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11.6 Reclassified Tags

This section consists of tags which were replaced by a tag with similar content but different environment (for example, a lemma which became a theorem).

Remark 11.6.0.35. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $n$ be an integer. Then an object $X \in \operatorname{\mathcal{C}}$ is $n$-truncated if and only if the right fibration $\operatorname{\mathcal{C}}_{/X} \rightarrow \operatorname{\mathcal{C}}$ is essentially $n$-categorical. This follows from the criterion of Corollary 5.1.5.18 (together with Remark 4.7.1.2).

Proof of Proposition 11.6.0.74. For every morphism $e: D \rightarrow D'$ in $\operatorname{\mathcal{D}}$, we can choose a natural transformation $\alpha : f \rightarrow e_{!} \circ f$ carrying each vertex of $K^{\triangleright }$ to a $U$-cocartesian morphism of $\operatorname{\mathcal{C}}$. It follows from Proposition 7.3.9.3 that if $f$ is a $U$-colimit diagram, then $e_{!} \circ f$ is also a $U$-colimit diagram, and therefore a colimit diagram in the $\infty $-category $\operatorname{\mathcal{C}}_{D'}$ (Proposition 11.6.0.174). This proves the necessity of condition $(\ast )$. For the converse, suppose that $f$ satisfies condition $(\ast )$; we wish to show that $f$ is a $U$-colimit diagram. By virtue of Proposition 7.1.6.19, this is equivalent to the assertion that for every object $C \in \operatorname{\mathcal{C}}$, the diagram of Kan complexes

11.10
\begin{equation} \begin{gathered}\label{equation:relative-colimit-by-fiber} \xymatrix@R =50pt@C=50pt{ \operatorname{Hom}_{\operatorname{Fun}(K^{\triangleright }, \operatorname{\mathcal{C}}) }( f, \underline{C}) \ar [r] \ar [d] & \operatorname{Hom}_{\operatorname{Fun}(K, \operatorname{\mathcal{C}}) }( f|_{K}, \underline{C}|_{K} ) \ar [d] \\ \operatorname{Hom}_{\operatorname{Fun}(K^{\triangleright }, \operatorname{\mathcal{D}}) }( U \circ f, U \circ \underline{C} ) \ar [r] & \operatorname{Hom}_{\operatorname{Fun}(K, \operatorname{\mathcal{D}}) }( U \circ f|_{K}, U \circ \underline{C}|_{K}) } \end{gathered} \end{equation}

is a homotopy pullback square, where $\underline{C} \in \operatorname{Fun}( K^{\triangleright }, \operatorname{\mathcal{C}})$ is the constant diagram taking the value $C$. Since $U$ is an inner fibration, the vertical maps in this diagram are Kan fibrations (Proposition 4.6.1.22 and Corollary 4.1.4.3). Using the criterion of Example 3.4.1.4, it will suffice to show that for every vertex $u \in \operatorname{Hom}_{\operatorname{Fun}(K^{\triangleright }, \operatorname{\mathcal{D}}) }( U \circ f, U \circ \underline{C} )$, the induced map

\[ \xymatrix@R =50pt@C=50pt{ \{ u\} \times _{\operatorname{Hom}_{\operatorname{Fun}(K^{\triangleright }, \operatorname{\mathcal{D}}) }( U \circ f, U \circ \underline{C} )} \operatorname{Hom}_{\operatorname{Fun}(K^{\triangleright }, \operatorname{\mathcal{C}}) }( f, \underline{C}) \ar [d]^{\theta _ u} \\ \{ u\} \times _{ \operatorname{Hom}_{\operatorname{Fun}(K, \operatorname{\mathcal{D}}) }( U \circ f|_{K}, U \circ \underline{C}|_{K}) } \operatorname{Hom}_{\operatorname{Fun}(K, \operatorname{\mathcal{C}}) }( f|_{K}, \underline{C}|_{K} )} \]

