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5.3.2 Homotopy Colimits of Simplicial Sets

Let $f_0: A \rightarrow A_0$ and $f_1: A \rightarrow A_1$ be morphisms of simplicial sets. Recall that the homotopy pushout of $A_0$ with $A_1$ along $A$ is defined to be the simplicial set

\[ A_{0} {\coprod }^{\mathrm{h}}_{A} A_1 = A_0 \coprod _{ (\{ 0 \} \times A)} ( \Delta ^1 \times A) \coprod _{( \{ 1\} \times A) } A_1 \]

(see Construction 3.4.2.2). This construction has two essential properties:

$(1)$

The formation of homotopy pushouts is compatible with weak homotopy equivalence. That is, if we are given a commutative diagram of simplicial sets

\[ \xymatrix@R =50pt@C=50pt{ A_0 \ar [d] & A \ar [l]_{f_0} \ar [r]^-{f_1} \ar [d] & A_1 \ar [d] \\ B_0 & B \ar [l]_{g_0} \ar [r]^-{g_1} & B_1, } \]

in which the vertical maps are weak homotopy equivalences, then the induced map $A_0 \coprod _{A}^{\mathrm{h}} A_1 \rightarrow B_{0} \coprod _{B}^{\mathrm{h} } B_1$ is also a weak homotopy equivalence (Corollary 3.4.2.15).

$(2)$

The homotopy pushout is equipped with a comparison map $A_0 \coprod _{A}^{ \mathrm{h} } A_1 \twoheadrightarrow A_0 \coprod _{A} A_1$, which is a weak homotopy equivalence if either $f_0: A_0 \rightarrow A$ or $f_1: A_1 \rightarrow A$ is a monomorphism (Corollary 3.4.2.13).

Our goal in this section is to introduce a variant of the homotopy pushout construction which can be applied to more general diagrams of simplicial sets. To every category $\operatorname{\mathcal{C}}$ and every functor $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{Set_{\Delta }}$, we introduce a simplicial set $ \underset { \longrightarrow }{\mathrm{holim}}(\mathscr {F} )$ which we refer to as the homotopy colimit of $\mathscr {F}$ (Construction 5.3.2.1). The homotopy colimit satisfies an analogue of property $(1)$: it is compatible both with weak homotopy equivalence (Proposition 5.3.2.18) and with categorical equivalence (Variant 5.3.2.19). Moreover, there is a natural epimorphism from the homotopy colimit $ \underset { \longrightarrow }{\mathrm{holim}}(\mathscr {F})$ to the usual colimit $\varinjlim (\mathscr {F})$ (Remark 5.3.2.9). We will see later that this map is often a weak homotopy equivalence (Corollary 7.5.6.14).

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.

Example 5.3.2.2 (Discrete Diagrams). Let $\operatorname{\mathcal{C}}$ be a category having only identity morphisms, and let $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{Set_{\Delta }}$ be a diagram of simplicial sets. Then the homotopy colimit $ \underset { \longrightarrow }{\mathrm{holim}}( \mathscr {F} )$ can be identified with the disjoint union $\coprod _{C \in \operatorname{\mathcal{C}}} \mathscr {F}(C)$.

Remark 5.3.2.3. Let $T: \operatorname{\mathcal{C}}' \rightarrow \operatorname{\mathcal{C}}$ be a functor between categories, let $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{Set_{\Delta }}$ be a diagram of simplicial sets indexed by $\operatorname{\mathcal{C}}$, and let $\mathscr {F}'$ denote the composition $\mathscr {F} \circ T$. Then we have a pullback diagram of simplicial sets

\[ \xymatrix@R =50pt@C=50pt{ \underset { \longrightarrow }{\mathrm{holim}}( \mathscr {F}' ) \ar [r] \ar [d]^{U'} & \underset { \longrightarrow }{\mathrm{holim}}( \mathscr {F} ) \ar [d]^{U} \\ \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}') \ar [r]^{ \operatorname{N}_{\bullet }(T) } & \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}), } \]

where $U$ and $U'$ denote the projection maps of Construction 5.3.2.1. In particular, for every object $C \in \operatorname{\mathcal{C}}$, we have a canonical isomorphism of simplicial sets

\[ \mathscr {F}(C) \simeq \{ C\} \times _{ \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) } \underset { \longrightarrow }{\mathrm{holim}}( \mathscr {F} ). \]

Example 5.3.2.4 (Constant Diagrams). Let $\operatorname{\mathcal{C}}$ be a category, let $X$ be a simplicial set, and let $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{Set_{\Delta }}$ be the constant diagram taking the value $X$. Combining Remark 5.3.2.3 with Example 5.3.2.2, we obtain a canonical isomorphism of simplicial sets $ \underset { \longrightarrow }{\mathrm{holim}}( \mathscr {F} ) \simeq \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) \times X$. In particular, if $X = \Delta ^0$, then the projection map $ \underset { \longrightarrow }{\mathrm{holim}}( \mathscr {F} ) \rightarrow \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$ is an isomorphism.

