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4.5.6 Isofibrant Diagrams

Let $\operatorname{\mathcal{C}}$ be a small category. Every diagram of simplicial sets $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{Set_{\Delta }}$ has a limit in the category $\operatorname{Set_{\Delta }}$, given concretely by the formula

\[ \varprojlim ( \mathscr {F} )(C)_ n = \varprojlim _{C \in \operatorname{\mathcal{C}}} \mathscr {F}(C)_{n}; \]

see Remark 1.1.1.13. Beware that, when using simplicial sets as a framework for higher category theory, this operation is badly behaved in general:

  • If each of the simplicial sets $\mathscr {F}(C)$ is an $\infty $-category, then the limit $\varprojlim (\mathscr {F})$ need not be an $\infty $-category.

  • If $\alpha : \mathscr {F} \rightarrow \mathscr {G}$ be a natural transformation between functors $\mathscr {F},\mathscr {G}: \operatorname{\mathcal{C}}\rightarrow \operatorname{Set_{\Delta }}$ which is a levelwise categorical equivalence (Definition 4.5.6.1), then the induced map $\varprojlim (\alpha ): \varprojlim (\mathscr {F} ) \rightarrow \varprojlim ( \mathscr {G} )$ need not be a categorical equivalence.

In this section, we will introduce the class of isofibrant diagrams $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{Set_{\Delta }}$ (Definition 4.5.6.3), and show that it does not suffer from these defects:

  • If $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{Set_{\Delta }}$ is an isofibrant diagram of simplicial sets, then the limit $\varprojlim (\mathscr {F})$ is an $\infty $-category (Corollary 4.5.6.11).

  • If $\alpha : \mathscr {F} \rightarrow \mathscr {G}$ is a levelwise categorical equivalence between isofibrant diagrams $\mathscr {F}, \mathscr {G}: \operatorname{\mathcal{C}}\rightarrow \operatorname{Set_{\Delta }}$, then the induced map $\varprojlim (\alpha ): \varprojlim (\mathscr {F} ) \rightarrow \varprojlim ( \mathscr {G} )$ is an equivalence of $\infty $-categories (Corollary 4.5.6.15).

We begin by introducing some terminology.

Definition 4.5.6.1. Let $\operatorname{\mathcal{C}}$ be a category and let $\alpha : \mathscr {F} \rightarrow \mathscr {G}$ be a natural transformation between diagrams $\mathscr {F}, \mathscr {G}: \operatorname{\mathcal{C}}\rightarrow \operatorname{Set_{\Delta }}$. We say that $\alpha $ is a levelwise categorical equivalence if, for every object $C \in \operatorname{\mathcal{C}}$, the induced map $\alpha _{C}: \mathscr {F}(C) \rightarrow \mathscr {G}(C)$ is a categorical equivalence of simplicial sets.

Remark 4.5.6.2. Definition 4.5.6.1 is a special case of a general convention. If $P$ is a property of morphisms of simplicial sets and $\alpha : \mathscr {F} \rightarrow \mathscr {G}$ is a natural transformation between diagrams $\mathscr {F}, \mathscr {G}: \operatorname{\mathcal{C}}\rightarrow \operatorname{Set_{\Delta }}$, then we say that $\alpha $ has the property $P$ levelwise if, for every object $C \in \operatorname{\mathcal{C}}$, the morphism of simplicial sets $\alpha _{C}: \mathscr {F}(C) \rightarrow \mathscr {G}(C)$ has the property $P$. For example, we say that $\alpha $ is a levelwise weak homotopy equivalence if, for every object $C \in \operatorname{\mathcal{C}}$, the morphism $\alpha _{C}: \mathscr {F}(C) \rightarrow \mathscr {G}(C)$ is a weak homotopy equivalence of simplicial sets.

Definition 4.5.6.3. Let $\operatorname{\mathcal{C}}$ be a small category. We say that a diagram $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{Set_{\Delta }}$ is isofibrant if it satisfies the following condition:

$(\ast )$

Let $\mathscr {E}: \operatorname{\mathcal{C}}\rightarrow \operatorname{Set_{\Delta }}$ be a functor and let $\mathscr {E}_0 \subseteq \mathscr {E}$ be a subfunctor for which the inclusion $\mathscr {E}_0 \hookrightarrow \mathscr {E}$ is a levelwise categorical equivalence. Then every natural transformation $\alpha _0: \mathscr {E}_0 \rightarrow \mathscr {F}$ admits an extension $\alpha : \mathscr {E} \rightarrow \mathscr {F}$.

