nLab category with arities

Idea

A category with arities is a category equipped with a “small presentation” of its objects.

Definition

A category with arities is a locally small category $\mathcal{E}$ equipped with a small full subcategory $\mathcal{A} \subseteq \mathcal{E}$ which is dense, in the sense that the restricted Yoneda embedding $\mathcal{E} \to [\mathcal{A}^{op},Set]$ is fully faithful.

A functor with arities or arity-respecting functor $F : (\mathcal{E},\mathcal{A}) \to (\mathcal{F},\mathcal{B})$ is a functor $F:\mathcal{E}\to\mathcal{F}$ such that the composite $\mathcal{E} \xrightarrow{F} \mathcal{F} \to [\mathcal{B}^{op},Set]$ preserves the density colimits for $\mathcal{A}$ in $\mathcal{E}$, i.e. such that for any $X\in\mathcal{E}$ and $B\in\mathcal{B}$ the canonical map $\colim^{A\in \mathcal{A} \atop A \to X} \mathcal{F}(B,F A) \to \mathcal{E}(B,F X)$ is an isomorphism of sets.

A natural transformation with arities is just an ordinary natural transformation between functors with arities.

This defines the 2-category of categories with arities. Note that since a natural transformation with arities is determined by its action on the generating objects $\mathcal{A}$, there is only a small set of natural transformations between any two functors with arities. Thus, even though the 2-category of categories with arities is “very large”, its hom-categories are locally small.

Related pages

• A monad in the 2-category of categories with arities is a monad with arities. In BMW it is claimed that the nerve theorem for monads with arities constructs an Eilenberg-Moore object in this 2-category.

References

• Clemens Berger, Paul-André Melliès, Mark Weber, Monads with Arities and their Associated Theories (2011) (arXiv:1101.3064)