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7.5.5 Categorical Limit Diagrams

The theory of homotopy limit diagrams introduced in §7.5.4 should be regarded as belonging to the “classical” homotopy theory of simplicial sets: for example, it is invariant under weak homotopy equivalence (Corollary 7.5.4.11). When using simplicial sets to model higher category theory (rather than homotopy theory), it is useful to work with slightly different class of diagrams.

Definition 7.5.5.1 (Categorical Limit Diagrams of $\infty $-Categories). Let $\operatorname{\mathcal{C}}$ be a small category and let $\overline{\mathscr {F}}: \operatorname{\mathcal{C}}^{\triangleleft } \rightarrow \operatorname{QCat}$ be a functor having restriction $\mathscr {F} = \overline{ \mathscr {F} }|_{\operatorname{\mathcal{C}}}$. We will say that $\overline{ \mathscr {F} }$ is a categorical limit diagram if the composite map

\[ \overline{ \mathscr {F} }( {\bf 0} ) \rightarrow \varprojlim ( \mathscr {F} ) \hookrightarrow \underset {\longleftarrow }{\mathrm{holim}}( \mathscr {F} ) \]

is a homotopy equivalence of Kan complexes; here ${\bf 0}$ denotes the initial object of the cone $\operatorname{\mathcal{C}}^{\triangleleft } \simeq \{ {\bf 0} \} \star \operatorname{\mathcal{C}}$, and the morphism on the right is the comparison map of Remark 7.5.2.10.

Example 7.5.5.2. Let $\operatorname{\mathcal{C}}$ be a category. A diagram of Kan complexes $\overline{\mathscr {F}}: \operatorname{\mathcal{C}}^{\triangleleft } \rightarrow \operatorname{Kan}$ is a categorical limit diagram (in the sense of Definition 7.5.5.1) if and only if it is a homotopy limit diagram (in the sense of Definition 7.5.4.1).

Example 7.5.5.3 (Limits of Isofibrant Diagrams). Let $\operatorname{\mathcal{C}}$ be a small category and let $\overline{\mathscr {F}}: \operatorname{\mathcal{C}}^{\triangleleft } \rightarrow \operatorname{Set_{\Delta }}$ be a limit diagram in the category of simplicial sets. Suppose that the diagram $\mathscr {F} = \overline{\mathscr {F}}|_{\operatorname{\mathcal{C}}}$ is isofibrant (Definition 4.5.6.3). Then $\overline{\mathscr {F}}$ is a categorical limit diagram of $\infty $-categories: this follows by combining Corollary 4.5.6.12 with Proposition 7.5.3.12.

Warning 7.5.5.4. Let $\operatorname{\mathcal{C}}$ be a category and let $\overline{\mathscr {F}}: \operatorname{\mathcal{C}}^{\triangleleft } \rightarrow \operatorname{QCat}$ be a diagram of $\infty $-categories. In general, the condition that $\overline{\mathscr {F}}$ is a categorical limit diagram (in the sense of Definition 7.5.5.1) is independent of the condition that it is a homotopy limit diagram (in the sense of Definition 7.5.4.8): see Exercises 4.5.2.10 and 4.5.2.11.

Remark 7.5.5.5. Let $\operatorname{\mathcal{C}}$ be a category, let $\overline{\mathscr {F}}: \operatorname{\mathcal{C}}^{\triangleleft } \rightarrow \operatorname{QCat}$ be a categorical limit diagram of $\infty $-categories, and define $\overline{\mathscr {F}}^{\simeq }: \operatorname{\mathcal{C}}^{\triangleleft } \rightarrow \operatorname{Kan}$ by the formula $\overline{\mathscr {F}}^{\simeq }(C) = \overline{\mathscr {F}}(C)^{\simeq }$. Then $\overline{\mathscr {F}}^{\simeq }$ is a homotopy limit diagram. This follows by combining Example 7.5.2.6 with Remark 4.5.1.20.