Example 4.5.6.4. Let $\operatorname{\mathcal{C}}= \{ X\} $ be a category having a single object and a single morphism. Then a diagram $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{Set_{\Delta }}$ is determined by the simplicial set $\mathscr {F}(X)$. The diagram $\mathscr {F}$ is isofibrant (in the sense of Definition 4.5.6.3) if and only if the simplicial set $\mathscr {F}(X)$ is an $\infty $-category.

Remark 4.5.6.5. Let $\operatorname{\mathcal{C}}$ be a small category and $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{Set_{\Delta }}$ be an isofibrant diagram. Then, for each object $C \in \operatorname{\mathcal{C}}$, the simplicial set $\mathscr {F}(C)$ is an $\infty $-category. That is, for $0 < i < n$, every morphism of simplicial sets $\sigma _0: \Lambda ^{n}_{i} \rightarrow \mathscr {F}(C)$ can be extended to an $n$-simplex of $\mathscr {F}(C)$. This follows by applying condition $(\ast )$ of Definition 4.5.6.3 to the functor

\[ \mathscr {E}: \operatorname{\mathcal{C}}\rightarrow \operatorname{Set_{\Delta }}\quad \quad \mathscr {E}(D) = \Delta ^ n \times \operatorname{Hom}_{\operatorname{\mathcal{C}}}(C,D), \]

together with the subfunctor $\mathscr {E}_0 \subseteq \mathscr {E}$ given by $\mathscr {E}_0(D) = \Lambda ^{n}_{i} \times \operatorname{Hom}_{\operatorname{\mathcal{C}}}(C,D)$.

We now give some more interesting examples of isofibrant diagrams.

Proposition 4.5.6.6. Let $(Q, \leq )$ be a well-founded partially ordered set (see Definition 5.4.1.1). Then a diagram of simplicial sets $\mathscr {F}: Q^{\operatorname{op}} \rightarrow \operatorname{Set_{\Delta }}$ is isofibrant if and only if, for each element $q \in Q$, the map

\[ \theta _{q}: \mathscr {F}(q) \rightarrow \varprojlim _{p < q} \mathscr {F}(p) \]

is an isofibration of simplicial sets.

Example 4.5.6.7 (Isofibrant Towers). Let $\mathscr {F}: \operatorname{\mathbf{Z}}_{\geq 0}^{\operatorname{op}} \rightarrow \operatorname{Set_{\Delta }}$ be a diagram, which we identify with a tower of simplicial sets

\[ \cdots \rightarrow \mathscr {F}(3) \rightarrow \mathscr {F}(2) \rightarrow \mathscr {F}(1) \rightarrow \mathscr {F}(0). \]

Then $\mathscr {F}$ is isofibrant (in the sense of Definition 4.5.6.3) if and only if each of the simplicial sets $\mathscr {F}(n)$ is an $\infty $-category and each of the transition functors $\mathscr {F}(n+1) \rightarrow \mathscr {F}(n)$ is an isofibration of $\infty $-categories.

Example 4.5.6.8 (Isofibrant Squares). A square diagram of $\infty $-categories

\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{E}}_{01} \ar [r]^-{F'_{1}} \ar [d]^{F'_{0}} & \operatorname{\mathcal{E}}_0 \ar [d]^{F_0} \\ \operatorname{\mathcal{E}}_1 \ar [r]^-{F_1} & \operatorname{\mathcal{E}}} \]

is isofibrant (when regarded as a functor $[1] \times [1] \rightarrow \operatorname{Set_{\Delta }}$) if and only if it satisfies the following conditions:

  • The functors $F_0: \operatorname{\mathcal{E}}_0 \rightarrow \operatorname{\mathcal{E}}$ and $F_1: \operatorname{\mathcal{E}}_1 \rightarrow \operatorname{\mathcal{E}}$ are isofibrations of $\infty $-categories.

  • The functor $(F'_1, F'_0): \operatorname{\mathcal{E}}_{01} \rightarrow \operatorname{\mathcal{E}}_0 \times _{\operatorname{\mathcal{E}}} \operatorname{\mathcal{E}}_1$ is an isofibration of $\infty $-categories.