is a homotopy equivalence of Kan complexes. Set $D' = U(C)$, so that $u$ can be identified with a morphism of simplicial sets $K^{\triangleright } \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{D}}}( D, D' )$, and the condition that $\theta _ u$ is a homotopy equivalence depends only on the homotopy class of $u$. Since the simplicial set $K^{\triangleright }$ is weakly contractible (Example 4.3.7.11), we may assume without loss of generality that $u: K^{\triangleright } \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{D}}}( D, D' )$ is the constant map taking the value $e$, for some morphism $e: D \rightarrow D'$ in $\operatorname{\mathcal{D}}$. In this case, we can use Proposition 5.1.3.11 to replace $\theta _{u}$ by the restriction map

\[ \operatorname{Hom}_{ \operatorname{Fun}( K^{\triangleright }, \operatorname{\mathcal{C}}_{D'})}( e_{!} \circ f, \underline{D} ) \rightarrow \operatorname{Hom}_{ \operatorname{Fun}( K, \operatorname{\mathcal{C}}_{D'})}( e_{!} \circ f|_{K}, \underline{D}|_{K} ), \]

which is a homotopy equivalence by virtue of assumption $(\ast )$ (see Proposition 7.1.6.19). $\square$

Remark 11.6.0.77. In the situation of Proposition 7.1.7.14, we can replace condition $(\ast )$ by the following:

$(\ast ')$

For every morphism $e: D \rightarrow D'$ in $\operatorname{\mathcal{D}}$, the morphism

\[ K^{\triangleleft } \xrightarrow { \overline{f} } \operatorname{\mathcal{C}}_{D} \hookrightarrow \Delta ^{1} \times _{\operatorname{\mathcal{D}}} \operatorname{\mathcal{C}} \]

is a colimit diagram in the $\infty $-category $\Delta ^1 \times _{\operatorname{\mathcal{D}}} \operatorname{\mathcal{C}}$.

See Corollary 7.1.6.12.

Variant 11.6.0.82. See Corollary 11.8.0.3. variant

Corollary 11.6.0.92. See Variant 7.4.4.14. Let $\lambda $ be an uncountable cardinal and let $\kappa = \mathrm{ecf}(\lambda )$ be the exponential cofinality of $\lambda $. Suppose we are given a diagram $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{QC}}$, where $\operatorname{\mathcal{C}}$ is a $\kappa $-small simplicial set. If the $\infty $-category $\mathscr {F}(C)$ is essentially $\lambda $-small for each $C \in \operatorname{\mathcal{C}}$, then the limit $\varprojlim (\mathscr {F} )$ is also essentially $\lambda $-small.

Proof. Using Proposition 4.9.5.7, we can choose a categorical equivalence $G: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$, where $\operatorname{\mathcal{D}}$ is a $\lambda $-small $\infty $-category (if $\kappa $ is uncountable, we can even arrange that $\operatorname{\mathcal{D}}$ is $\kappa $-small). Without loss of generality, we may assume that $\mathscr {F}$ is obtained as the restriction of the covariant transport representation of some cocartesian fibration $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{D}}$. Using Proposition 7.4.4.1, we can identify $\varprojlim ( \mathscr {F} )$ with a full subcategory of the $\infty $-category $\operatorname{Fun}_{ / \operatorname{\mathcal{D}}}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{E}})$. It will therefore suffice to show that the $\infty $-category $\operatorname{Fun}_{ / \operatorname{\mathcal{D}}}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{E}})$ is essentially $\lambda $-small (Corollary 4.9.5.16). By construction, we have a pullback diagram of simplicial sets

