symmetric monoidal (∞,1)-category of spectra
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.
For a category $C$ and a $C$-$C$ bimodule $H : C^{op} \times C \to Set$, an algebra for $H$ is given by a functor $X \colon D \to C$ and an extranatural transformation $\ast \to H(X,X)$, where $\ast \colon \mathbf{1} \to Set$ is constant at the point. $X$ is called the carrier of the algebra. A morphism $(X, \alpha) \to (Y, \beta)$ of $H$-algebras is given by a natural transformation $\phi \colon X \Rightarrow Y$ such that $H(X,\phi) \circ \alpha = H(\phi,Y) \circ \beta$.
If $D$ is the one-object category, an algebra $(X,\alpha)$ is given by an object $X$ in $C$ and an element $\alpha \in H(X, X)$. A morphism between two algebras $(X, \alpha)$ and $(Y, \beta)$ is then a morphism $m : X \to Y$ in $C$ such that $H(X, m) (\alpha) = H(m, Y) (\beta)$, these both being elements of $H(X, Y)$.
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 $Alg(H)$, together with its forgetful functor $U\colon Alg(H)\to C$, 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 $G\colon D\to C$ equipped with an extranatural transformation $\ast \to H(G-,G?)$. For such an extranatural transformation consists of, for every $d\in D$, an element $\xi_d \in H(G d,G d)$, such that for every morphism $v\colon d\to e$ in $D$, we have $H(id_d,v)(\xi_d) = H(v,id_e)(\xi_e)$. This is precisely the data of a functor $D\to Alg(H)$ lying over $C$.
One version of Yoneda's lemma says that for a profunctor $H \colon C ⇸ C$ there is a bijection between extranatural transformations $\ast \to H$ and natural transformations $\hom_C \to H$. So there are bijections
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 $Prof$ with carrier $C(1,X)$.
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 $Hom(F(-), ?)$, while a coalgebra for $F$ is an algebra for the bimodule $Hom(-, F(?))$.
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 $Hom_D(F(-), G(?))$. That is, it gives every object of $C$ the structure of an algebra for $Hom_D(F(-), G(?))$ 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 $Hom_C(1, ?) \times Hom_C(-, ?)$. If $C$ has binary coproducts, then this is of course the same as an initial algebra for the endofunctor $1+(-)$.
Last revised on July 4, 2021 at 16:49:56. See the history of this page for a list of all contributions to it.