Remark 7.5.5.6 (Homotopy Invariance). Let $\operatorname{\mathcal{C}}$ be a small category and let $\alpha : \overline{\mathscr {F}} \rightarrow \overline{\mathscr {G}}$ be a natural transformation between diagrams $\overline{\mathscr {F}}, \overline{\mathscr {G}}: \operatorname{\mathcal{C}}^{\triangleleft } \rightarrow \operatorname{QCat}$. Assume that, for every object $C \in \operatorname{\mathcal{C}}$, the induced map $\alpha _{C}: \overline{\mathscr {F}}(C) \rightarrow \overline{\mathscr {G}}(C)$ is an equivalence of $\infty $-categories. Then $\alpha $ determines a commutative diagram of $\infty $-categories

\[ \xymatrix@R =50pt@C=50pt{ \overline{\mathscr {F}}( {\bf 0} ) \ar [r] \ar [d] & \underset {\longleftarrow }{\mathrm{holim}}( \overline{\mathscr {F}}|_{\operatorname{\mathcal{C}}} ) \ar [d] \\ \overline{\mathscr {G}}( {\bf 0} ) \ar [r] & \underset {\longleftarrow }{\mathrm{holim}}( \overline{\mathscr {G}}|_{\operatorname{\mathcal{C}}} ), } \]

where the right vertical map is an equivalence (Remark 7.5.2.5). It follows that any two of the following conditions imply the third:

$(1)$

The functor $\overline{\mathscr {F}}$ is a categorical limit diagram.

$(2)$

The functor $\overline{\mathscr {G}}$ is a categorical limit diagram.

$(3)$

The natural transformation $\alpha $ induces an equivalence of $\infty $-categories $\overline{\mathscr {F}}( {\bf 0} ) \rightarrow \overline{\mathscr {G}}( {\bf 0} )$, where ${\bf 0}$ denotes the cone point of $\operatorname{\mathcal{C}}^{\triangleleft }$.

Proposition 7.5.5.7. Let $\operatorname{\mathcal{C}}$ be a category and let $\overline{\mathscr {F}}: \operatorname{\mathcal{C}}^{\triangleleft } \rightarrow \operatorname{QCat}$ be a functor. The following conditions are equivalent:

$(1)$

The functor $\overline{\mathscr {F}}$ is a categorical limit diagram, in the sense of Definition 7.5.5.1.

$(2)$

For every simplicial set $K$, the functor

\[ \overline{\mathscr {F}}^{K}: \operatorname{\mathcal{C}}^{\triangleleft } \rightarrow \operatorname{QCat}\quad \quad C \mapsto \operatorname{Fun}( K, \overline{\mathscr {F}}(C) ) \]

is a categorical limit diagram.

$(3)$

For every simplicial set $K$, the functor

\[ (\overline{\mathscr {F}}^{K})^{\simeq }: \operatorname{\mathcal{C}}^{\triangleleft } \rightarrow \operatorname{Kan}\quad \quad C \mapsto \operatorname{Fun}( K, \overline{\mathscr {F}}(C) )^{\simeq } \]

is a homotopy limit diagram.

$(4)$

The functor $( \overline{\mathscr {F}}^{\Delta ^1} )^{\simeq }: \operatorname{\mathcal{C}}^{\triangleleft } \rightarrow \operatorname{Kan}$ is a homotopy limit diagram.

Proof. The implication $(1) \Rightarrow (2)$ follows from Remarks 7.5.2.3 and 4.5.1.16, the implication $(2) \Rightarrow (3)$ from Remark 7.5.5.5, and the implication $(3) \Rightarrow (4)$ is immediate. Set $\mathscr {F} = \overline{\mathscr {F}}|_{\operatorname{\mathcal{C}}}$, and let ${\bf 0}$ denote the initial object of $\operatorname{\mathcal{C}}^{\triangleleft }$. Using Remark 7.5.2.3 and Example 7.5.2.6, we see that condition $(4)$ is equivalent to the requirement that the map $\overline{ \mathscr {F} }( {\bf 0} ) \rightarrow \underset {\longleftarrow }{\mathrm{holim}}( \mathscr {F} )$ induces a homotopy equivalence of Kan complexes

\[ \operatorname{Fun}( \Delta ^1, \overline{ \mathscr {F} }( {\bf 0} ))^{\simeq } \rightarrow \operatorname{Fun}( \Delta ^1, \underset {\longleftarrow }{\mathrm{holim}}( \mathscr {F} ))^{\simeq } \simeq \underset {\longleftarrow }{\mathrm{holim}}( ( \overline{\mathscr {F}}^{\Delta ^1} )^{\simeq } ). \]

The implication $(4) \Rightarrow (1)$ now follows from Theorem 4.5.7.1. $\square$

Corollary 7.5.5.8. Let $\operatorname{\mathcal{C}}$ be a category and let $\overline{\mathscr {F}}: \operatorname{\mathcal{C}}^{\triangleleft } \rightarrow \operatorname{QCat}$ be a functor. Then $\overline{\mathscr {F}}$ is a categorical limit diagram if and only if the induced functor of $\infty $-categories

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

is a limit diagram in the $\infty $-category $\operatorname{\mathcal{QC}}$ (in the sense of Definition 7.1.2.4).

