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7.5.2 Homotopy Limits of $\infty $-Categories

We now extend the definition of homotopy limit to diagrams taking values in the category $\operatorname{QCat}$.

Construction 7.5.2.1 (Homotopy Limits of $\infty $-Categories). Let $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{QCat}$ be a (strictly commutative) diagram of $\infty $-categories, let $\operatorname{N}_{\bullet }^{\mathscr {F}}(\operatorname{\mathcal{C}})$ denote the weighted nerve of $\mathscr {F}$ (Definition 5.3.3.1), and let $U: \operatorname{N}_{\bullet }^{\mathscr {F}}(\operatorname{\mathcal{C}}) \rightarrow \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$ the cocartesian fibration Proposition 5.3.3.15. We let $ \underset {\longleftarrow }{\mathrm{holim}}(\mathscr {F})$ denote the full subcategory

\[ \operatorname{Fun}^{\operatorname{CCart}}_{ / \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) }( \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}), \operatorname{N}_{\bullet }^{\mathscr {F}}(\operatorname{\mathcal{C}})) \subseteq \operatorname{Fun}_{ / \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) }( \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}), \operatorname{N}_{\bullet }^{\mathscr {F}}(\operatorname{\mathcal{C}})) \]

whose objects are functors $G: \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) \rightarrow \operatorname{N}_{\bullet }^{\mathscr {F}}(\operatorname{\mathcal{C}})$ which satisfy $U \circ G = \operatorname{id}_{ \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) }$ and which carry each morphism of $\operatorname{\mathcal{C}}$ to a $U$-cocartesian morphism of $\operatorname{N}_{\bullet }^{\mathscr {F}}(\operatorname{\mathcal{C}})$ (see Notation 5.3.1.10). We will refer to $ \underset {\longleftarrow }{\mathrm{holim}}(\mathscr {F})$ as the homotopy limit of the diagram $\mathscr {F}$.

Example 7.5.2.2 (Homotopy Limits of Kan Complexes). Let $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{Kan}$ be a (strictly commutative) diagram of Kan complexes. Then the projection map $U: \operatorname{N}_{\bullet }^{\mathscr {F}}(\operatorname{\mathcal{C}}) \rightarrow \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$ is a left fibration of simplicial sets (Corollary 5.3.3.16). It follows that every morphism of the $\infty $-category $\operatorname{N}_{\bullet }^{\mathscr {F}}(\operatorname{\mathcal{C}})$ is $U$-cocartesian, so the homotopy limit $ \underset {\longleftarrow }{\mathrm{holim}}(\mathscr {F})$ of Construction 7.5.2.1 coincides with the homotopy limit $ \underset {\longleftarrow }{\mathrm{holim}}(\mathscr {F} )$ of Construction 7.5.1.1.

Remark 7.5.2.3. Let $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{QCat}$ be a (strictly commutative) diagram of $\infty $-categories, let $K$ be a simplicial set, and let $\mathscr {F}^{K}: \operatorname{\mathcal{C}}\rightarrow \operatorname{QCat}$ denote the functor given on objects by the formula $\mathscr {F}^{K}(C) = \operatorname{Fun}(K, \mathscr {F}(C) )$. Then there is a canonical isomorphism of simplicial sets $ \underset {\longleftarrow }{\mathrm{holim}}( \mathscr {F}^{K} ) \simeq \operatorname{Fun}(K, \underset {\longleftarrow }{\mathrm{holim}}(\mathscr {F} ) )$ (see Remarks 5.3.3.5 and 5.3.1.19).

Proposition 7.5.2.4. Let $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{QCat}$ be a (strictly commutative) diagram of $\infty $-categories. Then the homotopy limit $ \underset {\longleftarrow }{\mathrm{holim}}(\mathscr {F})$ is an $\infty $-category. Moreover, $ \underset {\longleftarrow }{\mathrm{holim}}(\mathscr {F} )$ can be identified with a limit of the diagram

\[ \operatorname{N}_{\bullet }^{\operatorname{hc}}( \mathscr {F}): \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) \rightarrow \operatorname{N}_{\bullet }^{\operatorname{hc}}( \operatorname{QCat}) = \operatorname{\mathcal{QC}} \]

in the $\infty $-category $\operatorname{\mathcal{QC}}$.

