Kerodon

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).

Remark 7.5.2.5. Let $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{Cat}$ be a functor of ordinary categories. Then the category $ \underset {\longleftarrow }{\mathrm{holim}}(\mathscr {F})$ can be described more concretely as follows:

$(1)$

An $M \in \underset {\longleftarrow }{\mathrm{holim}}(\mathscr {F})$ is a rule which assigns to each object $C \in \operatorname{\mathcal{C}}$ an object $M(C) \in \mathscr {F})$, and to each morphism $u: C \rightarrow D$ in $\operatorname{\mathcal{C}}$ an isomorphism $M(u): \mathscr {F}(u)( M(C) ) \xrightarrow {\sim } M(D)$, subject to the following constraints:

• For each object $C \in \operatorname{\mathcal{C}}$, $M( \operatorname{id}_{C} )$ is the identity morphism from $M(C)$ to itself.

• For every pair of composable morphisms $u: C \rightarrow D$ and $v: D \rightarrow E$ of $\operatorname{\mathcal{C}}$, we have $M( v \circ u ) = M(v) \circ \mathscr {F}(v)( M(u) )$.

$(2)$

If $M$ and $N$ are objects of $ \underset {\longleftarrow }{\mathrm{holim}}(\mathscr {F})$, then a morphism $\alpha : M \rightarrow N$ in $ \underset {\longleftarrow }{\mathrm{holim}}(\mathscr {F})$ is a rule which assigns to each object $C \in \operatorname{\mathcal{C}}$ a morphism $\alpha _{C}: M(C) \rightarrow N(C)$ in the category $\mathscr {F}(C)$, subject to the following constraint:

• For every morphism $u: C \rightarrow D$ of $\operatorname{\mathcal{C}}$, the diagram

  \[ \xymatrix@C =50pt@R=50pt{ \mathscr {F}(u)( M(C) ) \ar [d]^{ \mathscr {F}(u)( \alpha _ C )} \ar [r]^-{ M(u) }_{\sim } & M(D) \ar [d]^{ \alpha _{D} } \\ \mathscr {F}(u)( N(C) ) \ar [r]^-{ N(u) }_{\sim } & N(D) } \]

  commutes (in the category $\mathscr {F}(D)$).

Remark 7.5.2.7 (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

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 Corollary 5.3.3.16), 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 5.3.3.20), so $ \underset {\longleftarrow }{\mathrm{holim}}(\alpha )$ is an equivalence of $\infty $-categories.

Remark 7.5.2.9. 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 7.5.2.10. 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 $\mathscr {F}(C)$.

Remark 7.5.2.12 (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

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 $).