Diamond operation

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 and , their diamond is the pushout of the diagram:

One has a canonical map for which the fiber of is and the fiber of is .

Right adjoints

Let be a simplicial set. The functor has a right adjoint (alternatively denoted ) and the functor has a right adjoint (alternatively denoted ). A special case is the terminal simplicial set, since is the category of pointed simplicial sets.

Properties

• For simplicial sets and , there is a unique morphism from the join of simplicial sets compatible with the maps and . It is a weak categorical equivalence, hence a weak equivalence of the Joyal model structure.
• For a simplicial set , the functors preserve weak categorical equivalences.

Literature

References