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7.5.3 The Homotopy Limit as a Derived Functor

Let $\operatorname{\mathcal{C}}$ be a small category. In general, the inverse limit functor $\varprojlim : \operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{QCat}) \rightarrow \operatorname{Set_{\Delta }}$ does not respect categorical equivalence: that is, if $\alpha : \mathscr {F} \rightarrow \mathscr {G}$ is a levelwise categorical equivalence of diagrams $\mathscr {F}, \mathscr {G}: \operatorname{\mathcal{C}}\rightarrow \operatorname{QCat}$, then the induced map $\varprojlim (\alpha ): \varprojlim (\mathscr {F} ) \rightarrow \varprojlim ( \mathscr {G} )$ need not be a categorical equivalence of simplicial sets. In §7.5.2 and §4.5.6, we discussed two different ways of addressing this point:

  • We can replace the limit $\varprojlim (\mathscr {F})$ by the homotopy limit $ \underset {\longleftarrow }{\mathrm{holim}}(\mathscr {F} )$ of Construction 7.5.2.1. If $\alpha : \mathscr {F} \rightarrow \mathscr {G}$ is a levelwise categorical equivalence of diagrams $\mathscr {F}, \mathscr {G}$, then Remark 7.5.2.5 guarantees that the induced map $ \underset {\longleftarrow }{\mathrm{holim}}(\alpha ): \underset {\longleftarrow }{\mathrm{holim}}(\mathscr {F} ) \rightarrow \underset {\longleftarrow }{\mathrm{holim}}(\mathscr {G})$ is an equivalence of $\infty $-categories.

  • We can restrict our attention to isofibrant diagrams of $\infty $-categories (Definition 4.5.6.3). If $\alpha : \mathscr {F} \rightarrow \mathscr {G}$ is a levelwise categorical equivalence between isofibrant diagrams, then Corollary 4.5.6.14 guarantees that the induced map $\varprojlim (\alpha ): \varprojlim (\mathscr {F}) \rightarrow \varprojlim (\mathscr {G})$ is an equivalence of $\infty $-categories.

In this section, we will show that these perspectives are closely related: if $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{QCat}$ is a diagram of $\infty $-categories, then the homotopy limit $ \underset {\longleftarrow }{\mathrm{holim}}( \mathscr {F} )$ can be identified with the limit of an isofibrant replacement for $\mathscr {F}$. More precisely, we show that there exists a canonical isomorphism $ \underset {\longleftarrow }{\mathrm{holim}}( \mathscr {F} ) \simeq \varprojlim ( \mathscr {F}^{+} )$, where $\mathscr {F}^{+}: \operatorname{\mathcal{C}}\rightarrow \operatorname{QCat}$ is an isofibrant diagram of simplicial sets equipped with a levelwise categorical equivalence $\alpha : \mathscr {F} \hookrightarrow \mathscr {F}^{+}$ (Construction 7.5.3.3 and Proposition 7.5.3.7). Moreover, we show that for any isofibrant diagram $\mathscr {G}: \operatorname{\mathcal{C}}\rightarrow \operatorname{QCat}$, the inclusion map $\varprojlim (\mathscr {G} ) \hookrightarrow \underset {\longleftarrow }{\mathrm{holim}}(\mathscr {G})$ is an equivalence of $\infty $-categories (Proposition 7.5.3.12). Consequently, if $\beta : \mathscr {F} \rightarrow \mathscr {G}$ is any levelwise categorical equivalence from $\mathscr {F}$ to an isofibrant diagram $\mathscr {G}$, then the maps

\[ \underset {\longleftarrow }{\mathrm{holim}}( \mathscr {F} ) \xrightarrow { \underset {\longleftarrow }{\mathrm{holim}}(\beta ) } \underset {\longleftarrow }{\mathrm{holim}}( \mathscr {G} ) \leftarrow \varprojlim ( \mathscr {G} ) \]

are equivalences of $\infty $-categories; in particular, the $\infty $-categories $ \underset {\longleftarrow }{\mathrm{holim}}( \mathscr {F} )$ and $\varprojlim (\mathscr {G} )$ are equivalent (see Remark 7.5.3.15).