Proof of Proposition 4.5.6.6. Suppose first that $\mathscr {F}: Q^{\operatorname{op}} \rightarrow \operatorname{Set_{\Delta }}$ is an isofibrant diagram. We will show that, for each element $q \in Q$, the induced map $\theta _{q}: \mathscr {F}(q) \rightarrow \varprojlim _{p < q} \mathscr {F}(p)$ is an isofibration of simplicial sets (for this step, we will not need to assume that $Q$ is well-founded). Fix a simplicial set $B$ and a simplicial subset $A \subseteq B$ for which the inclusion map $A \hookrightarrow B$ is a categorical equivalence; we wish to show that every lifting problem

4.40
\begin{equation} \begin{gathered}\label{equation:injective-isofibration-well-founded} \xymatrix@R =50pt@C=50pt{ A \ar [d] \ar [r] & \mathscr {F}(q) \ar [d]^{\theta _{q}} \\ B \ar [r] \ar@ {-->}[ur] & \varprojlim _{p < q} \mathscr {F}(p) } \end{gathered} \end{equation}

admits a solution. Define $\mathscr {B}: Q^{\operatorname{op}} \rightarrow \operatorname{Set_{\Delta }}$ by the formula $\mathscr {B}(p) = \left\{ \begin{array}{rl} B & \textnormal{ if }p \leq q \\ \emptyset & \textnormal{ otherwise, } \end{array}\right.$ and let $\mathscr {B}_0 \subseteq \mathscr {B}$ be the subfunctor given by the formula

\[ \mathscr {B}_0(p) = \left\{ \begin{array}{rl} B & \textnormal{ if $p < q$ } \\ A & \textnormal{ if }p=q \\ \emptyset & \textnormal{ otherwise. } \end{array}\right. \]

The lifting problem (4.40) can be identified with a natural transformation of functors $\alpha _0: \mathscr {B}_0 \rightarrow \mathscr {F}$. Since the inclusion $\mathscr {B}_0 \hookrightarrow \mathscr {B}$ is a levelwise categorical equivalence and $\mathscr {F}$ is isofibrant, we can extend $\alpha _0$ to a natural transformation $\alpha : \mathscr {B} \rightarrow \mathscr {F}$, which determines a solution to the lifting problem (4.40).

Now suppose that the partially ordered set $(Q, \leq )$ is well-founded and that for each $q \in Q$, the morphism $\theta _{q}$ is an isofibration of simplicial sets. We wish to show that the diagram $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{Set_{\Delta }}$ is isofibrant. Let $\mathscr {E}: \operatorname{\mathcal{C}}\rightarrow \operatorname{Set_{\Delta }}$ be a functor, let $\mathscr {E}_0 \subseteq \mathscr {E}$ be a subfunctor for which the inclusion $\mathscr {E}_0 \hookrightarrow \mathscr {E}$ is a levelwise categorical equivalence, and let $\alpha _0: \mathscr {E}_0 \rightarrow \mathscr {F}$ be a natural transformation; we wish to show that $\alpha _0$ can be extended to a natural transformation $\alpha : \mathscr {E} \rightarrow \mathscr {F}$.

For every downward-closed subset $P \subseteq Q$, let $\mathscr {E}^{P} \subseteq \mathscr {E}$ denote the subfunctor given by $\mathscr {E}^{P}(q) = \left\{ \begin{array}{rl} \mathscr {E}(q) & \textnormal{ if }q \in P \\ \emptyset & \textnormal{otherwise,} \end{array}\right.$, and set $\mathscr {E}^{P}_{0} = \mathscr {E}^{P} \cap \mathscr {E}_0$. Let $S$ denote the collection of pairs $(P, \alpha ^{P})$, where $P \subseteq Q$ is a downward-closed subset and $\alpha ^{P}: \mathscr {E}^{P} \rightarrow \mathscr {F}$ is a natural transformation satisfying $\alpha ^{P}|_{ \mathscr {E}^{P}_{0} } = \alpha _0|_{ \mathscr {E}^{P}_{0} }$. We regard $S$ as a partially ordered set, where $(P, \alpha ^{P}) \leq (P', \alpha ^{P'} )$ if $P$ is contained in $P'$ and $\alpha ^{P}$ is equal to the restriction $\alpha ^{P'}|_{ \mathscr {E}^{P}}$. The partially ordered set $S$ satisfies the hypotheses of Zorn's lemma, and therefore contains a maximal element $(P, \alpha ^{P})$. To complete the proof, it will suffice to show that $P = Q$, so that $\alpha ^{P}: \mathscr {E} \rightarrow \mathscr {F}$ is an extension of $\alpha _0$. Assume otherwise. Since $Q$ is well-founded, the complement $Q \setminus P$ contains a minimal element $q$. Set $P' = P \cup \{ q\} $. Since $\theta _{q}$ is an isofibration of simplicial sets, the lifting problem