Proof. By virtue of Example 5.7.5.6, the functor $\operatorname{N}_{\bullet }^{\operatorname{hc}}( \mathscr {F} )$ is a covariant transport representation for the cocartesian fibration $U: \operatorname{N}_{\bullet }^{\mathscr {F}}(\operatorname{\mathcal{C}}) \rightarrow \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$. Proposition 7.5.2.4 is therefore a special case of Corollary 7.4.1.9. $\square$

Remark 7.5.2.5 (Homotopy Invariance). Let $\operatorname{\mathcal{C}}$ be a category and let $\alpha : \mathscr {F} \rightarrow \mathscr {G}$ be a natural transformation between functors $\mathscr {F}, \mathscr {G}: \operatorname{\mathcal{C}}\rightarrow \operatorname{QCat}$. Then $\alpha $ determines a commutative diagram of $\infty $-categories

\[ \xymatrix@R =50pt@C=50pt{ \operatorname{N}_{\bullet }^{\mathscr {F}}(\operatorname{\mathcal{C}}) \ar [dr]^{U} \ar [rr]^{T} & & \operatorname{N}_{\bullet }^{\mathscr {G}}(\operatorname{\mathcal{C}}) \ar [dl]_{V} \\ & \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}). & } \]

The functor $T$ carries $U$-cocartesian morphisms of $\operatorname{N}_{\bullet }^{\mathscr {F}}(\operatorname{\mathcal{C}})$ to $V$-cocartesian morphisms of $\operatorname{N}_{\bullet }^{\mathscr {G}}(\operatorname{\mathcal{C}})$ (see Proposition 5.3.3.15), and therefore induces a functor of $\infty $-categories $ \underset {\longleftarrow }{\mathrm{holim}}(\alpha ): \underset {\longleftarrow }{\mathrm{holim}}(\mathscr {F}) \rightarrow \underset {\longleftarrow }{\mathrm{holim}}(\mathscr {G} )$. If $\alpha $ is a levelwise categorical equivalence, then $T$ is an equivalence of cocartesian fibrations over $\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$ (Corollary 5.3.3.17), so $ \underset {\longleftarrow }{\mathrm{holim}}(\alpha )$ is an equivalence of $\infty $-categories.

Example 7.5.2.6 (Homotopy Limits of Cores). Let $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{QCat}$ be a diagram of $\infty $-categories, and let $\mathscr {F}^{\simeq }: \operatorname{\mathcal{C}}\rightarrow \operatorname{Kan}$ be the functor given on objects by the formula $\mathscr {F}^{\simeq }(C) = \mathscr {F}(C)^{\simeq }$. Then the inclusion map $\mathscr {F}^{\simeq } \hookrightarrow \mathscr {F}$ induces a monomorphism of simplicial sets $ \underset {\longleftarrow }{\mathrm{holim}}( \mathscr {F}^{\simeq } ) \rightarrow \underset {\longleftarrow }{\mathrm{holim}}( \mathscr {F} )$, whose image is the core of the $\infty $-category $ \underset {\longleftarrow }{\mathrm{holim}}(\mathscr {F})$ (see Example 5.3.3.18 and Remark 5.3.1.20). In other words, there is a canonical isomorphism of Kan complexes $ \underset {\longleftarrow }{\mathrm{holim}}( \mathscr {F}^{\simeq } ) \simeq \underset {\longleftarrow }{\mathrm{holim}}( \mathscr {F})^{\simeq }$.

Remark 7.5.2.7. Let $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{QCat}$ be a diagram of $\infty $-categories and let $\mathscr {F}_0 = \mathscr {F}|_{\operatorname{\mathcal{C}}_0}$ be the restriction of $\mathscr {F}$ to a subcategory $\operatorname{\mathcal{C}}_0 \subseteq \operatorname{\mathcal{C}}$. Suppose that the inclusion $\operatorname{N}_{\bullet }( \operatorname{\mathcal{C}}_0 ) \hookrightarrow \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$ is left anodyne (this condition is satisfied, for example, if the inclusion map $\operatorname{\mathcal{C}}_0 \hookrightarrow \operatorname{\mathcal{C}}$ has a right adjoint: see Corollary 7.2.3.7). Then the restriction map $ \underset {\longleftarrow }{\mathrm{holim}}( \mathscr {F} ) \rightarrow \underset {\longleftarrow }{\mathrm{holim}}( \mathscr {F}_0)$ is a trivial Kan fibration of $\infty $-categories (see Proposition 5.3.1.21).