Example 5.3.2.5 (Set-Valued Functors). Let $\operatorname{\mathcal{C}}$ be a category and let $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{Set}$ be a diagram of sets indexed by $\operatorname{\mathcal{C}}$. Let us abuse notation by identifying $\mathscr {F}$ with a diagram of simplicial sets (by identifying each of the sets $\mathscr {F}(C)$ as a discrete simplicial set). Then there is a canonical isomorphism of simplicial sets

\[ \underset { \longrightarrow }{\mathrm{holim}}( \mathscr {F} ) \simeq \operatorname{N}_{\bullet }( \int _{\operatorname{\mathcal{C}}} \mathscr {F} ). \]

Here $\int _{\operatorname{\mathcal{C}}} \mathscr {F}$ denotes the category of elements of of the functor $\mathscr {F}$ (Construction 5.2.6.1).

Example 5.3.2.6 (Corepresentable Functors). Let $\operatorname{\mathcal{C}}$ be a category and let $h^{C}: \operatorname{\mathcal{C}}\rightarrow \operatorname{Set}$ be the functor corepresented by an object $C \in \operatorname{\mathcal{C}}$, given by $h^{C}(D) = \operatorname{Hom}_{\operatorname{\mathcal{C}}}(C,D)$. Let us abuse notation by regarding $h^{C}$ as a functor from $\operatorname{\mathcal{C}}$ to the category of simplicial sets (by identifying each morphism set $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(C,D)$ with the corresponding discrete simplicial set). Combining Examples 5.3.2.5 and 5.2.6.5, we obtain a canonical isomorphism of simplicial sets $ \underset { \longrightarrow }{\mathrm{holim}}( h^{C} ) \simeq \operatorname{N}_{\bullet }( \operatorname{\mathcal{C}}_{C/ } )$.

Remark 5.3.2.7. Let $\operatorname{\mathcal{C}}$ be a category, let $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{Set_{\Delta }}$ be a diagram of simplicial sets indexed by $\operatorname{\mathcal{C}}$, and let $\operatorname{\mathcal{C}}_0 \subseteq \operatorname{\mathcal{C}}$ be a full subcategory. Suppose that, for every object $C \in \operatorname{\mathcal{C}}$ which does not belong to $\operatorname{\mathcal{C}}_0$, the simplicial set $\mathscr {F}(C)$ is empty. Then the image of the projection map $ \underset { \longrightarrow }{\mathrm{holim}}(\mathscr {F} ) \rightarrow \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$ is contained in $\operatorname{N}_{\bullet }( \operatorname{\mathcal{C}}_0 )$. Setting $\mathscr {F}_0 = \mathscr {F}|_{\operatorname{\mathcal{C}}_0}$, we deduce that the canonical map

\[ \underset { \longrightarrow }{\mathrm{holim}}( \mathscr {F}_0 ) \simeq \operatorname{N}_{\bullet }( \operatorname{\mathcal{C}}_0 ) \times _{ \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) } \underset { \longrightarrow }{\mathrm{holim}}(\mathscr {F} ) \hookrightarrow \underset { \longrightarrow }{\mathrm{holim}}( \mathscr {F} ) \]

is an isomorphism.

Remark 5.3.2.8 (Functoriality). Let $\operatorname{\mathcal{C}}$ be a category. Then the formation of homotopy colimits determines a functor

\[ \underset { \longrightarrow }{\mathrm{holim}}: \operatorname{Fun}( \operatorname{\mathcal{C}}, \operatorname{Set_{\Delta }}) \rightarrow (\operatorname{Set_{\Delta }})_{ / \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) } \quad \quad \mathscr {F} \mapsto \underset { \longrightarrow }{\mathrm{holim}}(\mathscr {F} ). \]

Moreover, this functor preserves small limits and colimits.