\[ \xymatrix@R =50pt@C=50pt{ \mathscr {E}_0(q) \ar [rr]^{\alpha _0} \ar [d] & & \mathscr {F}(q) \ar [d]^{\theta _{q}} \\ \mathscr {E}(q) \ar [r] \ar@ {-->}[urr] & \varprojlim _{p < q} \mathscr {E}(p) \ar [r]^-{\alpha ^{P} } & \varprojlim _{ p < q} \mathscr {F}(p) } \]

admits a solution in the category of simplicial sets. This solution determines a natural transformation $\alpha ^{P'}: \mathscr {E}^{P'} \rightarrow \mathscr {F}$ satisfying $\alpha ^{P'}|_{ \mathscr {E}^{P} } = \alpha ^{P}$ and $\alpha ^{P'}|_{ \mathscr {E}^{P'}_{0} } = \alpha _{0} |_{ \mathscr {E}^{P'}_{0} }$, contradicting the maximality of the pair $(P, \alpha ^{P})$. $\square$

We now record some useful properties of isofibrant diagrams of simplicial sets. Fix a small category $\operatorname{\mathcal{C}}$, and let us regard $\operatorname{Fun}( \operatorname{\mathcal{C}}, \operatorname{Set_{\Delta }})$ as equipped with the simplicial enrichment described in Example 2.4.2.2. For every simplicial set $K$, we let $\underline{K}$ denote the constant functor $\operatorname{\mathcal{C}}\rightarrow \operatorname{Set_{\Delta }}$ taking the value $K$.

Proposition 4.5.6.9. Let $\operatorname{\mathcal{C}}$ be a small category and let $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{Set_{\Delta }}$ be an isofibrant diagram of simplicial sets. For every functor $\mathscr {E}: \operatorname{\mathcal{C}}\rightarrow \operatorname{Set_{\Delta }}$ and every subfunctor $\mathscr {E}_0 \subseteq \mathscr {E}$, the restriction map

\[ \theta : \operatorname{Hom}_{ \operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{Set_{\Delta }})}( \mathscr {E}, \mathscr {F} )_{\bullet } \rightarrow \operatorname{Hom}_{ \operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{Set_{\Delta }}) }( \mathscr {E}_0, \mathscr {F} )_{\bullet } \]

is an isofibration of simplicial sets. If the inclusion $\mathscr {E}_0 \hookrightarrow \mathscr {E}$ is a levelwise categorical equivalence, then $\theta $ is a trival Kan fibration.

Proof. Let $B$ be a simplicial set and let $A \subseteq B$ be a simplicial subset. We wish to show that every lifting problem

4.41
\begin{equation} \begin{gathered}\label{equation:isofibrant-mapping-space} \xymatrix@R =50pt@C=50pt{ A \ar [r] \ar [d] & \operatorname{Hom}_{ \operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{Set_{\Delta }}) }( \mathscr {E}, \mathscr {F})_{\bullet } \ar [d]^{\theta } \\ B \ar [r] \ar@ {-->}[ur] & \operatorname{Hom}_{ \operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{Set_{\Delta }}) }( \mathscr {E}_0, \mathscr {F})_{\bullet } } \end{gathered} \end{equation}

admits a solution, provided that either the inclusion map $A \hookrightarrow B$ is a categorical equivalence or the inclusion $\mathscr {E}_0 \hookrightarrow \mathscr {E}$ is a levelwise categorical equivalence. Unwinding the definitions, we see that the diagram (4.41) determines a natural transformation

\[ \alpha _0: ( \underline{A} \times \mathscr {E} ) \coprod _{ ( \underline{A} \times \mathscr {E}_0 ) } ( \underline{B} \times \mathscr {E}_0) \rightarrow \mathscr {F}, \]

and that solutions to (4.41) can be identified with extensions of $\alpha _0$ to a natural transformation $\alpha : \underline{B} \times \mathscr {E} \rightarrow \mathscr {F}$. By virtue of our assumption that $\mathscr {F}$ is isofibrant, we are reduced to proving that the inclusion map

\[ ( \underline{A} \times \mathscr {E} ) \coprod _{ ( \underline{A} \times \mathscr {E}_0 ) } ( \underline{B} \times \mathscr {E}_0) \hookrightarrow \underline{B} \times \mathscr {E} \]

is a levelwise categorical equivalence, which follows from Corollary 4.5.4.15. $\square$

