Segal maps, named after Graeme Segal, appear in the definition of the Segal conditions on a simplicial object.
In the context of higher category theory they were used by Charles Rezk to define Segal categories and complete Segal spaces.
Let be a bisimplicial set. Assume for simplicity that is fibrant with respect to the Reedy model structure on the functor category . The -th Segal map of is the canonical morphism of simplicial sets
Here the right-hand side is the limit of the diagram
where there are copies of .
More explicitly, this morphism is induced by the morphisms (), which are induced by the morphisms in , the simplex category, which map and .
If is not Reedy fibrant, then one must replace the limit above with a homotopy limit.
Last revised on June 9, 2021 at 21:24:21. See the history of this page for a list of all contributions to it.