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.
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:
Let be a small category. The functor has a right adjoint (alternatively denoted ) and the functor also has a right adjoint (alternatively denoted ). A special case is the terminal small category, since is the category of pointed small categories.