In higher category theory in mathematics, the diamond operation of simplicial sets is an operation taking two simplicial sets to construct another simplicial set. It is closely related to the join of simplicial sets and used in an alternative construction of the twisted diagonal.
Definition.
For simplicial set formula_1 and formula_2, their "diamond" formula_3 is the pushout of the diagram:
 formula_4
One has a canonical map formula_5 for which the fiber of formula_6 is formula_1 and the fiber of formula_8 is formula_2.
Right adjoints.
Let formula_2 be a simplicial set. The functor formula_11 has a right adjoint formula_12 (alternatively denoted formula_13) and the functor formula_14 has a right adjoint formula_15 (alternatively denoted formula_16). A special case is formula_17 the terminal simplicial set, since formula_18 is the category of pointed simplicial sets.