Proof. By virtue of Corollary 7.5.4.6, the diagram $\operatorname{N}_{\bullet }^{\operatorname{hc}}( \overline{\mathscr {F}} )$ is a limit diagram in the $\infty $-category $\operatorname{\mathcal{QC}}$ if and only if, for every $\infty $-category $\operatorname{\mathcal{E}}$, the diagram of Kan complexes

\[ \operatorname{\mathcal{C}}^{\triangleleft } \rightarrow \operatorname{Kan}\quad \quad C \mapsto \operatorname{Hom}_{ \operatorname{QCat}}( \operatorname{\mathcal{E}}, \overline{\mathscr {F}}(C) )_{\bullet } = \operatorname{Fun}(\operatorname{\mathcal{E}}, \overline{\mathscr {F}}(C) )^{\simeq } \]

is a homotopy limit diagram. Using Proposition 7.5.5.7, we see that this is equivalent to the requirement that $\overline{\mathscr {F}}$ is a categorical limit diagram. $\square$

Corollary 7.5.5.9. Let $\operatorname{\mathcal{C}}$ be a small category and let $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{QCat}$ be an isofibrant diagram of $\infty $-categories Then $\mathscr {F}$ has a limit in the category $\operatorname{QCat}$, which is preserved by the inclusion functor $\operatorname{N}_{\bullet }( \operatorname{QCat}) \hookrightarrow \operatorname{N}_{\bullet }^{\operatorname{hc}}( \operatorname{QCat}) = \operatorname{\mathcal{QC}}$.

Corollary 7.5.5.10. Suppose we are given a commutative diagram of $\infty $-categories

7.50
\begin{equation} \begin{gathered}\label{equation:categorical-pullback-as-categorical-limit} \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{C}}_{01} \ar [r] \ar [d] & \operatorname{\mathcal{C}}_{0} \ar [d] \\ \operatorname{\mathcal{C}}_{1} \ar [r] & \operatorname{\mathcal{C}}, } \end{gathered} \end{equation}

which we identify with a functor $\overline{\mathscr {F}}: [1] \times [1] \rightarrow \operatorname{\mathcal{QC}}$. The following conditions are equivalent:

$(1)$

The diagram (7.50) is a categorical pullback square, in the sense of Definition 4.5.2.7.

$(2)$

The functor $\overline{\mathscr {F}}$ is a categorical limit diagram, in the sense of Definition 7.5.5.1.

Proof. Using Proposition 4.5.2.12, we can restate $(2)$ as follows:

$(2')$

For every simplicial set $K$, the diagram of Kan complexes

\[ \xymatrix@R =50pt@C=50pt{ \operatorname{Fun}(K,\operatorname{\mathcal{C}}_{01})^{\simeq } \ar [r] \ar [d] & \operatorname{Fun}(K,\operatorname{\mathcal{C}}_{0})^{\simeq } \ar [d] \\ \operatorname{Fun}(K,\operatorname{\mathcal{C}}_{1})^{\simeq } \ar [r] & \operatorname{Fun}(K,\operatorname{\mathcal{C}})^{\simeq } } \]

is a homotopy pullback square.

The equivalence $(1) \Leftrightarrow (2')$ follows by combining Propositions 7.5.4.13, and 7.5.5.7. $\square$

We now extend the scope of Definition 7.5.5.1 to arbitrary diagrams of simplicial sets.

Definition 7.5.5.11 (Categorical Limit Diagrams of Simplicial Sets). Let $\operatorname{\mathcal{C}}$ be a category. We say that a functor $\overline{\mathscr {F}}: \operatorname{\mathcal{C}}^{\triangleleft } \rightarrow \operatorname{Set_{\Delta }}$ is a categorical limit diagram if there exists a levelwise categorical equivalence $\alpha : \overline{\mathscr {F}} \rightarrow \overline{\mathscr {G}}$, where $\overline{\mathscr {G}}: \operatorname{\mathcal{C}}^{\triangleleft } \rightarrow \operatorname{QCat}$ is a categorical limit diagram (in the sense of Definition 7.5.5.1).

Remark 7.5.5.12. Let $\operatorname{\mathcal{C}}$ be a category and let $\overline{\mathscr {F}}: \operatorname{\mathcal{C}}^{\triangleleft } \rightarrow \operatorname{QCat}$ be a functor. The following conditions are equivalent:

$(1)$

The functor $\overline{\mathscr {F}}$ is a categorical limit diagram in the sense of Definition 7.5.5.1: that is, it induces an equivalence of $\infty $-categories $\overline{\mathscr {F}}( {\bf 0} ) \rightarrow \underset {\longleftarrow }{\mathrm{holim}}( \overline{\mathscr {F}}|_{\operatorname{\mathcal{C}}} )$.