11.11
\begin{equation} \begin{gathered}\label{equation:limit-essential-smallness} \xymatrix@C =50pt@R=50pt{ \operatorname{Fun}_{ / \operatorname{\mathcal{D}}} ( \operatorname{\mathcal{C}}, \operatorname{\mathcal{E}}) \ar [r] \ar [d] & \operatorname{Fun}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{E}}) \ar [d]^{U \circ } \\ \{ G \} \ar [r] & \operatorname{Fun}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{D}}) } \end{gathered} \end{equation}

where the vertical maps are cocartesian fibrations (Theorem 5.2.1.1), and therefore isofibrations (Proposition 5.1.4.9). It follows that (11.11) is also a categorical pullback square (Corollary 4.5.3.28). Using Corollary 4.9.5.19, we are reduced to proving that the $\infty $-categories $\operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{E}})$ and $\operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})$ are essentially $\lambda $-small, which follows from Remark 4.9.5.13. $\square$

Corollary 11.6.0.98. Let $n$ be an integer, let $\operatorname{\mathcal{C}}$ be a simplicial set and let $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{S}}$ be a diagram. Suppose that, for every vertex $C \in \operatorname{\mathcal{C}}$, the Kan complex $\mathscr {F}(C)$ is $n$-truncated. Then the limit $\varprojlim ( \mathscr {F} )$ is an $n$-truncated Kan complex.$\infty $-category.

Example 11.6.0.103. Let $(A, \leq )$ be a partially ordered set. Combining Exercise 11.6.0.101 with Corollary 9.1.2.8, we see that the $\infty $-category $\operatorname{N}_{\bullet }(A)$ is filtered if and only if the partially ordered set $(A, \leq )$ is directed.

Exercise 11.6.0.119. See Proposition 1.2.4.7.

Let $0 \leq i \leq n$ be integers. For $j \in [n] \setminus \{ i\} $, we can regard the map $\delta ^{j}_{n}$ of Construction 1.1.1.4 of simplicial sets from $\Delta ^{n-1}$ to the horn $\Lambda ^{n}_{i} \subseteq \Delta ^{n}$. Show that, for any simplicial set $S_{\bullet }$, the construction

\[ ( f: \Lambda ^{n}_{i} \rightarrow S_{\bullet } ) \mapsto \{ f \circ \delta ^{j}_{n} \} _{j \in [n] \setminus \{ i\} } \]

determines an injection $\operatorname{Hom}_{\operatorname{Set_{\Delta }}}( \Lambda ^{n}_{i}, S_{\bullet } ) \rightarrow \prod _{ j \in [n] \setminus \{ i\} } S_{n-1}$, whose image is the collection of “incomplete” sequences $( \sigma _0, \ldots , \sigma _{i-1}, \bullet , \sigma _{i+1}, \ldots , \sigma _ n)$ satisfying $d^{n-1}_ j(\sigma _ k) = d^{n-1}_{k-1}( \sigma _{j})$ for $j, k \in [n] \setminus \{ i\} $ with $j < k$.

Remark 11.6.0.131. The contents of this tag are now at Example 4.6.6.8.

Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $F: K \rightarrow \operatorname{\mathcal{C}}$ be a diagram. The slice diagonal morphism $\delta _{/F}: \operatorname{\mathcal{C}}_{/F} \hookrightarrow \operatorname{\mathcal{C}}\vec{\times }_{ \operatorname{Fun}(K, \operatorname{\mathcal{C}}) } \{ F\} $ carries each $n$-simplex of $\operatorname{\mathcal{C}}_{/F}$ to an $n$-simplex $\sigma $ of the oriented fiber product $\operatorname{\mathcal{C}}\vec{\times }_{ \operatorname{Fun}(K, \operatorname{\mathcal{C}}) } \{ F\} $, which we can identify with a map $\Delta ^0 \diamond K \rightarrow \operatorname{Fun}( \Delta ^ n, \operatorname{\mathcal{C}})$. It is not difficult to see that this map factors (uniquely) through the comparison map $c: \Delta ^0 \diamond K \twoheadrightarrow K^{\triangleleft }$ of Notation 4.6.2.3, and can therefore also be viewed as an $n$-simplex of the simplicial set $\operatorname{Fun}(K^{\triangleleft }, \operatorname{\mathcal{C}}) \times _{ \operatorname{Fun}(K,\operatorname{\mathcal{C}})} \{ F\} $. Consequently, $\delta _{/F}$ factors as a composition