Remark 5.3.2.9 (Comparison with the Colimit). Let $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{Set_{\Delta }}$ be a diagram of simplicial sets and let $\{ t_{C}: \mathscr {F}(C) \rightarrow X \} _{C \in \operatorname{\mathcal{C}}}$ be a collection of morphisms which exhibit $X$ as a colimit of the diagram $\mathscr {F}$. The morphisms $t_{C}$ then determine a natural transformation $t_{\bullet }: \mathscr {F} \rightarrow \underline{X}$, where $\underline{X}: \operatorname{\mathcal{C}}\rightarrow \operatorname{Set_{\Delta }}$ denotes the constant functor taking the value $X$. Using Example 5.3.2.4, we obtain a morphism of simplicial sets

\[ \theta : \underset { \longrightarrow }{\mathrm{holim}}(\mathscr {F} ) \xrightarrow { \underset { \longrightarrow }{\mathrm{holim}}(t_{\bullet }) } \underset { \longrightarrow }{\mathrm{holim}}( \underline{X} ) \simeq \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) \times X \rightarrow X, \]

which we will refer to as the comparison map. Note that, for every vertex $C \in \operatorname{\mathcal{C}}$, the restriction of $\theta $ to the fiber $\{ C\} \times _{ \operatorname{N}_{\bullet }( \operatorname{\mathcal{C}}) } \underset { \longrightarrow }{\mathrm{holim}}(\mathscr {F} )$ can be identified with the morphism $t_{C}$. Since $X$ is the union of the images of the morphisms $t_{C}$, it follows that the comparison map $\theta : \underset { \longrightarrow }{\mathrm{holim}}(\mathscr {F} ) \twoheadrightarrow \varinjlim (\mathscr {F})$ is an epimorphism of simplicial sets.

Example 5.3.2.10 (Disjoint Unions). Let $I$ be a set, which we regard as a category having only identity morphisms. Let $\mathscr {F}: I \rightarrow \operatorname{Set_{\Delta }}$ be a functor, which we identify with a collection of simplicial sets $\{ X_ i \} _{i \in I}$. Then the comparison map

\[ \underset { \longrightarrow }{\mathrm{holim}}( \mathscr {F} ) \twoheadrightarrow \varinjlim ( \mathscr {F} ) = \coprod _{i \in I} X_ i \]

is an isomorphism of simplicial sets.

Notation 5.3.2.11 (The Mapping Simplex). Suppose we are given a diagram of simplicial sets

\[ X(0) \xrightarrow {f(1)} X(1) \xrightarrow { f(1) } X(2) \xrightarrow {f(3)} \cdots \xrightarrow {f(n)} X(n), \]

which we will identify with a functor $\mathscr {F}: [n] \rightarrow \operatorname{Set_{\Delta }}$. We denote the homotopy colimit $ \underset { \longrightarrow }{\mathrm{holim}}(\mathscr {F} )$ by $ \underset { \longrightarrow }{\mathrm{holim}}( X(0) \rightarrow \cdots \rightarrow X(n) )$, and refer to it as the mapping simplex of the diagram $\mathscr {F}$.

Let $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{Set_{\Delta }}$ be any diagram of simplicial sets and suppose we are given an $n$-simplex of $\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$, corresponding to a diagram $C_0 \rightarrow \cdots \rightarrow C_ n$ in the category $\operatorname{\mathcal{C}}$. By virtue of Remark 5.3.2.3, the fiber product $\Delta ^ n \times _{ \operatorname{N}_{\bullet }( \operatorname{\mathcal{C}}) } \underset { \longrightarrow }{\mathrm{holim}}(\mathscr {F} )$ can be identified with the mapping simplex of the diagram $\mathscr {F}(C_0) \rightarrow \cdots \rightarrow \mathscr {F}(C_ n)$. When $n=0$, this mapping simplex can be identified with the simplicial set $\mathscr {F}(C_0)$ (Example 5.3.2.4). For larger values of $n$, the mapping simplex can be computed recursively:

Remark 5.3.2.12. Let $n \geq 1$ and let $\mathscr {F}: [n] \rightarrow \operatorname{Set_{\Delta }}$ be a diagram of simplicial sets which we denote by

\[ X(0) \rightarrow X(1) \rightarrow X(2) \rightarrow \cdots \rightarrow X(n). \]

Let $\mathscr {F}': [n] \rightarrow \operatorname{Set_{\Delta }}$ denote the constant diagram taking the value $X(0)$. Let $\mathscr {F}_0 \subseteq \mathscr {F}$ be the subfunctor given by the diagram

\[ \emptyset \rightarrow X(1) \rightarrow X(2) \rightarrow \cdots \rightarrow X(n), \]

and define $\mathscr {F}'_0 \subseteq \mathscr {F}'$ similarly, so that we have a pushout diagram

\[ \xymatrix@R =50pt@C=50pt{ \mathscr {F}'_0 \ar [r] \ar [d] & \mathscr {F}_0 \ar [d] \\ \mathscr {F}' \ar [r] & \mathscr {F} } \]

