In higher category theory in mathematics, the join of simplicial sets is an operation making the category of simplicial sets into a monoidal category. In particular, it takes two simplicial sets to construct another simplicial set. It is closely related to the diamond operation and used in the construction of the twisted diagonal. Under the nerve construction, it corresponds to the join of categories and under the geometric realization, it corresponds to the join of topological spaces.
Definition.
For natural numbers formula_1, one has the identity:
 formula_2
which can be extended by colimits to a functor a functor formula_3, which together with the empty simplicial set as unit element makes the category of simplicial sets formula_4 into a monoidal category. For simplicial set formula_5 and formula_6, their "join" formula_7 is the simplicial set:
 formula_8
A formula_9-simplex formula_10 therefore either factors over formula_5 or formula_6 or splits into a formula_13-simplex formula_14 and a formula_15-simplex formula_16 with formula_17 and formula_18.
One has canonical morphisms formula_19, which combine into a canonical morphism formula_20 through the universal property of the coproduct. One also has a canonical morphism formula_21 of terminal maps, for which the fiber of formula_22 is formula_5 and the fiber of formula_24 is formula_6.
For a simplicial set formula_5, one further defines its "left cone" and "right cone" as:
 formula_27
 formula_28
Right adjoint.
Let formula_6 be a simplicial set. The functor formula_30 has a right adjoint formula_31 (alternatively denoted formula_32) and the functor formula_33 also has a right adjoint formula_34 (alternatively denoted formula_35). A special case is formula_36 the terminal simplicial set, since formula_37 is the category of pointed simplicial sets.
Let formula_38 be a category and formula_39 be an object. Let formula_40 be the terminal category (with the notation taken from the terminal object of the simplex category), then there is an associated functor formula_41, which with the nerve induces a morphism formula_42. For every simplicial set formula_43, one has by additionally using the adjunction between the join of categories and slice categories:
 formula_44
Hence according to the Yoneda lemma, one has (with the alternative notation, which here better underlines the result):
 formula_45
Examples.
One has:
 formula_46
 formula_47
 formula_48
 formula_72
 For every simplicial set formula_43, one has:
 formula_74
 so the claim follows from the Yoneda lemma.