\[ \operatorname{\mathcal{C}}_{/F} \xrightarrow { \delta '_{/F} } \operatorname{Fun}(K^{\triangleleft }, \operatorname{\mathcal{C}}) \times _{ \operatorname{Fun}(K,\operatorname{\mathcal{C}})} \{ F\} \xrightarrow {\iota } \operatorname{\mathcal{C}}\vec{\times }_{ \operatorname{Fun}(K, \operatorname{\mathcal{C}}) } \{ F\} , \]

where $\iota $ is a monomorphism of simplicial sets given by precomposition with $c$. Since $c$ is a categorical equivalence of simplicial sets (Theorem 4.6.2.8), the functor $\iota $ is an equivalence of $\infty $-categories: this follows by applying Corollary 4.5.3.31 to the diagram

\[ \xymatrix@R =50pt@C=50pt{ \operatorname{Fun}( K^{\triangleleft }, \operatorname{\mathcal{C}}) \ar [rr]^{ \circ c} \ar [dr] & & \operatorname{Fun}( \Delta ^0 \diamond K, \operatorname{\mathcal{C}}) \ar [dl] \\ & \operatorname{Fun}(K, \operatorname{\mathcal{C}}), & } \]

since the vertical maps are isofibrations (Corollary 4.4.5.3). It follows from Theorem 4.6.4.19 that the functor

\[ \delta '_{/F}: \operatorname{\mathcal{C}}_{/F} \rightarrow \operatorname{Fun}(K^{\triangleleft },\operatorname{\mathcal{C}}) \times _{\operatorname{Fun}(K,\operatorname{\mathcal{C}})} \{ F\} \]

is also an equivalence of $\infty $-categories. Similarly, the coslice diagonal morphism $\delta _{F/}$ factors through an equivalence of $\infty $-categories

\[ \delta '_{F/}: \operatorname{\mathcal{C}}_{F/} \rightarrow \{ F\} \times _{ \operatorname{Fun}(K,\operatorname{\mathcal{C}}) } \operatorname{Fun}(K^{\triangleright }, \operatorname{\mathcal{C}}). \]

Exercise 11.6.0.138. The contents of this tag are now at Remark 1.1.1.7 and Proposition 1.1.1.9. Let $C_{\bullet }$ be a semisimplicial object of a category $\operatorname{\mathcal{C}}$. Show that the face operators of Notation 11.6.0.125 satisfy the following condition:

$(\ast )$

For $n \geq 2$ and $0 \leq i < j \leq n$, we have $d^{n-1}_{i} \circ d^{n}_{j} = d^{n-1}_{j-1} \circ d^{n}_{i}$ (as a map from $C_{n}$ to $C_{n-2}$).

Conversely, show that any collection of objects $\{ C_ n \} _{n \geq 0}$ and morphisms $\{ d^{n}_ i: C_ n \rightarrow C_{n-1} \} _{0 \leq i \leq n}$, satisfying $(\ast )$ determines a unique semisimplicial object of $\operatorname{\mathcal{C}}$.

Proposition 11.6.0.213. Let $f: X_{} \rightarrow S_{}$ be a morphism of simplicial sets. Then $f$ is a Kan fibration if and only if it is both a left fibration and a right fibration.

Variant 11.6.0.214 (Strictly Unitary $2$-Categories). The contents of this tag can be found in Definition 2.2.7.1.

