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:Lurie 2009, Definition 4.2.1.1Cisinksi 2019, 4.2.1.
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 ).Lurie 2009, after Corollary 4.2.1.4.Cisinski 2019, 4.2.1. 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 .Cisinski 2019, Proposition 4.2.2. It is a weak categorical equivalence, hence a weak equivalence of the Joyal model structure.Lurie 2009, Proposition 4.2.1.2.Cisinksi 2019, Proposition 4.2.3. For a simplicial set , the functors preserve weak categorical equivalences.Lurie 2009, Corollary 4.2.1.3.Cisinski 2019, Proposition 4.2.4.
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