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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).

Corollary 11.6.0.6. 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 4.7.5.5, 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 4.7.5.14). By construction, we have a pullback diagram of simplicial sets

11.8
\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 (11.8) is also a categorical pullback square (Corollary 4.5.2.27). Using Corollary 4.7.5.17, 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 4.7.5.11. $\square$

Corollary 11.6.0.12. 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 11.6.0.17. Let $(A, \leq )$ be a partially ordered set. Combining Exercise 11.6.0.15 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 11.6.0.33. 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 11.6.0.45. 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}}\operatorname{\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}}\operatorname{\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 4.5.8.3, 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

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

where $\iota $ is a monomorphism of simplicial sets given by precomposition with $c$. Since $c$ is a categorical equivalence of simplicial sets (Theorem 4.5.8.8), the functor $\iota $ is an equivalence of $\infty $-categories: this follows by applying Corollary 4.5.2.32 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 4.6.4.17 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 11.6.0.52. 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 11.6.0.39 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 11.6.0.129. 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 11.6.0.130 (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 11.6.0.134).

Example 11.6.0.132. 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 11.6.0.130). Then Proposition 2.2.1.16 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 11.6.0.133. 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 $\mathrm{h} \mathit{\operatorname{\mathcal{C}}}$ 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 $u: \operatorname{\mathcal{C}}\rightarrow \operatorname{N}_{\bullet }( \mathrm{h} \mathit{\operatorname{\mathcal{C}}} )$ of Construction 1.4.5.6 exhibits $\mathrm{h} \mathit{\operatorname{\mathcal{C}}}$ as a homotopy category of $\operatorname{\mathcal{C}}$, by virtue of Proposition 1.4.5.7.

Example 11.6.0.134. 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 2.2.1.15).

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 11.6.0.130), 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 11.6.0.135. 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 11.6.0.136. 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 11.6.0.137. 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 11.6.0.138. 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 11.6.0.139. 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 $\mathrm{h} \mathit{S} \rightarrow \mathrm{h} \mathit{\operatorname{Kan}}$ carries every morphism in $\mathrm{h} \mathit{S}$ to an invertible morphism in $\mathrm{h} \mathit{\operatorname{Kan}}$. This is clear, since the homotopy category $\mathrm{h} \mathit{S}$ is a groupoid (Proposition 1.4.6.10).

Remark 11.6.0.140. 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.