Corollary 4.5.6.10. Let $\operatorname{\mathcal{C}}$ be a small category and let $\mathscr {E}, \mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{Set_{\Delta }}$ be diagrams of simplicial sets. If $\mathscr {F}$ is isofibrant, then the simplicial set $\operatorname{Hom}_{ \operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{Set_{\Delta }}) }( \mathscr {E}, \mathscr {F} )_{\bullet }$ is an $\infty $-category.

Proof. Apply Proposition 4.5.6.9 in the special case $\mathscr {E}_0 = \emptyset $. $\square$

Corollary 4.5.6.11. Let $\operatorname{\mathcal{C}}$ be a small category and let $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{Set_{\Delta }}$ be an isofibrant diagram of simplicial sets. Then the limit $\varprojlim ( \mathscr {F} )$ is an $\infty $-category.

Proof. Apply Corollary 4.5.6.10 in the special case $\mathscr {E} = \underline{ \Delta ^{0} }$. $\square$

Proposition 4.5.6.12. Let $\operatorname{\mathcal{C}}$ be a small category, let $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{Set_{\Delta }}$ be an isofibrant diagram, and let $\alpha : \mathscr {E} \rightarrow \mathscr {E}'$ be a natural transformation between diagrams $\mathscr {E}, \mathscr {E}': \operatorname{\mathcal{C}}\rightarrow \operatorname{Set_{\Delta }}$. If $\alpha $ is a levelwise categorical equivalence, then precomposition with $\alpha $ induces an equivalence of $\infty $-categories

\[ \operatorname{Hom}_{\operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{Set_{\Delta }}) }( \mathscr {E}', \mathscr {F})_{\bullet } \rightarrow \operatorname{Hom}_{\operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{Set_{\Delta }}) }( \mathscr {E}, \mathscr {F})_{\bullet }. \]

Proof. Using Exercise 3.1.7.10, we can choose a contractible Kan complex $X$ containing a pair of vertices $x,y \in X$ with $x \neq y$. Evaluation at the vertices $x$ and $y$ determine trivial Kan fibrations of $\infty $-categories

\[ \operatorname{ev}_{x}, \operatorname{ev}_{y}: \operatorname{Fun}(X, \operatorname{Hom}_{\operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{Set_{\Delta }}) }( \mathscr {E}, \mathscr {F})_{\bullet } ) \rightarrow \operatorname{Hom}_{\operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{Set_{\Delta }}) }( \mathscr {E}, \mathscr {F})_{\bullet }. \]

Form a pullback diagram

\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{D}}\ar [r]^-{ T } \ar [d]^{U} & \operatorname{Fun}(X, \operatorname{Hom}_{\operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{Set_{\Delta }}) }( \mathscr {E}, \mathscr {F})_{\bullet } ) \ar [d]^{ \operatorname{ev}_{x} } \\ \operatorname{Hom}_{\operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{Set_{\Delta }}) }( \mathscr {E}', \mathscr {F})_{\bullet } \ar [r]^-{ \circ \alpha } & \operatorname{Hom}_{\operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{Set_{\Delta }}) }( \mathscr {E}, \mathscr {F})_{\bullet },} \]

so that $U$ is also a trivial Kan fibration and therefore an equivalence of $\infty $-categories. It will therefore suffice to show that $\operatorname{ev}_{x} \circ T$ is an equivalence of $\infty $-categories. Since the functors $\operatorname{ev}_{x}$ and $\operatorname{ev}_{y}$ are isomorphic, this is equivalent to the requirement that $\operatorname{ev}_{y} \circ T$ is an equivalence of $\infty $-categories. In fact, the functor $\operatorname{ev}_{y} \circ T$ is a trivial Kan fibration: this follows by applying Proposition 4.5.6.9 to the levelwise categorical equivalence

\[ \underline{ \{ y\} } \times \mathscr {E} \hookrightarrow ( \underline{X} \times \mathscr {E} ) \coprod _{ ( \underline{ \{ x\} } \times \mathscr {E} ) } \mathscr {E}'. \]
$\square$