in the category $\operatorname{Fun}( [n], \operatorname{Set_{\Delta }})$. Applying Remark 5.3.2.8, we deduce that the induced diagram of simplicial sets

\[ \xymatrix@R =50pt@C=50pt{ \underset { \longrightarrow }{\mathrm{holim}}(\mathscr {F}'_0) \ar [r] \ar [d] & \underset { \longrightarrow }{\mathrm{holim}}(\mathscr {F}_0) \ar [d] \\ \underset { \longrightarrow }{\mathrm{holim}}(\mathscr {F}') \ar [r] & \underset { \longrightarrow }{\mathrm{holim}}(\mathscr {F}) } \]

is also a pushout square. Using Example 5.3.2.4 and Remark 5.3.2.7, we can rewrite this diagram as

\[ \xymatrix@R =50pt@C=50pt{ \operatorname{N}_{\bullet }( \{ 1 < 2 < \cdots < n \} ) \times X(0) \ar [r] \ar [d] & \underset { \longrightarrow }{\mathrm{holim}}( X(1) \rightarrow \cdots \rightarrow X(n) ) \ar [d] \\ \Delta ^ n \times X(0) \ar [r] & \underset { \longrightarrow }{\mathrm{holim}}( X(0) \rightarrow \cdots \rightarrow X(n) ). } \]

Example 5.3.2.13 (The Mapping Cylinder). Let $f: X \rightarrow Y$ be a morphism of simplicial sets, which we identify with a diagram $\mathscr {F}: [1] \rightarrow \operatorname{Set_{\Delta }}$. We will denote the homotopy colimit $ \underset { \longrightarrow }{\mathrm{holim}}( \mathscr {F} )$ by $ \underset { \longrightarrow }{\mathrm{holim}}( f: X \rightarrow Y)$ and refer to it as the mapping cylinder of the morphism $f$. Applying Remark 5.3.2.12, we obtain an isomorphism of simplicial sets

\[ \underset { \longrightarrow }{\mathrm{holim}}(f: X \rightarrow Y) \simeq (\Delta ^1 \times X) \coprod _{ ( \{ 1\} \times X) } Y; \]

that is, the mapping cylinder $ \underset { \longrightarrow }{\mathrm{holim}}(f: X \rightarrow Y)$ can be identified with the homotopy pushout $X \coprod _{X}^{\mathrm{h}} Y$ of Construction 3.4.2.2.

Remark 5.3.2.14. Let $n$ be a nonnegative integer, and suppose we are given a diagram of simplicial sets

\[ X(0) \rightarrow X(1) \rightarrow X(2) \rightarrow \cdots \rightarrow X(n). \]

For each integer $0 \leq i \leq n$, let $\Delta ^ n_{\geq i}$ denote the nerve of the linearly ordered set $\{ i < i+1 < \cdots < n \} $, which we regard as a simplicial subset of $\Delta ^ n$. Applying Remark 5.3.2.12 repeatedly, we can identify the mapping simplex $ \underset { \longrightarrow }{\mathrm{holim}}( X(0) \rightarrow \cdots \rightarrow X(n) )$ with the iterated pushout

\[ (\Delta ^ n \times X(0)) \coprod _{ ( \Delta ^{n}_{\geq 1} \times X(0)) } (\Delta ^{n}_{\geq 1} \times X(1) ) \coprod _{ ( \Delta ^{n}_{\geq 2} \times X(1))} \cdots \coprod _{ (\{ n\} \times X(n-1))} (\{ n\} \times X(n)). \]

Example 5.3.2.15 (Homotopy Quotients). Let $G$ be a group and let $BG$ denote the associated groupoid (consisting of a single object with automorphism group $G$). Let $X$ be a simplicial set equipped with an action of $G$, which we identify with a functor $\mathscr {F}: BG \rightarrow \operatorname{Set_{\Delta }}$. We will denote the homotopy colimit $ \underset { \longrightarrow }{\mathrm{holim}}(\mathscr {F} )$ by $X_{\mathrm{h}G}$, and refer to it as the homotopy quotient of $X$ by the action of $G$.

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 $A_0 \coprod A_1$).

Exercise 5.3.2.17. Let $\operatorname{\mathcal{C}}$ be a category and let $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{Set_{\Delta }}$ be a diagram of simplicial sets with the following properties:

  • For every object $C \in \operatorname{\mathcal{C}}$, the simplicial set $\mathscr {F}(C)$ is a Kan complex.

  • For every morphism $u: C \rightarrow C'$ in $\operatorname{\mathcal{C}}$, the induced map $\mathscr {F}(u): \mathscr {F}(C) \rightarrow \mathscr {F}(C')$ is a Kan fibration.