We say that a $2$-category $\operatorname{\mathcal{C}}$ is strictly unitary if, for every $1$-morphism $f: X \rightarrow Y$ in $\operatorname{\mathcal{C}}$, we have equalities

\[ \operatorname{id}_{Y} \circ f = f = f \circ \operatorname{id}_{X}, \]

and the left and right unit constraints $\lambda _{f}$, $\rho _{f}$ are the identity $2$-morphisms from $f$ to itself. Every strict $2$-category is strictly unitary, but the converse is false: we will see later that every $2$-category is isomorphic (in an appropriate sense) to a strictly unitary $2$-category (see Example 11.6.0.218).

Example 11.6.0.216. The contents of this tag can now be found in Remark 2.2.7.3.

Let $\operatorname{\mathcal{C}}$ be a strictly unitary $2$-category (Variant 11.6.0.214). Then Proposition 2.2.1.15 can be formulated more simply as follows: for every pair of composable $1$-morphisms $X \xrightarrow {f} Y \xrightarrow {g} Z$, the associativity constraints $\alpha _{ \operatorname{id}_ Z, g, f}$ and $\alpha _{g,f,\operatorname{id}_ X}$ are equal to the identity (as $2$-morphisms from $g \circ f$ to itself).

Example 11.6.0.217. The contents of this tag are now contained in Proposition 1.4.5.7.

Let $\operatorname{\mathcal{C}}$ be an $\infty $-category. Then the homotopy category $\operatorname {h}\! \mathit{\operatorname{\mathcal{C}}}$ constructed in Definition 1.4.5.3 is also a homotopy category of $\operatorname{\mathcal{C}}$ in the sense of Definition 1.3.6.1. More precisely, the map $u: \operatorname{\mathcal{C}}\rightarrow \operatorname{N}_{\bullet }( \operatorname {h}\! \mathit{\operatorname{\mathcal{C}}} )$ of Construction 1.4.5.6 exhibits $\operatorname {h}\! \mathit{\operatorname{\mathcal{C}}}$ as a homotopy category of $\operatorname{\mathcal{C}}$, by virtue of Proposition 1.4.5.7.

Example 11.6.0.218. The contents of this tag are now contained in Remark 2.2.7.6 and Proposition 2.2.7.7.

Let $\operatorname{\mathcal{C}}$ be any $2$-category. Then the left and right unit constraints on $\operatorname{\mathcal{C}}$ determine a twisting cochain $\{ \mu _{g,f} \} $, given concretely by the formula

\[ \mu _{g,f} = \begin{cases} \lambda _{f}: g \circ f \Rightarrow f & \text{ if } g = \operatorname{id}_{Y} \\ \rho _{g}: g \circ f \Rightarrow g & \text{ if } f = \operatorname{id}_ Y \\ \operatorname{id}_{g \circ f}: g \circ f \Rightarrow g \circ f & \text{ otherwise. } \end{cases} \]

Note that this prescription is consistent, since $\lambda _{f} = \rho _{g}$ in the special case where $f = g = \operatorname{id}_{Y}$ (Corollary 2.2.1.14).

Let $\operatorname{\mathcal{C}}'$ be the twist of $\operatorname{\mathcal{C}}$ with respect to the cocycle $\{ \mu _{g,f} \} $. Then $\operatorname{\mathcal{C}}'$ is a strictly unitary $2$-category (in the sense of Variant 11.6.0.214), and Exercise 2.2.6.9 supplies a strictly unitary isomorphism of $2$-categories $\operatorname{\mathcal{C}}\simeq \operatorname{\mathcal{C}}'$ In particular, for every $2$-category $\operatorname{\mathcal{C}}$, there exists a strictly unitary isomorphism of $2$-categories $\operatorname{\mathcal{C}}\simeq \operatorname{\mathcal{C}}'$, where $\operatorname{\mathcal{C}}'$ is strictly unitary.

