Kerodon

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Construction 5.3.2.1. Let $\operatorname{\mathcal{C}}$ be a category and let $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{Set_{\Delta }}$ be a functor. For every integer $n \geq 0$, we let $\underset { \longrightarrow }{\mathrm{holim}}(\mathscr {F})_{n}$ denote the set of all ordered pairs $( \sigma , \tau )$, where $\sigma : [n] \rightarrow \operatorname{\mathcal{C}}$ is an $n$-simplex of the nerve $\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$ and $\tau$ is an $n$-simplex of the simplicial set $\mathscr {F}( \sigma (0) )$.

If $(\sigma , \tau )$ is an element of $\underset { \longrightarrow }{\mathrm{holim}}( \mathscr {F})_{n}$ and $\alpha : [m] \rightarrow [n]$ is a nondecreasing function of linearly ordered sets, we set $\alpha ^{\ast }( \sigma , \tau ) = (\sigma \circ \alpha , \tau ') \in \underset { \longrightarrow }{\mathrm{holim}}( \mathscr {F} )_{m}$, where $\tau '$ is given by the composite map

$\Delta ^{m} \xrightarrow { \alpha } \Delta ^{n} \xrightarrow { \tau } \mathscr {F}( \sigma (0) ) \rightarrow \mathscr {F}( (\sigma \circ \alpha )(0) ).$

By means of this construction, the assignment $[n] \mapsto \underset { \longrightarrow }{\mathrm{holim}}( \mathscr {F} )_{n}$ determines a simplicial set $\underset { \longrightarrow }{\mathrm{holim}}( \mathscr {F} ) = \underset { \longrightarrow }{\mathrm{holim}}( \mathscr {F} )_{\bullet }$ which we will refer to as the homotopy colimit of the diagram $\mathscr {F}$. Note that the construction $(\sigma , \tau ) \mapsto \sigma$ determines a morphism of simplicial sets $U: \underset { \longrightarrow }{\mathrm{holim}}(\mathscr {F}) \rightarrow \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$, which we will refer to as the projection map.