\left(H\right)$ lying over $C$.

## Coalgebras in Prof

One version of Yoneda's lemma says that for a profunctor $K:C⇸C$ there is a bijection between extranatural transformations $*\to K$ and natural transformations ${\mathrm{hom}}_{C}\to K$. So there are bijections

$\begin{array}{c}*\phantom{\rule{mediummathspace}{0ex}}\stackrel{¨}{\to }\phantom{\rule{mediummathspace}{0ex}}H\left(X,X\right)\\ {\mathrm{hom}}_{D}⇒H\left(X,X\right)\\ C\left(1,X\right)⇒H\circ C\left(1,X\right)\end{array}$\array{ \ast \: {\ddot\to} \: H(X,X) \\ \hom_D \Rightarrow H(X,X) \\ C(1,X) \Rightarrow H \circ C(1,X) }

where the last holds by the usual properties of representable profunctors (see e.g. proarrow equipment). This exhibits each $H$-algebra on $X$ in the above sense as a $H$-coalgebra in $\mathrm{Prof}$ with carrier $C\left(1,X\right)$.

## Examples

• Algebras and coalgebras for endofunctors are special cases of algebras for bimodules; specifically, an algebra for an endofunctor $F$ is an algebra for the bimodule $\mathrm{Hom}\left(F\left(-\right),?\right)$, while a coalgebra for $F$ is an algebra for the bimodule $\mathrm{Hom}\left(-,F\left(?\right)\right)$.

• A natural transformation between functors $F$ and $G$ from $C$ to $D$ is a section of the forgetful functor into $C$ from the category of algebras for the $C-C$ bimodule ${\mathrm{Hom}}_{D}\left(F\left(-\right),G\left(?\right)\right)$. That is, it gives every object of $C$ the structure of an algebra for ${\mathrm{Hom}}_{D}\left(F\left(-\right),G\left(?\right)\right)$ in such a way as that every morphism of $C$ has the property of being an algebra morphism between the algebras on its domain and codomain.

• A natural numbers object (in the weak, unparametrized sense) in a category $C$ with terminal object $1$ is an initial object in the category of algebras for the bimodule ${\mathrm{Hom}}_{C}\left(1,?\right)×{\mathrm{Hom}}_{C}\left(-,?\right)$. If $C$ has binary coproducts, then this is of course the same as an initial algebra for the endofunctor $1+\left(-\right)$.

Revised on April 26, 2011 05:12:16 by Urs Schreiber (82.113.99.7)