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 and the functor has a right adjoint . 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