nLab Segal map

Idea

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.

Definition

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

$$X_n \to (X_1)^{\times_{X_0} (n)} = X_1 \times_{X_0} \cdots \times_{X_0} X_1.$$

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

$$X_1 \stackrel{d_1}{\to} X_0 \stackrel{d_0}{\leftarrow} X_1 \stackrel{d_1}{\to} \cdots \stackrel{d_1}{\to} X_0 \stackrel{d_0}{\leftarrow} X_1$$

where there are $n$ copies of $X_1$.

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

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

References

• Julie Bergner, Three models for the homotopy theory of homotopy theories, Topology 46 (2007), 397-436, arXiv:math/0504334.