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 formula_1 and formula_2, their "join" formula_3 is the small category with:
 formula_4
 formula_5
The join defines a functor formula_6, which together with the empty category as unit element makes the category of small categories formula_7 into a monoidal category.
For a small category formula_1, one further defines its "left cone" and "right cone" as:
 formula_9
 formula_10
Right adjoints.
Let formula_2 be a small category. The functor formula_12 has a right adjoint formula_13 (alternatively denoted formula_14) and the functor formula_15 also has a right adjoint formula_16 (alternatively denoted formula_17). A special case is formula_18 the terminal small category, since formula_19 is the category of pointed small categories.