Remark 7.5.2.8. Let $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{QCat}$ be a diagram of $\infty $-categories. Arguing as in Remark 7.5.1.6, we can identify the homotopy limit $ \underset {\longleftarrow }{\mathrm{holim}}( \mathscr {F} )$ with a simplicial subset of the product $\prod _{C \in \operatorname{\mathcal{C}}} \operatorname{Fun}( \operatorname{N}_{\bullet }( \operatorname{\mathcal{C}}_{/C} ), \mathscr {F}(C) )$, whose $n$-simplices are collections of maps $\{ \sigma _{C}: \Delta ^ n \times \operatorname{N}_{\bullet }( \operatorname{\mathcal{C}}_{/C} ) \rightarrow \mathscr {F}(C) \} $ which satisfy the following pair of conditions:

$(\ast )$

For every morphism $f: C \rightarrow D$ in the category $\operatorname{\mathcal{C}}$, the diagram of simplicial sets

\[ \xymatrix@R =50pt@C=50pt{ \Delta ^{n} \times \operatorname{N}_{\bullet }( \operatorname{\mathcal{C}}_{/C} ) \ar [r]^-{\circ f} \ar [d]^{ \sigma _{C} } & \Delta ^ n \times \operatorname{N}_{\bullet }( \operatorname{\mathcal{C}}_{/D} ) \ar [d]^{ \sigma _ D } \\ \mathscr {F}(C) \ar [r]^-{ \mathscr {F}(f) } & \mathscr {F}(D) } \]

is commutative.

$(\ast ')$

For every object $C \in \operatorname{\mathcal{C}}$ and every integer $0 \leq i \leq n$, the composite map

\[ \{ i\} \times \operatorname{N}_{\bullet }( \operatorname{\mathcal{C}}_{/C} ) \hookrightarrow \Delta ^ n \times \operatorname{N}_{\bullet }( \operatorname{\mathcal{C}}_{/C} ) \xrightarrow { \sigma _{C} } \mathscr {F}(C) \]

carries every morphism in the category $\operatorname{\mathcal{C}}_{/C}$ to an isomorphism in the $\infty $-category $\mathscr {F}(C)$.

Example 7.5.2.9 (Duality with Homotopy Colimits). Let $\operatorname{\mathcal{C}}$ be a category, let $\mathscr {F}: \operatorname{\mathcal{C}}^{\operatorname{op}} \rightarrow \operatorname{Set_{\Delta }}$ be a diagram of simplicial sets, and let $W$ denote the collection of horizontal edges of the homotopy colimit $ \underset { \longrightarrow }{\mathrm{holim}}( \mathscr {F}^{\operatorname{op}} )$ (see Definition 5.3.4.1). Let $\operatorname{\mathcal{D}}$ be an $\infty $-category and let $\operatorname{\mathcal{D}}^{\mathscr {F}}: \operatorname{\mathcal{C}}\rightarrow \operatorname{QCat}$ denote the functor given by the formula $\operatorname{\mathcal{D}}^{\mathscr {F}}(C) = \operatorname{Fun}( \mathscr {F}(C), \operatorname{\mathcal{D}})$. Arguing as in Example 7.5.1.7, we obtain a canonical isomorphism

\[ \theta : \operatorname{Fun}_{ / \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) }( \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}), \operatorname{N}_{\bullet }^{\operatorname{\mathcal{D}}^{\mathscr {F}} }(\operatorname{\mathcal{C}}) )^{\operatorname{op}} \simeq \operatorname{Fun}( \underset { \longrightarrow }{\mathrm{holim}}(\mathscr {F}^{\operatorname{op}}), \operatorname{\mathcal{D}}^{\operatorname{op}} ). \]

Unwinding the definitions, we see that $\theta $ restricts to an isomorphism of $\infty $-categories $ \underset {\longleftarrow }{\mathrm{holim}}( \operatorname{\mathcal{D}}^{\mathscr {F}} )^{\operatorname{op}} \simeq \operatorname{Fun}( \underset { \longrightarrow }{\mathrm{holim}}(\mathscr {F}^{\operatorname{op}})[W^{-1}], \operatorname{\mathcal{D}}^{\operatorname{op}} )$.