We begin with some elementary observations.

Proposition 7.5.3.1. Let $\operatorname{\mathcal{C}}$ be a small category and let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$ be an isofibration of $\infty $-categories. Then the weak transport representation

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

of Construction 5.3.1.1 is an isofibrant diagram of simplicial sets.

Proof. Let $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{Set_{\Delta }}$ be a functor and let $\mathscr {F}_0 \subseteq \mathscr {F}$ be a subfunctor for which the inclusion $\mathscr {F}_0 \hookrightarrow \mathscr {F}$ is a levelwise categorical equivalence. We wish to show that every natural transformation $\alpha _0: \mathscr {F}_0 \rightarrow \operatorname{wTr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}}$ admits an extension $\alpha : \mathscr {F} \rightarrow \operatorname{wTr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}}$. Using Corollary 5.3.2.23, we can reformulate this as a lifting problem

\[ \xymatrix@R =50pt@C=50pt{ \underset { \longrightarrow }{\mathrm{holim}}( \mathscr {F}_0 ) \ar [r] \ar [d] & \operatorname{\mathcal{E}}\ar [d]^{U} \\ \underset { \longrightarrow }{\mathrm{holim}}( \mathscr {F} ) \ar [r] \ar@ {-->}[ur] & \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) } \]

in the category of simplicial sets. Since $U$ is an isofibration, we are reduced to showing that the inclusion map $ \underset { \longrightarrow }{\mathrm{holim}}( \mathscr {F}_0 ) \hookrightarrow \underset { \longrightarrow }{\mathrm{holim}}( \mathscr {F} )$ is a categorical equivalence (Proposition 4.5.5.1), which is a special case of Variant 5.3.2.19. $\square$

Corollary 7.5.3.2. Let $\operatorname{\mathcal{C}}$ be a small category and let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$ be a cocartesian fibration of $\infty $-categories. Then the strict transport representation

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

of Construction 5.3.1.5 is an isofibrant diagram of simplicial sets.

Proof. Let $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{Set_{\Delta }}$ be a functor and let $\mathscr {F}_0 \subseteq \mathscr {F}$ be a subfunctor for which the inclusion $\mathscr {F}_0 \hookrightarrow \mathscr {F}$ is a levelwise categorical equivalence. Suppose we are given a natural transformation $\alpha _0: \mathscr {F}_0 \rightarrow \operatorname{sTr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}}$. It follows from Proposition 7.5.3.1 that $\alpha _0$ can be extended to a natural transformation $\alpha : \mathscr {F} \rightarrow \operatorname{wTr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}}$. To complete the proof, it will suffice to show that $\alpha $ factors through the subfunctor $\operatorname{sTr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}}$. Equivalently, we must show that for each object $C \in \operatorname{\mathcal{C}}$, the lifting problem

\[ \xymatrix@R =50pt@C=50pt{ \mathscr {F}_0(C) \ar [r] \ar [d] & \operatorname{sTr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}}(C) \ar [d] \\ \mathscr {F}(C) \ar [r] \ar@ {-->}[ur] & \operatorname{wTr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}}(C) } \]

admits a (unique) solution. This is clear, since the left vertical map is a categorical equivalence and $\operatorname{sTr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}}(C)$ is a replete subcategory of $\operatorname{wTr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}}(C)$ (see Remark 5.3.1.15). $\square$

Construction 7.5.3.3 (Explicit Isofibrant Replacement). Let $\operatorname{\mathcal{C}}$ be a small category and 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 $\mathscr {F}$-weighted nerve of $\operatorname{\mathcal{C}}$ (Definition 5.3.3.1), and let $\mathscr {F}^{+} = \operatorname{sTr}_{ \operatorname{N}_{\bullet }^{\mathscr {F}}(\operatorname{\mathcal{C}}) / \operatorname{\mathcal{C}}}$ denote the strict transport representation of the projection map $\operatorname{N}_{\bullet }^{\mathscr {F}}(\operatorname{\mathcal{C}}) \rightarrow \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$. It follows from Remark 5.3.4.10 that there is a unique natural transformation $\alpha : \mathscr {F} \rightarrow \mathscr {F}^{+}$ for which the diagram of simplicial sets