$(2)$

The functor $\overline{\mathscr {F}}$ is a categorical limit diagram in the sense of Definition 7.5.5.11: that is, there exists a levelwise categorical equivalence $\alpha : \overline{\mathscr {F}} \rightarrow \overline{\mathscr {G}}$, where $\overline{\mathscr {G}}: \operatorname{\mathcal{C}}^{\triangleleft } \rightarrow \operatorname{QCat}$ induces an equivalence of $\infty $-categories $\overline{\mathscr {G}}( {\bf 0} ) \rightarrow \underset {\longleftarrow }{\mathrm{holim}}( \overline{\mathscr {G}}|_{\operatorname{\mathcal{C}}} )$.

The implication $(1) \Rightarrow (2)$ is immediate, and the reverse implication follows from Remark 7.5.5.6.

Proposition 7.5.5.13. Let $\operatorname{\mathcal{C}}$ be a category and let $\overline{\mathscr {F}}: \operatorname{\mathcal{C}}^{\triangleleft } \rightarrow \operatorname{Set_{\Delta }}$ be a functor. The following conditions are equivalent:

$(1)$

The functor $\overline{\mathscr {F}}$ is a categorical limit diagram. That is, there exists a categorical limit diagram $\overline{\mathscr {F}}': \operatorname{\mathcal{C}}^{\triangleleft } \rightarrow \operatorname{QCat}$ and a levelwise categorical equivalence $\alpha : \overline{\mathscr {F}} \rightarrow \overline{\mathscr {F}}'$.

$(2)$

Let $\overline{\mathscr {F}}': \operatorname{\mathcal{C}}^{\triangleleft } \rightarrow \operatorname{QCat}$ be any functor. If there exists a levelwise categorical equivalence $\alpha : \overline{\mathscr {F}} \rightarrow \overline{\mathscr {F}}'$, then $\overline{\mathscr {F}}'$ is a categorical limit diagram.

Proof. We proceed as in the proof of Proposition 7.5.4.10. Using Proposition 4.1.3.2, we can choose a functor $\overline{\mathscr {G}}: \operatorname{\mathcal{C}}^{\triangleleft } \rightarrow \operatorname{QCat}$ and a natural transformation $\beta : \overline{\mathscr {F}} \rightarrow \overline{\mathscr {G}}$ for which the morphism of simplicial sets $\beta _{C}: \overline{\mathscr {F}}(C) \rightarrow \overline{\mathscr {G}}(C)$ is inner anodyne for each object $C \in \operatorname{\mathcal{C}}^{\triangleleft }$. We will show that $(1)$ and $(2)$ are equivalent to the following:

$(3)$

The functor $\overline{\mathscr {G}}$ is a categorical limit diagram.

The implications $(2) \Rightarrow (3) \Rightarrow (1)$ are immediate. To prove the reverse implications, suppose we are given another functor $\overline{\mathscr {F}}': \operatorname{\mathcal{C}}^{\triangleleft } \rightarrow \operatorname{QCat}$ and a levelwise categorical equivalence $\alpha : \overline{\mathscr {F}} \rightarrow \overline{\mathscr {F}}'$. We will show that $\overline{\mathscr {F}}'$ is a categorical limit diagram if and only if $\overline{\mathscr {G}}$ is a categorical limit diagram.

Applying Proposition 4.1.3.2 again, we can choose a functor $\overline{\mathscr {G}}': \operatorname{\mathcal{C}}^{\triangleleft } \rightarrow \operatorname{QCat}$ and a commutative diagram

\[ \xymatrix@R =50pt@C=50pt{ \overline{\mathscr {F}} \ar [r]^-{\alpha } \ar [d]^{\beta } & \overline{\mathscr {F}}' \ar [d]^{\beta '} \\ \overline{\mathscr {G}} \ar [r]^-{\alpha '} & \overline{\mathscr {G}}' } \]

with the property that, for every object $C \in \operatorname{\mathcal{C}}^{\triangleleft }$, the induced morphism of simplicial sets $\overline{\mathscr {F}}'(C) \coprod _{ \overline{\mathscr {F}}(C) } \overline{\mathscr {G}}(C) \rightarrow \overline{\mathscr {G}}'(C)$ is inner anodyne (and, in particular, a categorical equivalence). Applying Proposition 4.5.4.11, we see that the diagram of simplicial sets