Remark 7.5.2.10 (Comparison with the Limit). Let $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{QCat}$ be a diagram of $\infty $-categories and let $X = \varprojlim ( \mathscr {F} )$ denote the limit of $\mathscr {F}$, formed in the category of simplicial sets. Let $\underline{X}: \operatorname{\mathcal{C}}\rightarrow \operatorname{Set_{\Delta }}$ denote the constant functor taking the value $X$. We then have a tautological map $\underline{X} \rightarrow \mathscr {F}$. The induced morphism of simplicial sets

\[ X \times \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) \simeq \operatorname{N}_{\bullet }^{ \underline{X} }(\operatorname{\mathcal{C}}) \rightarrow \operatorname{N}_{\bullet }^{\mathscr {F}}(\operatorname{\mathcal{C}}) \]

determines a comparison map $\iota : X = \varprojlim ( \mathscr {F} ) \rightarrow \underset {\longleftarrow }{\mathrm{holim}}( \mathscr {F} )$. Note that $\iota $ is a monomorphism of simplicial sets (since each of the projection maps $X = \varprojlim (\mathscr {F}) \rightarrow \mathscr {F}(C)$ factor through $\iota $).

Proposition 7.5.2.11. Let $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{QCat}$ be a diagram of $\infty $-categories, and suppose that the category $\operatorname{\mathcal{C}}$ has an initial object. Then the comparison map $\iota : \varprojlim (\mathscr {F}) \hookrightarrow \underset {\longleftarrow }{\mathrm{holim}}(\mathscr {F})$ of Remark 7.5.2.10 is an equivalence of $\infty $-categories.

Proof. Let $C \in \operatorname{\mathcal{C}}$ be an initial object, so that the inclusion map $\{ C\} \rightarrow \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$ is left anodyne (Corollary 4.6.6.25). Applying Remark 7.5.2.7, we see that evaluation at $C$ induces an equivalence of $\infty $-categories $\operatorname{ev}_{C}: \underset {\longleftarrow }{\mathrm{holim}}(\mathscr {F}) \rightarrow \mathscr {F}(C)$. Our assumption that $C$ is initial also guarantees that the composition $(\operatorname{ev}_ C \circ \iota ): \varprojlim (\mathscr {F}) \rightarrow \mathscr {F}(C)$ is an isomorphism of simplicial sets, so that $\varprojlim (\mathscr {F})$ is an $\infty $-category and $\iota $ is an equivalence of $\infty $-categories. $\square$

Example 7.5.2.12. Let $I$ be a set, which we regard as a category having only identity morphisms. Let $\mathscr {F}: I \rightarrow \operatorname{QCat}$ be a diagram, which we view as a collection of $\infty $-categories $\{ \operatorname{\mathcal{C}}_ i \} _{i \in I}$ indexed by $I$. Then the comparison morphism

\[ \prod _{i \in I} \operatorname{\mathcal{C}}_ i = \varprojlim (\mathscr {F}) \rightarrow \underset {\longleftarrow }{\mathrm{holim}}(\mathscr {F}) \]

of Remark 7.5.2.10 is an isomorphism.

Exercise 7.5.2.13 (Homotopy Limits of Sets). Let $\operatorname{\mathcal{C}}$ be a category and let $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{Set}$ be a diagram in the category of sets. Let us abuse notation by identifying $\operatorname{Set}$ with the full subcategory of $\operatorname{Kan}$ spanned by the constant simplicial sets. Show that the comparison map $\varprojlim (\mathscr {F}) \hookrightarrow \underset {\longleftarrow }{\mathrm{holim}}(\mathscr {F})$ of Remark 7.5.2.10 is an isomorphism.

Beware that the comparison morphism of Remark 7.5.2.10 is not an isomorphism in general.