\[ \xymatrix@R =50pt@C=50pt{ & \underset { \longrightarrow }{\mathrm{holim}}( \mathscr {F}^{+} ) \ar [dr]^{ \lambda _{u} } & \\ \underset { \longrightarrow }{\mathrm{holim}}( \mathscr {F} ) \ar [ur]^{ \underset { \longrightarrow }{\mathrm{holim}}(\alpha ) } \ar [rr]^{ \lambda _{t} } & & \operatorname{N}_{\bullet }^{\mathscr {F}}(\operatorname{\mathcal{C}}), } \]

is commutative, where $\lambda _{u}$ denotes the universal scaffold of Construction 5.3.4.7 and $\lambda _{t}$ denotes the taut scaffold of Construction 5.3.4.11. We will refer to $\mathscr {F}^{+}$ as the isofibrant replacement of $\mathscr {F}$.

Proposition 7.5.3.4. Let $\operatorname{\mathcal{C}}$ be a small category, let $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{QCat}$ be a diagram of $\infty $-categories, and let $\alpha : \mathscr {F} \rightarrow \mathscr {F}^{+}$ be the natural transformation of Construction 7.5.3.3. Then $\mathscr {F}^{+}: \operatorname{\mathcal{C}}\rightarrow \operatorname{QCat}$ is an isofibrant diagram, and $\alpha $ is a levelwise categorical equivalence. Moreover, $\alpha $ is also a monomorphism.

Proof. It follows from Corollary 7.5.3.2 that the diagram $\mathscr {F}^{+}$ is isofibrant. To see that $\alpha $ is a monomorphism, we observe that for each object $C \in \operatorname{\mathcal{C}}$, the functor

\[ \alpha _{C}: \mathscr {F}(C) \rightarrow \mathscr {F}^{+}(C) = \operatorname{Fun}^{ \operatorname{CCart}}_{ / \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) }( \operatorname{N}_{\bullet }( \operatorname{\mathcal{C}}_{C/ }), \operatorname{N}_{\bullet }^{\mathscr {F}}(\operatorname{\mathcal{C}})) \]

has a left inverse, given by the evaluation map

\[ \operatorname{ev}_{C}: \operatorname{Fun}^{ \operatorname{CCart}}_{ / \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) }( \operatorname{N}_{\bullet }( \operatorname{\mathcal{C}}_{C/ }), \operatorname{N}_{\bullet }^{\mathscr {F}}(\operatorname{\mathcal{C}})) \rightarrow \{ C\} \times _{ \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) } \operatorname{N}_{\bullet }^{\mathscr {F}}(\operatorname{\mathcal{C}}) \simeq \mathscr {F}(C). \]

Since $\operatorname{ev}_{C}$ is a trivial Kan fibration (Proposition 5.3.1.7), it follows that $\alpha _{C}$ is an equivalence of $\infty $-categories. $\square$

Corollary 7.5.3.5 (Existence of Isofibrant Replacements). Let $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{Set_{\Delta }}$ be a diagram of simplicial sets. Then there exists a monomorphism of diagrams $\alpha : \mathscr {F} \hookrightarrow \mathscr {G}$, where $\alpha $ is a levelwise categorical equivalence and $\mathscr {G}: \operatorname{\mathcal{C}}\rightarrow \operatorname{QCat}$ is an isofibrant diagram of $\infty $-categories.

Proof. Using Proposition 4.1.3.2, we can reduce to the case where $\mathscr {F}$ is a diagram of $\infty $-categories. In this case, we can take $\alpha $ to be the natural transformation $\mathscr {F} \hookrightarrow \mathscr {F}^{+}$ of Construction 7.5.3.3 (Proposition 7.5.3.4). $\square$

Variant 7.5.3.6. Let $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{Set_{\Delta }}$ be a diagram of simplicial sets. Then there exists a monomorphism of diagrams $\alpha : \mathscr {F} \hookrightarrow \mathscr {G}$, where $\alpha $ is a levelwise weak homotopy equivalence and $\mathscr {G}: \operatorname{\mathcal{C}}\rightarrow \operatorname{Kan}$ is an isofibrant diagram of Kan complexes.

