Segal map




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 XX be a bisimplicial set. Assume for simplicity that XX is fibrant with respect to the Reedy model structure on the functor category [Δ op,SSet][\Delta^{op}, SSet]. The nn-th Segal map of XX is the canonical morphism of simplicial sets

X n(X 1) × X 0(n)=X 1× X 0× X 0X 1. X_n \to (X_1)^{\times_{X_0} (n)} = X_1 \times_{X_0} \cdots \times_{X_0} X_1.

Here the right-hand side is the limit of the diagram

X 1d 1X 0d 0X 1d 1d 1X 0d 0X 1 X_1 \stackrel{d_1}{\to} X_0 \stackrel{d_0}{\leftarrow} X_1 \stackrel{d_1}{\to} \cdots \stackrel{d_1}{\to} X_0 \stackrel{d_0}{\leftarrow} X_1

where there are nn copies of X 1X_1.

More explicitly, this morphism is induced by the morphisms a i:X nX 1a_i: X_n \to X_1 (0in10 \le i \le n-1), which are induced by the morphisms α i:[1][n]\alpha^i: [1] \to [n] in Δ\Delta, the simplex category, which map 0i0 \mapsto i and 1i+11 \mapsto i+1.

If XX is not Reedy fibrant, then one must replace the limit above with a homotopy limit.


Last revised on June 9, 2021 at 17:24:21. See the history of this page for a list of all contributions to it.