Example 7.5.2.14. Let $[1]$ denote the linearly ordered set $\{ 0 < 1 \} $ and let $\mathscr {F}: [1] \rightarrow \operatorname{QCat}$ be a diagram, which we identify with a functor of $\infty $-categories $T: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$. Then the homotopy limit $ \underset {\longleftarrow }{\mathrm{holim}}(\mathscr {F})$ of Construction 7.5.1.1 can be identified with the homotopy fiber product

\[ \operatorname{\mathcal{C}}\times ^{\mathrm{h}}_{\operatorname{\mathcal{D}}} \operatorname{\mathcal{D}}= \operatorname{\mathcal{C}}\times _{ \operatorname{Fun}( \{ 0\} , \operatorname{\mathcal{D}}) } \operatorname{Isom}(\operatorname{\mathcal{D}}) \]

of Construction 4.5.2.1. Under this identification, the comparison morphism $\varprojlim (\mathscr {F}) \rightarrow \underset {\longleftarrow }{\mathrm{holim}}(\mathscr {F})$ of Remark 7.5.2.10 corresponds to the monomorphism

\[ \operatorname{\mathcal{C}}\simeq \operatorname{\mathcal{C}}\times _{\operatorname{\mathcal{D}}} \operatorname{\mathcal{D}}\hookrightarrow \operatorname{\mathcal{C}}\times _{\operatorname{\mathcal{D}}}^{\mathrm{h}} \operatorname{\mathcal{D}} \]

of Proposition 3.4.0.7. This morphism is usually not an isomorphism of simplicial sets, though it is always an equivalence of $\infty $-categories (Proposition 7.5.2.11).

Example 7.5.2.15. Let $\operatorname{\mathcal{K}}$ be the partially ordered set depicted in the diagram

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

and suppose we are given a functor $\mathscr {F}: \operatorname{\mathcal{K}}\rightarrow \operatorname{QCat}$, which we depict as a diagram of $\infty $-categories

\[ \operatorname{\mathcal{C}}_0 \xrightarrow {T_0} \operatorname{\mathcal{C}}\xleftarrow {T_1} \operatorname{\mathcal{C}}_1. \]

Then the homotopy limit $ \underset {\longleftarrow }{\mathrm{holim}}(\mathscr {F})$ can be identified with the iterated homotopy pullback $\operatorname{\mathcal{C}}_0 \times ^{\mathrm{h}}_{\operatorname{\mathcal{C}}} (\operatorname{\mathcal{C}}_1 \times _{\operatorname{\mathcal{C}}}^{\mathrm{h}} \operatorname{\mathcal{C}})$. Applying Corollary 4.5.2.18, we see that the equivalence $\operatorname{\mathcal{C}}_1 \hookrightarrow \operatorname{\mathcal{C}}_1 \times _{\operatorname{\mathcal{C}}}^{\mathrm{h}} \operatorname{\mathcal{C}}$ of Example 7.5.2.14 induces an equivalence of $\infty $-categories

\[ \operatorname{\mathcal{C}}_0 \times ^{\mathrm{h}}_{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_1 \hookrightarrow \operatorname{\mathcal{C}}_0 \times ^{\mathrm{h}}_{\operatorname{\mathcal{C}}} (\operatorname{\mathcal{C}}_1 \times _{\operatorname{\mathcal{C}}}^{\mathrm{h}} \operatorname{\mathcal{C}}) \simeq \underset {\longleftarrow }{\mathrm{holim}}(\mathscr {F}). \]

In particular, the comparison map $\varprojlim (\mathscr {F}) \rightarrow \underset {\longleftarrow }{\mathrm{holim}}(\mathscr {F})$ is a categorical equivalence of simplicial sets if and only if the inclusion $\operatorname{\mathcal{C}}_0 \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_1 \hookrightarrow \operatorname{\mathcal{C}}_0 \times _{\operatorname{\mathcal{C}}}^{\mathrm{h}} \operatorname{\mathcal{C}}_1$ is a categorical equivalence of simplicial sets. This condition is satisfied if either $T_0$ or $T_1$ is a isofibration of $\infty $-categories (Corollary 4.5.2.22), but not in general.