nLab
category with arities

Categories with arities

Context

Algebra

2-Category theory

Categories 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 𝒜\mathcal{A} \subseteq \mathcal{E} which is dense, in the sense that the restricted Yoneda embedding [𝒜 op,Set]\mathcal{E} \to [\mathcal{A}^{op},Set] is fully faithful.

A functor with arities or arity-respecting functor F:(,𝒜)(,)F : (\mathcal{E},\mathcal{A}) \to (\mathcal{F},\mathcal{B}) is a functor F:F:\mathcal{E}\to\mathcal{F} such that the composite F[ op,Set]\mathcal{E} \xrightarrow{F} \mathcal{F} \to [\mathcal{B}^{op},Set] preserves the density colimits for 𝒜\mathcal{A} in \mathcal{E}, i.e. such that for any XX\in\mathcal{E} and BB\in\mathcal{B} the canonical map colim A𝒜AX(B,FA)(B,FX)\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 𝒜\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.

References

Created on December 30, 2018 at 01:24:56. See the history of this page for a list of all contributions to it.