Show that the projection map $ \underset { \longrightarrow }{\mathrm{holim}}(\mathscr {F}) \rightarrow \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$ is a left fibration of simplicial sets.

We now apply the preceding analysis to study the homotopy invariance properties of Construction 5.3.2.1.

Proposition 5.3.2.18. Let $\operatorname{\mathcal{C}}$ be a category and let $\alpha : \mathscr {F} \rightarrow \mathscr {G}$ be a levelwise weak homotopy equivalence between diagrams $\mathscr {F}, \mathscr {G}: \operatorname{\mathcal{C}}\rightarrow \operatorname{Set_{\Delta }}$. Then the induced map $ \underset { \longrightarrow }{\mathrm{holim}}(\alpha ): \underset { \longrightarrow }{\mathrm{holim}}(\mathscr {F} ) \rightarrow \underset { \longrightarrow }{\mathrm{holim}}( \mathscr {G} )$ is a weak homotopy equivalence of simplicial sets.

Proof. By virtue of Proposition 3.4.2.16, it will suffice to show that for every $n$-simplex $\Delta ^ n \rightarrow \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$, the induced map $\Delta ^{n} \times _{ \operatorname{N}_{\bullet }( \operatorname{\mathcal{C}}) } \underset { \longrightarrow }{\mathrm{holim}}(\mathscr {F} ) \rightarrow \Delta ^{n} \times _{ \operatorname{N}_{\bullet }( \operatorname{\mathcal{C}}) } \underset { \longrightarrow }{\mathrm{holim}}(\mathscr {G})$ is a weak homotopy equivalence. Using Remark 5.3.2.3, we are reduced to proving Proposition 5.3.2.18 in the special case where $\operatorname{\mathcal{C}}$ is the linearly ordered set $[n] = \{ 0 < 1 < \cdots < n \} $. We now proceed by induction on $n$. If $n = 0$, the desired result follows immediately from Example 5.3.2.2. Let us therefore assume that $n > 0$. Let $\mathscr {F}'$ denote the restriction of $\mathscr {F}$ to the full subcategory $\{ 1 < 2 < \cdots < n \} $ and define $\mathscr {G}'$ similarly. The natural transformation $\alpha $ determines a commutative diagram of simplicial sets

\[ \xymatrix@R =50pt@C=50pt{ \Delta ^ n \times \mathscr {F}(0) \ar [d] & \operatorname{N}_{\bullet }( \{ 1 < \cdots < n \} ) \times \mathscr {F}(0) \ar [r] \ar [d] \ar [l] & \underset { \longrightarrow }{\mathrm{holim}}( \mathscr {F}' ) \ar [d] \\ \Delta ^ n \times \mathscr {G}(0) & \operatorname{N}_{\bullet }( \{ 1 < \cdots < n \} ) \times \mathscr {G}(0) \ar [l] \ar [r] & \underset { \longrightarrow }{\mathrm{holim}}( \mathscr {G}' ), } \]

where the left horizontal maps are monomorphisms, the right vertical map is a weak homotopy equivalence by virtue of our inductive hypothesis, and the other vertical maps are weak homotopy equivalences by virtue of our assumption on $\alpha $. The desired result now follows by combining Corollary 3.4.2.14 with Remark 5.3.2.12. $\square$

Using exactly the same argument, we see that the formation of homotopy colimits is compatible with categorical equivalence:

Variant 5.3.2.19. Let $\operatorname{\mathcal{C}}$ be a category and let $\alpha : \mathscr {F} \rightarrow \mathscr {G}$ be a levelwise categorical equivalence between diagrams $\mathscr {F}, \mathscr {G}: \operatorname{\mathcal{C}}\rightarrow \operatorname{Set_{\Delta }}$. Then the induced map $ \underset { \longrightarrow }{\mathrm{holim}}(\alpha ): \underset { \longrightarrow }{\mathrm{holim}}(\mathscr {F} ) \rightarrow \underset { \longrightarrow }{\mathrm{holim}}( \mathscr {G} )$ is a categorical equivalence of simplicial sets.