\[ \xymatrix@R =50pt@C=50pt{ \overline{\mathscr {F}}(C) \ar [r]^-{\alpha _ C} \ar [d]^{\beta _ C} & \overline{\mathscr {F}}'(C) \ar [d]^{\beta '_{C}} \\ \overline{\mathscr {G}}(C) \ar [r]^-{\alpha '_{C}} & \overline{\mathscr {G}}'(C) } \]

is a categorical pushout square. Since $\alpha _ C$ and $\beta _ C$ are categorical equivalences, it follows that $\alpha '_ C$ and $\beta '_{C}$ are also categorical equivalences (Proposition 4.5.4.10). Applying Remark 7.5.5.6, we see that $\overline{ \mathscr {F} }'$ and $\overline{ \mathscr {G} }$ are categorical limit diagrams if and only if $\overline{ \mathscr {G} }'$ is a categorical limit diagram. $\square$

Corollary 7.5.5.14. Let $\operatorname{\mathcal{C}}$ be a category, let $\overline{\mathscr {F}}, \overline{\mathscr {F}}': \operatorname{\mathcal{C}}^{\triangleleft } \rightarrow \operatorname{Set_{\Delta }}$ be functors, and let $\alpha : \overline{\mathscr {F}} \rightarrow \overline{\mathscr {F}}'$ be a natural transformation. Suppose that, for every object $C \in \operatorname{\mathcal{C}}$, the induced map $\alpha _{C}: \overline{\mathscr {F}}(C) \rightarrow \overline{\mathscr {F}}'(C)$ is a categorical equivalence of simplicial sets. Then any two of the following conditions imply the third:

$(1)$

The functor $\overline{\mathscr {F}}$ is a categorical limit diagram.

$(2)$

The functor $\overline{\mathscr {F}}'$ is a categorical limit diagram.

$(3)$

The natural transformation $\alpha $ induces a categorical equivalence of simplicial sets $\overline{\mathscr {F}}( {\bf 0} ) \rightarrow \overline{\mathscr {F}}'( {\bf 0} )$, where ${\bf 0}$ denotes the initial object of $\operatorname{\mathcal{C}}^{\triangleleft }$.

Proof. Using Proposition 4.1.3.2, we can choose functors $\overline{\mathscr {G}}, \overline{\mathscr {G}}': \operatorname{\mathcal{C}}^{\triangleleft } \rightarrow \operatorname{QCat}$ and a commutative diagram

\[ \xymatrix@R =50pt@C=50pt{ \overline{\mathscr {F}} \ar [r]^-{\alpha } \ar [d] & \overline{\mathscr {F}}' \ar [d] \\ \overline{\mathscr {G}} \ar [r]^-{\beta } & \overline{\mathscr {G}}', } \]

where the vertical maps are levelwise categorical equivalences. By virtue of Proposition 7.5.5.13, we can replace $\alpha $ by the natural transformation $\beta : \overline{\mathscr {G}} \rightarrow \overline{\mathscr {G}}'$. In this case, the desired result follows from Remark 7.5.5.6. $\square$

Corollary 7.5.5.15. Let $\operatorname{\mathcal{C}}$ be a small category and let $\overline{ \mathscr {F} }: \operatorname{\mathcal{C}}^{\triangleleft } \rightarrow \operatorname{Set_{\Delta }}$ be a functor. Let $\overline{ \mathscr {F} }^{\operatorname{op}}: \operatorname{\mathcal{C}}^{\triangleleft } \rightarrow \operatorname{Set_{\Delta }}$ be the functor given on objects by $\overline{ \mathscr {F} }^{\operatorname{op}}(C) = \overline{\mathscr {F}}(C)^{\operatorname{op}}$. Then $\overline{\mathscr {F}}$ is a categorical limit diagram if and only if $\overline{\mathscr {F}}^{\operatorname{op}}$ is a categorical limit diagram.

Proof. Using Proposition 4.1.3.2, we can choose a functor $\overline{\mathscr {G}}: \operatorname{\mathcal{C}}^{\triangleleft } \rightarrow \operatorname{QCat}$ and a levelwise categorical equivalence $\alpha : \overline{\mathscr {F}} \rightarrow \overline{\mathscr {G}}$. By virtue of Corollary 7.5.5.14, it will suffice to show that $\overline{\mathscr {G}}$ is a categorical limit diagram if and only if $\overline{\mathscr {G}}^{\operatorname{op}}$ is a categorical limit diagram. This follows by combining Proposition 7.5.5.7 with Corollary 7.5.4.12. $\square$