Example 5.3.2.16 (0369)

Example 5.3.2.16. Let $\operatorname{\mathcal{C}}$ be the partially ordered set depicted in the diagram

\[ \bullet \leftarrow \bullet \rightarrow \bullet \]

and suppose we are given a functor $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{Set_{\Delta }}$, which we identify with a diagram of simplicial sets

\[ A_0 \xleftarrow {f_0} A \xrightarrow {f_1} A_1. \]

The homotopy colimit $ \underset { \longrightarrow }{\mathrm{holim}}(\mathscr {F} )$ can be identified with the iterated homotopy pushout

\[ (A {\coprod }_{A}^{\mathrm{h}} A_0) {\coprod }_{A}^{\mathrm{h} } A_1. \]

In particular, the comparison map $q_0: A {\coprod }_{A}^{\mathrm{h}} A_0 \twoheadrightarrow A {\coprod }_{A} A_0 \simeq A_0$ induces an epimorphism of simplicial sets

\[ q: \underset { \longrightarrow }{\mathrm{holim}}( \mathscr {F} ) \rightarrow A_0 {\coprod }_{A}^{\mathrm{h}} A_1. \]

Note that $q_0$ is always a weak homotopy equivalence of simplicial sets (Corollary 3.4.2.13), so that $q$ is also a weak homotopy equivalence (Corollary 3.4.2.14). Beware that $q$ is never an isomorphism, except in the trivial case where the simplicial set $A$ is empty (in which case the homotopy colimit $ \underset { \longrightarrow }{\mathrm{holim}}(\mathscr {F})$ and the homotopy pushout $A_0 {\coprod }_{A}^{\mathrm{h}} A_1$ can both be identified with the disjoint union $A_0 \coprod A_1$).