Proposition 11.6.0.219. Let $i: A_{} \hookrightarrow B_{}$ be an anodyne morphism of simplicial sets and let $f: X_{} \rightarrow S_{}$ be a Kan fibration. Then the induced map

\[ \operatorname{Fun}( B_{}, X_{} ) \rightarrow \operatorname{Fun}( B_{}, S_{} ) \times _{ \operatorname{Fun}( A_{}, S_{} )} \operatorname{Fun}( A_{}, X_{} ) \]

is a trivial Kan fibration.

Lemma 11.6.0.220. Let $f: A_{} \hookrightarrow B_{}$ and $f': A'_{} \hookrightarrow B'_{}$ be monomorphisms of simplicial sets. If either $f$ is anodyne, then the induced map

\[ (A_{} \times B'_{} ) \coprod _{ A_{} \times A'_{} } ( B_{} \times A'_{} ) \hookrightarrow B_{} \times B'_{} \]

is anodyne.

Proposition 11.6.0.221. Let $f: X_{} \rightarrow S_{}$ be a morphism of simplicial sets. The following conditions are equivalent:

$(1)$

The morphism $f$ is a right fibration.

$(2)$

For every pair of integers $0 < i \leq n$, every lifting problem

\[ \xymatrix@R =50pt@C=50pt{ \Lambda ^{n}_{i} \ar [r]^-{\sigma _0} \ar [d] & X_{} \ar [d]^{f} \\ \Delta ^{n} \ar@ {-->}[ur]^-{\sigma } \ar [r]^-{ \overline{\sigma } } & S_{} } \]

admits a solution (indicated by the dotted arrow).

Lemma 11.6.0.222. Let $q: X \rightarrow S$ be a left fibration of simplicial sets, where $S$ is a Kan complex. Suppose that $q$ induces a surjection $\pi _0(q): \pi _0(X) \rightarrow \pi _0(S)$. Then $q$ is surjective on vertices.

Proof. Fix a vertex $s \in S$. Since $\pi _0(q)$ is surjective, there exists a vertex $x \in X$ for which $q(x)$ and $s$ belong to the same connected component of $\pi _0(S)$. Since $S$ is a Kan complex, we can choose an edge $e: q(x) \rightarrow s$ in the simplicial set $S$. Our assumption that $q$ is a left fibration guarantees that we can write $e = q(\overline{e})$ for some edge $\overline{e}: x \rightarrow \overline{s}$ of the simplicial set $X$. In particular, there exists a vertex $\overline{s} \in X$ satisfying $q( \overline{s} ) = s$. $\square$

Remark 11.6.0.223. Assuming Theorem 5.2.2.20, one can give a more direct proof of ***. Let $q: X \rightarrow S$ be a left fibration of simplicial sets, where $S$ is a Kan complex. To show that $q$ is a Kan fibration, it will suffice (by virtue of Theorem 5.2.2.20) to show that the covariant transport functor $\operatorname {h}\! \mathit{S} \rightarrow \operatorname {h}\! \mathit{\operatorname{Kan}}$ carries every morphism in $\operatorname {h}\! \mathit{S}$ to an invertible morphism in $\operatorname {h}\! \mathit{\operatorname{Kan}}$. This is clear, since the homotopy category $\operatorname {h}\! \mathit{S}$ is a groupoid (Proposition 1.4.6.10).

Remark 11.6.0.224. Assuming Theorem 5.2.2.20, one can give a more direct proof of Proposition 4.4.2.14. Let $q: X \rightarrow S$ be a left fibration of simplicial sets with the property that, for every vertex $s \in S$, the fiber $X_{s}$ is a contractible Kan complex. For each edge $e: s \rightarrow s'$ of $S$, the covariant transport $e_{!}: X_{s} \rightarrow X_{s'}$ is a morphism between contractible Kan complexes, and is therefore automatically a homotopy equivalence. Applying Theorem 5.2.2.20, we deduce that $q$ is a Kan fibration. Since the fibers of $q$ are contractible, Proposition 3.3.7.6 guarantees that $q$ is a trivial Kan fibration.