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6.5 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).

Proposition 6.5.0.4. 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 6.5.0.5 (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 6.5.0.9).

Example 6.5.0.7. 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 6.5.0.5). 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 6.5.0.8. The contents of this tag are now contained in Proposition 1.3.5.7.

Let $\operatorname{\mathcal{C}}$ be an $\infty $-category. Then the homotopy category $\mathrm{h} \mathit{\operatorname{\mathcal{C}}}$ constructed in Definition 1.3.5.3 is also a homotopy category of $\operatorname{\mathcal{C}}$ in the sense of Definition 1.2.5.1. More precisely, the map $u: \operatorname{\mathcal{C}}\rightarrow \operatorname{N}_{\bullet }( \mathrm{h} \mathit{\operatorname{\mathcal{C}}} )$ of Construction 1.3.5.6 exhibits $\mathrm{h} \mathit{\operatorname{\mathcal{C}}}$ as a homotopy category of $\operatorname{\mathcal{C}}$, by virtue of Proposition 1.3.5.7.

Example 6.5.0.9. 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 6.5.0.5), 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 6.5.0.10. 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 6.5.0.11. 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 6.5.0.12. 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).