Proof. Using Proposition 3.1.7.1, we can reduce to the case where $\mathscr {F}$ is a diagram of Kan complexes. In this case, we can again take $\alpha $ to be the natural transformation $\mathscr {F} \hookrightarrow \mathscr {F}^{+}$ of Construction 7.5.3.3 (since $\alpha $ is a levelwise categorical equivalence, it follows that $\mathscr {F}^{+}$ is also a diagram of Kan complexes: see Remark 4.5.1.21). $\square$

Proposition 7.5.3.7. Let $\operatorname{\mathcal{C}}$ be a small category, let $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{QCat}$ be a diagram, and let $\mathscr {F}^{+}: \operatorname{\mathcal{C}}\rightarrow \operatorname{QCat}$ be the isofibrant replacement of Construction 7.5.3.3. Then there is a canonical isomorphism of simplicial sets $\theta : \underset {\longleftarrow }{\mathrm{holim}}( \mathscr {F} ) \xrightarrow {\sim } \varprojlim (\mathscr {F}^{+} )$, which is characterized by the following requirement: for each object $C \in \operatorname{\mathcal{C}}$, the composition

\begin{eqnarray*} \operatorname{Fun}_{ / \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) }^{\operatorname{CCart}}( \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}), \operatorname{N}_{\bullet }^{\mathscr {F}}(\operatorname{\mathcal{C}}) ) & = & \underset {\longleftarrow }{\mathrm{holim}}( \mathscr {F} ) \\ & \xrightarrow {\theta } & \varprojlim ( \mathscr {F}^{+} ) \\ & \rightarrow & \mathscr {F}^{+}(C) \\ & = & \operatorname{Fun}_{ / \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) }^{\operatorname{CCart}}( \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}_{C/}), \operatorname{N}_{\bullet }^{\mathscr {F}}(\operatorname{\mathcal{C}}) ) \end{eqnarray*}

is given by precomposition with the projection map $\operatorname{\mathcal{C}}_{C/} \rightarrow \operatorname{\mathcal{C}}$.

Proposition 7.5.3.7 is a consequence of the following concrete assertion:

Lemma 7.5.3.8. Let $\operatorname{\mathcal{C}}$ be a category. Then the collection of projection maps $\{ \operatorname{N}_{\bullet }( \operatorname{\mathcal{C}}_{C/} ) \rightarrow \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) \} _{ C \in \operatorname{\mathcal{C}}}$ exhibit $\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$ as the colimit of the diagram

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

Proof. Fix an integer $n \geq 0$; we wish to show that the canonical map

\[ \rho : \varinjlim _{C \in \operatorname{\mathcal{C}}^{\operatorname{op}} } \operatorname{N}_{n}( \operatorname{\mathcal{C}}_{C/} ) \rightarrow \operatorname{N}_{n}( \operatorname{\mathcal{C}}) \]

is an isomorphism in the category of sets. Let $\sigma $ be an $n$-simplex of $\operatorname{N}_{n}(\operatorname{\mathcal{C}})$, given by a diagram

\[ X_0 \rightarrow X_1 \rightarrow \cdots \rightarrow X_ n \]

in the category $\operatorname{\mathcal{C}}$. Then the fiber $\rho ^{-1} \{ \sigma \} $ can be identified with the colimit

\[ \varinjlim _{C \in \operatorname{\mathcal{C}}^{\operatorname{op}} } \operatorname{Hom}_{\operatorname{\mathcal{C}}}( C, X_0 ), \]

formed in the category of sets. This colimit consists of a single element, represented by the identity morphism $\operatorname{id}_{ X_0} \in \operatorname{Hom}_{\operatorname{\mathcal{C}}}( X_0, X_0)$. $\square$

