Join (category theory)
In category theory in mathematics, the join of categories is an operation making the category of small categories into a monoidal category. In particular, it takes two small categories to construct another small category. Under the nerve construction, it corresponds to the join of simplicial sets.
Definition
For small categories and, their join is the small category with:The join defines a functor, which together with the empty category as unit element makes the category of small categories into a monoidal category.
For a small category, one further defines its left cone and right cone as:
Right adjoints
Let be a small category. The functor has a right adjoint and the functor also has a right adjoint . A special case is the terminal small category, since is the category of pointed small categories.Properties
- The join is associative. For small categories, and, one has:
- :
- The join reverses under the dual category. For small categories and, one has:
- :
- Under the nerve, the join of categories becomes the join of simplicial sets. For small categories and, one has:
- :