Proof. By virtue of Corollary 4.5.7.3, it will suffice to show that for every $n$-simplex $\Delta ^ n \rightarrow \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$, the induced map $\Delta ^{n} \times _{ \operatorname{N}_{\bullet }( \operatorname{\mathcal{C}}) } \underset { \longrightarrow }{\mathrm{holim}}(\mathscr {F} ) \rightarrow \Delta ^{n} \times _{ \operatorname{N}_{\bullet }( \operatorname{\mathcal{C}}) } \underset { \longrightarrow }{\mathrm{holim}}(\mathscr {G})$ is a categorical equivalence of simplicial sets. Using Remark 5.3.2.3, we are reduced to proving Variant 5.3.2.19 in the special case where $\operatorname{\mathcal{C}}$ is the linearly ordered set $[n] = \{ 0 < 1 < \cdots < n \} $. We now proceed by induction on $n$. If $n = 0$, the desired result follows immediately from Example 5.3.2.2. Let us therefore assume that $n > 0$. Let $\mathscr {F}'$ denote the restriction of $\mathscr {F}$ to the full subcategory $\{ 1 < 2 < \cdots < n \} $ and define $\mathscr {G}'$ similarly. The natural transformation $\alpha $ determines a commutative diagram of simplicial sets

\[ \xymatrix@R =50pt@C=50pt{ \Delta ^ n \times \mathscr {F}(0) \ar [d] & \operatorname{N}_{\bullet }( \{ 1 < \cdots < n \} ) \times \mathscr {F}(0) \ar [r] \ar [d] \ar [l] & \underset { \longrightarrow }{\mathrm{holim}}( \mathscr {F}' ) \ar [d] \\ \Delta ^ n \times \mathscr {G}(0) & \operatorname{N}_{\bullet }( \{ 1 < \cdots < n \} ) \times \mathscr {G}(0) \ar [l] \ar [r] & \underset { \longrightarrow }{\mathrm{holim}}( \mathscr {G}' ), } \]

where the left horizontal maps are monomorphisms, the right vertical map is a categorical equivalence by virtue of our inductive hypothesis, and the other vertical maps are categorical equivalences by virtue of our assumption on $\alpha $. The desired result now follows by combining Corollary 4.5.4.14 with Remark 5.3.2.12. $\square$

The homotopy colimit of Construction 5.3.2.1 can be characterized by a universal mapping property.

Construction 5.3.2.20. Let $\operatorname{\mathcal{C}}$ be a category and let $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{Set_{\Delta }}$ be a diagram of simplicial sets indexed by $\operatorname{\mathcal{C}}$. For each object $C \in \operatorname{\mathcal{C}}$, we let

\[ f_{C}: \operatorname{N}_{\bullet }( \operatorname{\mathcal{C}}_{C/} ) \times \mathscr {F}(C) \rightarrow \underset { \longrightarrow }{\mathrm{holim}}( \mathscr {F} ) \]

denote the morphism of simplicial sets given on $n$-simplices by the formula $f_{C}( \sigma , \tau ) = ( \overline{\sigma }, \overline{\tau } )$, where $\overline{\sigma }$ denotes the image of $\sigma $ in $\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$ and $\overline{\tau }$ denote the image of $\tau $ under the map $\mathscr {F}(C) \rightarrow \mathscr {F}( \overline{\sigma }(0) )$. Note that we can identify $f_{C}$ with a morphism of simplicial sets

\[ u_{\mathscr {F},C}: \mathscr {F}(C) \rightarrow \operatorname{Fun}_{ / \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})}( \operatorname{N}_{\bullet }( \operatorname{\mathcal{C}}_{C/}), \underset { \longrightarrow }{\mathrm{holim}}( \mathscr {F} ) ) = \operatorname{wTr}_{ \underset { \longrightarrow }{\mathrm{holim}}(\mathscr {F}) / \operatorname{\mathcal{C}}}(C). \]

This morphism depends functorially on $C$: that is, the collection $u_{\mathscr {F}} = \{ u_{\mathscr {F},C} \} _{C \in \operatorname{\mathcal{C}}}$ is a natural transformation from $\mathscr {F}$ to the functor

\[ \operatorname{\mathcal{C}}\rightarrow \operatorname{Set_{\Delta }}\quad \quad C \mapsto \operatorname{wTr}_{ \underset { \longrightarrow }{\mathrm{holim}}(\mathscr {F}) / \operatorname{\mathcal{C}}}. \]

For every pair of functors $\mathscr {F}, \mathscr {G}: \operatorname{\mathcal{C}}\rightarrow \operatorname{Set_{\Delta }}$, let $\operatorname{Hom}_{ \operatorname{Fun}( \operatorname{\mathcal{C}}, \operatorname{Set_{\Delta }}) }( \mathscr {F}, \mathscr {G} )_{\bullet }$ denote the simplicial set parametrizing natural transformations from $\mathscr {F}$ to $\mathscr {G}$ (Example 2.4.2.2), described concretely by the formula

\[ \operatorname{Hom}_{ \operatorname{Fun}( \operatorname{\mathcal{C}}, \operatorname{Set_{\Delta }}) }( \mathscr {F}, \mathscr {G} )_{n} = \operatorname{Hom}_{ \operatorname{Fun}( \operatorname{\mathcal{C}}, \operatorname{Set_{\Delta }}) }( \mathscr {F}, \mathscr {G}^{\Delta ^ n} ). \]

Here $\mathscr {G}^{\Delta ^ n}: \operatorname{\mathcal{C}}\rightarrow \operatorname{Set_{\Delta }}$ denotes the functor given by $\mathscr {G}^{\Delta ^ n}(C) = \operatorname{Fun}( \Delta ^ n, \mathscr {G}(C) )$.