Remark 7.5.3.9. Let $\operatorname{\mathcal{C}}$ be a category, let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$ is a morphism of simplicial sets, and let $\operatorname{wTr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}}: \operatorname{\mathcal{C}}\rightarrow \operatorname{Set_{\Delta }}$ denote the weak transport representation of Construction 5.3.1.1, given on objects by the formula $\operatorname{wTr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}}(C) = \operatorname{Fun}_{ / \operatorname{N}_{\bullet }( \operatorname{\mathcal{C}}) }( \operatorname{N}_{\bullet }( \operatorname{\mathcal{C}}_{C/} ), \operatorname{\mathcal{E}})$. Then Lemma 7.5.3.8 supplies a canonical isomorphism of simplicial sets

\[ \operatorname{Fun}_{ / \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) }( \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}), \operatorname{\mathcal{E}}) \xrightarrow {\sim } \varprojlim ( \operatorname{wTr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}} ). \]

Variant 7.5.3.10. Let $\operatorname{\mathcal{C}}$ be a category, let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$ be a cocartesian fibration of $\infty $-categories, and let $\operatorname{sTr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}}: \operatorname{\mathcal{C}}\rightarrow \operatorname{Set_{\Delta }}$ denote the strict transport representation of Construction 5.3.1.5, given on objects by the formula $\operatorname{sTr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}}(C) = \operatorname{Fun}^{\operatorname{CCart}}_{ / \operatorname{N}_{\bullet }( \operatorname{\mathcal{C}}) }( \operatorname{N}_{\bullet }( \operatorname{\mathcal{C}}_{C/} ), \operatorname{\mathcal{E}})$. Then the isomorphism of Remark 7.5.3.9 restricts to an isomorphism of simplicial sets

\[ \operatorname{Fun}^{\operatorname{CCart}}_{ / \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) }( \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}), \operatorname{\mathcal{E}}) \xrightarrow {\sim } \varprojlim ( \operatorname{sTr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}} ). \]

Proof of Proposition 7.5.3.7. Apply Variant 7.5.3.10 in the special case where $\operatorname{\mathcal{E}}= \operatorname{N}_{\bullet }^{\mathscr {F}}(\operatorname{\mathcal{C}})$ is the $\mathscr {F}$-weighted nerve of the category $\operatorname{\mathcal{C}}$. $\square$

Remark 7.5.3.11. In the situation of Proposition 7.5.3.7, the isomorphism $\theta : \underset {\longleftarrow }{\mathrm{holim}}(\mathscr {F} ) \xrightarrow {\sim } \varprojlim ( \mathscr {F}^{+} )$ fits into a commutative diagram

\[ \xymatrix@R =50pt@C=50pt{ & \underset {\longleftarrow }{\mathrm{holim}}( \mathscr {F} ) \ar [dr]^{\theta }_{\sim } & \\ \varprojlim ( \mathscr {F} ) \ar [ur]^{ \iota } \ar [rr]^{ \varprojlim ( \alpha ) } & & \varprojlim ( \mathscr {F}^{+} ), } \]

where $\iota $ is the comparison map of Remark 7.5.2.10 and $\alpha : \mathscr {F} \hookrightarrow \mathscr {F}^{+}$ is the natural transformation appearing in Construction 7.5.3.3.

Proposition 7.5.3.12. Let $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{QCat}$ be an isofibrant diagram of $\infty $-categories. Then the inclusion map $\iota : \varprojlim ( \mathscr {F} ) \hookrightarrow \underset {\longleftarrow }{\mathrm{holim}}( \mathscr {F} )$ is an equivalence of $\infty $-categories.

Proof. Let $\alpha : \mathscr {F} \hookrightarrow \mathscr {F}^{+}$ be the isofibrant replacement of Construction 7.5.3.3. By virtue of Proposition 7.5.3.7 (and Remark 7.5.3.11), it will suffice to show that the limit $\varprojlim (\alpha ): \varprojlim ( \mathscr {F} ) \hookrightarrow \varprojlim ( \mathscr {F}^{+} )$ is an equivalence of $\infty $-categories. This is a special case of Corollary 4.5.6.15, since $\alpha $ is a levelwise categorical equivalence between isofibrant diagrams (Proposition 7.5.3.4). $\square$