Corollary 4.5.6.13. Let $\operatorname{\mathcal{C}}$ be a small category and let $\alpha : \mathscr {E} \rightarrow \mathscr {F}$ be a levelwise categorical equivalence of isofibrant diagrams $\mathscr {E}, \mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{Set_{\Delta }}$. Then $\alpha $ admits a homotopy inverse: that is, there is a natural transformation $\beta : \mathscr {F} \rightarrow \mathscr {E}$ such that $\alpha \circ \beta $ and $\beta \circ \alpha $ are isomorphic to $\operatorname{id}_{ \mathscr {F} }$ and $\operatorname{id}_{ \mathscr {E} }$ as objects of the $\infty $-categories $\operatorname{Hom}_{ \operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{Set_{\Delta }})} ( \mathscr {F}, \mathscr {F} )_{\bullet }$ and $\operatorname{Hom}_{ \operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{Set_{\Delta }})} ( \mathscr {E}, \mathscr {E} )_{\bullet }$, respectively.

Proof. Since $\mathscr {E}$ is isofibrant, Proposition 4.5.6.12 guarantees that the functor

\[ \operatorname{Hom}_{ \operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{Set_{\Delta }})} ( \mathscr {F}, \mathscr {E} )_{\bullet } \xrightarrow { \circ \alpha } \operatorname{Hom}_{ \operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{Set_{\Delta }})} ( \mathscr {E}, \mathscr {E} )_{\bullet } \]

is an equivalence of $\infty $-categories. In particular, there exists a natural transformation $\beta : \mathscr {F} \rightarrow \mathscr {E}$ such that $\beta \circ \alpha $ is isomorphic to $\operatorname{id}_{ \mathscr {E} }$ (when viewed as an object of the $\infty $-category $\operatorname{Hom}_{ \operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{Set_{\Delta }})} ( \mathscr {E}, \mathscr {E} )_{\bullet }$). To complete the proof, it will suffice to show that $\beta $ is also a right homotopy inverse to $\alpha $: that is, the composition $\alpha \circ \beta $ is isomorphic to $\operatorname{id}_{ \mathscr {F}}$ (when viewed as an object of the $\infty $-category $\operatorname{Hom}_{ \operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{Set_{\Delta }})} ( \mathscr {F}, \mathscr {F} )_{\bullet }$).

For each object $C \in \operatorname{\mathcal{C}}$, the functor $\beta _{C}: \mathscr {F}(C) \rightarrow \mathscr {E}(C)$ is a left homotopy inverse of the functor $\alpha _{C}: \mathscr {E}(C) \rightarrow \mathscr {F}(C)$. Since $\alpha _ C$ is an equivalence of $\infty $-categories, it follows that $\beta _{C}$ is also an equivalence of $\infty $-categories. Allowing $C$ to vary, we conclude that $\beta $ is a levelwise categorical equivalence. We can therefore repeat the preceding argument to obtain a natural transformation $\gamma : \mathscr {E} \rightarrow \mathscr {F}$ such that $\gamma \circ \beta $ is isomorphic to $\operatorname{id}_{ \mathscr {F} }$. We then have isomorphisms

\[ \alpha \simeq (\gamma \circ \beta ) \circ \alpha = \gamma \circ (\beta \circ \alpha ) \simeq \gamma \]

in the $\infty $-category $\operatorname{Hom}_{ \operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{Set_{\Delta }})} ( \mathscr {E}, \mathscr {F} )_{\bullet }$, so that $\alpha \circ \beta $ is also isomorphic to $\operatorname{id}_{ \mathscr {F}}$. $\square$

Corollary 4.5.6.14. Let $\operatorname{\mathcal{C}}$ be a small category, let $\mathscr {E}: \operatorname{\mathcal{C}}\rightarrow \operatorname{Set_{\Delta }}$ be a diagram of simplicial sets, and let $\alpha : \mathscr {F} \rightarrow \mathscr {G}$ be a levelwise categorical equivalence between diagrams $\mathscr {F}, \mathscr {G}: \operatorname{\mathcal{C}}\rightarrow \operatorname{Set_{\Delta }}$. If $\mathscr {F}$ and $\mathscr {G}$ are isofibrant, then composition with $\alpha $ induces an equivalence of $\infty $-categories

\[ \operatorname{Hom}_{\operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{Set_{\Delta }}) }( \mathscr {E}, \mathscr {F})_{\bullet } \rightarrow \operatorname{Hom}_{\operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{Set_{\Delta }}) }( \mathscr {E}, \mathscr {G})_{\bullet }. \]