Proposition 5.3.2.21. Let $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{Set_{\Delta }}$ be a diagram of simplicial sets, let $\operatorname{\mathcal{E}}$ be a simplicial set, and define $\mathscr {G}: \operatorname{\mathcal{C}}\rightarrow \operatorname{Set_{\Delta }}$ by the formula $\mathscr {G}( C) = \operatorname{Fun}( \operatorname{N}_{\bullet }( \operatorname{\mathcal{C}}_{C/} ), \operatorname{\mathcal{E}})$. Then composition with the natural transformation $u_{\mathscr {F}}$ of Construction 5.3.2.20 induces an isomorphism of simplicial sets

\[ \Phi _{\mathscr {F}}: \operatorname{Fun}( \underset { \longrightarrow }{\mathrm{holim}}( \mathscr {F} ), \operatorname{\mathcal{E}}) \rightarrow \operatorname{Hom}_{ \operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{Set_{\Delta }}) }( \mathscr {F}, \mathscr {G} )_{\bullet }. \]

Proof. For every object $C \in \operatorname{\mathcal{C}}$, let $h^{C}: \operatorname{\mathcal{C}}\rightarrow \operatorname{Set_{\Delta }}$ denote the functor corepresented by $C$ (given by $h^{C}(D) = \operatorname{Hom}_{\operatorname{\mathcal{C}}}(C,D)$, regarded as a discrete simplicial set). For every simplicial set $K$, let $\underline{K}: \operatorname{\mathcal{C}}\rightarrow \operatorname{Set_{\Delta }}$ denote the constant functor taking the value $K$. For every functor $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{Set_{\Delta }}$ we have a coequalizer diagram

\[ \coprod _{ C \rightarrow D} h^{D} \times \underline{ \mathscr {F}(C) } \rightrightarrows \coprod _{ C } h^{C} \times \underline{ \mathscr {F}(C) } \rightarrow \mathscr {F} \]

in the category $\operatorname{Fun}( \operatorname{\mathcal{C}}, \operatorname{Set_{\Delta }})$. Note that, if we regard the simplicial set $\operatorname{\mathcal{E}}$ as fixed, then the construction $\mathscr {F} \mapsto \Phi _{\mathscr {F}}$ carries colimits in $\operatorname{Fun}( \operatorname{\mathcal{C}}, \operatorname{Set_{\Delta }})$ to limits in the arrow category $\operatorname{Fun}( [1], \operatorname{Set_{\Delta }})$. We can therefore assume without loss of generality that the functor $\mathscr {F}$ factors as a product $h^{C} \times \underline{K}$, for some object $C \in \operatorname{\mathcal{C}}$ and some simplicial set $K$.

Fix an integer $n \geq 0$; we wish to show that $\Phi _{\mathscr {F}}$ induces a bijection from $n$-simplices of $\operatorname{Fun}( \underset { \longrightarrow }{\mathrm{holim}}( \mathscr {F} ), \operatorname{\mathcal{E}})$ to $n$-simplices of $\operatorname{Hom}_{ \operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{Set_{\Delta }}) }( \mathscr {F}, \mathscr {G} )_{\bullet }$. Replacing $\operatorname{\mathcal{E}}$ by the simplicial set $\operatorname{Fun}(K \times \Delta ^ n, \operatorname{\mathcal{E}})$, we are reduced to proving that Construction 5.3.2.20 induces a bijection

\[ \Phi _0: \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( \underset { \longrightarrow }{\mathrm{holim}}( h^{C} ), \operatorname{\mathcal{E}}) \rightarrow \operatorname{Hom}_{ \operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{Set_{\Delta }})}( h^{C}, \mathscr {G} ). \]

Let $\mathscr {G}_0: \operatorname{\mathcal{C}}\rightarrow \operatorname{Set}$ denote the functor given on objects by the formula

\[ \mathscr {G}_0(C) = \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( \Delta ^0, \mathscr {G}(C) ) = \operatorname{Hom}_{ \operatorname{Set_{\Delta }}}( \operatorname{N}_{\bullet }( \operatorname{\mathcal{C}}_{C/} ), \operatorname{\mathcal{E}}). \]