Example 7.5.3.13 (Towers of Isofibrations). Suppose we are given a tower of $\infty $-categories

\[ \cdots \rightarrow \operatorname{\mathcal{C}}(3) \rightarrow \operatorname{\mathcal{C}}(2) \rightarrow \operatorname{\mathcal{C}}(1) \rightarrow \operatorname{\mathcal{C}}(0), \]

which we identify with a functor $\mathscr {F}: \operatorname{\mathbf{Z}}_{\geq 0}^{\operatorname{op}} \rightarrow \operatorname{QCat}$. If each of the transition functors $\operatorname{\mathcal{C}}(n+1) \rightarrow \operatorname{\mathcal{C}}(n)$ is an isofibration, then the comparison map $\varprojlim _{n} \operatorname{\mathcal{C}}(n) = \varprojlim (\mathscr {F}) \hookrightarrow \underset {\longleftarrow }{\mathrm{holim}}(\mathscr {F})$ is an equivalence of $\infty $-categories. This follows by combining Example 4.5.6.7 with Proposition 7.5.3.12.

Warning 7.5.3.14. Let $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{QCat}$ be a strictly commutative diagram of $\infty $-categories and let $\alpha : \mathscr {F} \hookrightarrow \mathscr {F}^{+}$ denote the isofibrant replacement of Construction 7.5.3.3, and let $\theta : \underset {\longleftarrow }{\mathrm{holim}}( \mathscr {F} ) \xrightarrow {\sim } \varprojlim ( \mathscr {F}^{+} )$ be the isomorphism of Proposition 7.5.3.7. We then have a diagram of simplicial sets

\[ \xymatrix@R =50pt@C=50pt{ \varprojlim ( \mathscr {F} ) \ar [r]^-{ \varprojlim ( \alpha ) } \ar [d] & \varprojlim ( \mathscr {F}^{+} ) \ar [d] \\ \underset {\longleftarrow }{\mathrm{holim}}( \mathscr {F} ) \ar [r]_{ \underset {\longleftarrow }{\mathrm{holim}}(\alpha ) } \ar [ur]^{\theta }_{\sim } & \underset {\longleftarrow }{\mathrm{holim}}( \mathscr {F}^{+} ), } \]

where the outer square and the upper left triangle are commutative (Remark 7.5.3.11). Beware that the lower right triangle is usually not commutative. That is, $ \underset {\longleftarrow }{\mathrm{holim}}( \mathscr {F} )$ and $\varprojlim ( \mathscr {F}^{+} )$ are isomorphic when viewed as abstract simplicial sets, but do not coincide when identified with simplicial subsets of $ \underset {\longleftarrow }{\mathrm{holim}}( \mathscr {F}^{+} )$.

Remark 7.5.3.15 (The Homotopy Limit as a Right Derived Functor). The results of this section can be interpreted in the language of model categories. For every small category $\operatorname{\mathcal{C}}$, the category $\operatorname{Fun}( \operatorname{\mathcal{C}}, \operatorname{Set_{\Delta }})$ can be equipped with a model structure in which the cofibrations are monomorphisms and the weak equivalences are levelwise categorical equivalences (see Example ). The inverse limit functor

\[ \varprojlim : \operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{Set_{\Delta }}) \rightarrow \operatorname{Set_{\Delta }} \]

then admit a right derived functor $\underset {\longleftarrow }{\mathrm{Rlim}}: \operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{Set_{\Delta }}) \rightarrow \operatorname{Set_{\Delta }}$, which carries a diagram $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{Set_{\Delta }}$ to the limit of a fibrant replacement of $\mathscr {F}$. It follows from Propositions 7.5.3.4 and 7.5.3.7 that, when restricted to the subcategory $\operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{QCat}) \subset \operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{Set_{\Delta }})$, the functor $\underset {\longleftarrow }{\mathrm{Rlim}}$ is (categorically) equivalent to the homotopy limit functor $ \underset {\longleftarrow }{\mathrm{holim}}: \operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{QCat}) \rightarrow \operatorname{QCat}$ of Construction 7.5.2.1. We will return to this point in §.