Corollary 4.5.6.15. Let $\operatorname{\mathcal{C}}$ be a small category, and let $\alpha : \mathscr {F} \rightarrow \mathscr {G}$ be a levelwise categorical equivalence between isofibrant diagrams $\mathscr {F}, \mathscr {G}: \operatorname{\mathcal{C}}\rightarrow \operatorname{Set_{\Delta }}$. Then the induced map $\varprojlim (\alpha ): \varprojlim ( \mathscr {F} ) \rightarrow \varprojlim ( \mathscr {G} )$ is an equivalence of $\infty $-categories.

Proof. Apply Corollary 4.5.6.14 in the special case $\mathscr {E} = \underline{ \Delta ^{0} }$. $\square$

Example 4.5.6.16. Suppose we are given a commutative diagram of $\infty $-categories

\[ \xymatrix@R =50pt@C=50pt{ \cdots \ar [r] & \operatorname{\mathcal{C}}(3) \ar [d] \ar [r] & \operatorname{\mathcal{C}}(2) \ar [r] \ar [d] & \operatorname{\mathcal{C}}(1) \ar [r] \ar [d] & \operatorname{\mathcal{C}}(0) \ar [d] \\ \cdots \ar [r] & \operatorname{\mathcal{D}}(3) \ar [r] & \operatorname{\mathcal{D}}(2) \ar [r] & \operatorname{\mathcal{D}}(1) \ar [r] & \operatorname{\mathcal{D}}(0), } \]

where the horizontal maps are isofibrations and the vertical maps are equivalences of $\infty $-categories. Then the induced map $\varprojlim \operatorname{\mathcal{C}}(n) \rightarrow \varprojlim \operatorname{\mathcal{D}}(n)$ is an equivalence of $\infty $-categories. This follows by combining Example 4.5.6.7, Corollary 4.5.6.11, and Corollary 4.5.6.15.

Proposition 4.5.6.17. Let $\operatorname{\mathcal{C}}$ be a small category and let $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{Set_{\Delta }}$ be an isofibrant diagram. Suppose that, for every object $C \in \operatorname{\mathcal{C}}$, the simplicial set $\mathscr {F}(C)$ is a Kan complex. Then, for every diagram $\mathscr {E}: \operatorname{\mathcal{C}}\rightarrow \operatorname{Set_{\Delta }}$, the simplicial set $X = \operatorname{Hom}_{ \operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{Set_{\Delta }}) }( \mathscr {E}, \mathscr {F} )_{\bullet }$ is a Kan complex.

Proof. By virtue of Corollary 4.5.6.10, the simplicial set $X$ is an $\infty $-category. Define $\mathscr {F}^{\Delta ^{1}}: \operatorname{\mathcal{C}}\rightarrow \operatorname{Set_{\Delta }}$ by the formula $\mathscr {F}^{ \Delta ^{1} }(C) = \operatorname{Fun}( \Delta ^1, \mathscr {F}(C) )$. Then $\mathscr {F}^{ \Delta ^1 }$ is also an isofibrant diagram. Moreover, our assumption that each $\mathscr {F}(C)$ is a Kan complex guarantees that the diagonal embedding $\mathscr {F} \hookrightarrow \mathscr {F}^{\Delta ^1}$ is a levelwise categorical equivalence. Applying Corollary 4.5.6.14, we deduce that the diagonal map $X \hookrightarrow \operatorname{Fun}( \Delta ^1, X)$ is an equivalence of $\infty $-categories. In particular, every morphism of $X$ is isomorphic (as an object of the $\infty $-category $\operatorname{Fun}( \Delta ^1, X)$ ) to an identity morphism, and is therefore an isomorphism (Example 4.4.1.13). Applying Proposition 4.4.2.1, we deduce that $X$ is a Kan complex. $\square$

Corollary 4.5.6.18. Let $\operatorname{\mathcal{C}}$ be a small category and let $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{Set_{\Delta }}$ be an isofibrant diagram. Suppose that, for every object $C \in \operatorname{\mathcal{C}}$, the simplicial set $\mathscr {F}(C)$ is a Kan complex. Then the simplicial set $\varprojlim ( \mathscr {F} )$ is a Kan complex.

