# Kerodon

$\Newextarrow{\xRightarrow}{5,5}{0x21D2}$ $\newcommand\empty{}$

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 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 $A_0 \coprod A_1$).