Under the identification of $ \underset { \longrightarrow }{\mathrm{holim}}( h^{C} ) \simeq \operatorname{N}_{\bullet }( \operatorname{\mathcal{C}}_{C/} )$ of Example 5.3.2.6, the function $\Phi _0$ corresponds to the bijection $\mathscr {G}_0(C) \simeq \operatorname{Hom}_{ \operatorname{Fun}( \operatorname{\mathcal{C}}, \operatorname{Set}) }( h^{C}, \mathscr {G}_0 )$ supplied by Yoneda's lemma. $\square$

Corollary 5.3.2.22. Let $\operatorname{\mathcal{C}}$ be a small category. Then the homotopy colimit functor

\[ \operatorname{Fun}( \operatorname{\mathcal{C}}, \operatorname{Set_{\Delta }}) \rightarrow \operatorname{Set_{\Delta }}\quad \quad \mathscr {F} \mapsto \underset { \longrightarrow }{\mathrm{holim}}(\mathscr {F} ) \]

admits a right adjoint, given by the construction

\[ \operatorname{Set_{\Delta }}\rightarrow \operatorname{Fun}( \operatorname{\mathcal{C}}, \operatorname{Set_{\Delta }}) \quad \quad \operatorname{\mathcal{E}}\mapsto (C \mapsto \operatorname{Fun}( \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}_{C/} ), \operatorname{\mathcal{E}}) ). \]

Corollary 5.3.2.23. Let $\operatorname{\mathcal{C}}$ be a category, let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$ be a morphism of simplicial sets, and let $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{Set_{\Delta }}$ be a functor. Then composition with the natural transformation $u_{\mathscr {F}}$ of Construction 5.3.2.20 induces an isomorphism of simplicial sets

\[ \operatorname{Fun}_{/ \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) }( \underset { \longrightarrow }{\mathrm{holim}}( \mathscr {F} ), \operatorname{\mathcal{E}}) \rightarrow \operatorname{Hom}_{ \operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{Set_{\Delta }}) }( \mathscr {F}, \operatorname{wTr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}} )_{\bullet }, \]

where $\operatorname{wTr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}}$ is the weak transport representation of Construction 5.3.1.1.

Proof. Define $\mathscr {G}, \mathscr {H}: \operatorname{\mathcal{C}}\rightarrow \operatorname{Set_{\Delta }}$ by the formulae $\mathscr {G}( C) = \operatorname{Fun}( \operatorname{N}_{\bullet }( \operatorname{\mathcal{C}}_{C/} ), \operatorname{\mathcal{E}})$ and $\mathscr {H}(C) = \operatorname{Fun}( \operatorname{N}_{\bullet }( \operatorname{\mathcal{C}}_{C/} ), \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) )$. We have a commutative diagram of simplicial sets

\[ \xymatrix@R =50pt@C=50pt{ \operatorname{Fun}( \underset { \longrightarrow }{\mathrm{holim}}( \mathscr {F} ), \operatorname{\mathcal{E}}) \ar [r] \ar [d]^{U \circ } & \operatorname{Hom}_{ \operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{Set_{\Delta }}) }( \mathscr {F}, \mathscr {G} )_{\bullet } \ar [d]^{U \circ } \\ \operatorname{Fun}( \underset { \longrightarrow }{\mathrm{holim}}( \mathscr {F} ), \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) ) \ar [r] & \operatorname{Hom}_{ \operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{Set_{\Delta }}) }( \mathscr {F}, \mathscr {H} )_{\bullet }, } \]

where the horizontal maps are isomorphisms by virtue of Proposition 5.3.2.21. Corollary 5.3.2.23 follows by restricting to fibers of the vertical maps. $\square$

Corollary 5.3.2.24. Let $\operatorname{\mathcal{C}}$ be a small category. Then the homotopy colimit functor

\[ \underset { \longrightarrow }{\mathrm{holim}}: \operatorname{Fun}( \operatorname{\mathcal{C}}, \operatorname{Set_{\Delta }}) \rightarrow (\operatorname{Set_{\Delta }})_{ / \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) } \]

admits a right adjoint, given by the functor

\[ (\operatorname{Set_{\Delta }})_{ / \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) } \rightarrow \operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{Set_{\Delta }}) \quad \quad (U: \operatorname{\mathcal{E}}\rightarrow \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) ) \mapsto (\operatorname{wTr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}}: \operatorname{\mathcal{C}}\rightarrow \operatorname{Set_{\Delta }}) \]

of Construction 5.3.1.1.