Proof. Apply Proposition 4.5.6.17 in the special case $\mathscr {E} = \underline{ \Delta ^{0} }$. $\square$

Corollary 4.5.6.19. Let $\operatorname{\mathcal{C}}$ be a small category, let $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{Set_{\Delta }}$ be an isofibrant diagram, and define $\mathscr {F}^{\simeq }: \operatorname{\mathcal{C}}\rightarrow \operatorname{Set_{\Delta }}$ by the formula $\mathscr {F}^{\simeq }(C) = \mathscr {F}(C)^{\simeq }$. Then $\mathscr {F}^{\simeq }$ is also an isofibrant diagram. Moreover, the inclusion map $\varprojlim ( \mathscr {F}^{\simeq } ) \hookrightarrow \varprojlim ( \mathscr {F} )$ restricts to an isomorphism of $\varprojlim ( \mathscr {F}^{\simeq } )$ with the core of the $\infty $-category $\varprojlim ( \mathscr {F} )$.

Proof. We first show that the diagram $\mathscr {F}^{\simeq }$ is isofibrant. Let $\mathscr {E}: \operatorname{\mathcal{C}}\rightarrow \operatorname{Set_{\Delta }}$ be a functor and let $\mathscr {E}_0 \subseteq \mathscr {E}$ be a subfunctor for which the inclusion $\mathscr {E}_0 \hookrightarrow \mathscr {E}$ is a levelwise categorical equivalence. Suppose we are given a natural transformation $\alpha _0: \mathscr {E}_0 \rightarrow \mathscr {F}^{\simeq }$. Our assumption that $\mathscr {F}$ is isofibrant guarantees that $\alpha _0$ can be extended to a natural transformation $\alpha : \mathscr {E} \rightarrow \mathscr {F}$. We claim that $\alpha $ automatically factors through the functor $\mathscr {F}^{\simeq }$: that is, for every object $C \in \operatorname{\mathcal{C}}$, the map $\alpha _{C}: \mathscr {E}(C) \rightarrow \mathscr {F}(C)$ factors through the core of $\mathscr {F}(C)$. This follows from the observation that the lifting problem

\[ \xymatrix@R =50pt@C=50pt{ \mathscr {E}_0(C) \ar [d] \ar [r]^-{ \alpha _0} & \mathscr {F}(C)^{\simeq } \ar [d] \\ \mathscr {E}(C) \ar [r]^-{ \alpha } \ar@ {-->}[ur] & \mathscr {F}(C) } \]

has a (unique) solution, since the inclusion $\mathscr {F}(C)^{\simeq } \hookrightarrow \mathscr {F}(C)$ is an isofibration (Proposition 4.4.3.6).

We now prove the second assertion. Let $X$ denote the core of the $\infty $-category $\varprojlim ( \mathscr {F} )$. For every object $C \in \operatorname{\mathcal{C}}$, the projection map $\varprojlim ( \mathscr {F} ) \rightarrow \mathscr {F}(C)$ carries $X$ into the core of $\mathscr {F}(C)$. It follows that $X$ is contained in the inverse limit $\varprojlim ( \mathscr {F}^{\simeq } )$. The reverse inclusion follows from Corollary 4.4.3.17, since the simplicial set $\varprojlim ( \mathscr {F}^{\simeq } )$ is a Kan complex (Corollary 4.5.6.18). $\square$

Corollary 4.5.6.20. Suppose we are given an inverse system of $\infty $-categories

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

where each of the transition functors $\operatorname{\mathcal{C}}(n) \rightarrow \operatorname{\mathcal{C}}(n-1)$ is an isofibration. Then the limit $\operatorname{\mathcal{C}}= \varprojlim _{n} \operatorname{\mathcal{C}}(n)$ is an $\infty $-category, whose core $\operatorname{\mathcal{C}}^{\simeq }$ is the inverse limit $\varprojlim _{n} \operatorname{\mathcal{C}}(n)^{\simeq }$. In other words, a morphism of $\operatorname{\mathcal{C}}$ is an isomorphism if and only if its image in each $\operatorname{\mathcal{C}}(n)$ is an isomorphism.