Proposition 8.5.0.1. The contents of this tag are now at Corollary 7.1.7.20.

## 8.5 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).

Remark 8.5.0.2. The contents of this tag are now at Proposition 7.1.7.19.

Proposition 8.5.0.3. The contents of this tag are now split between Corollaries 7.2.2.3 and 7.2.2.8.

Proposition 8.5.0.4. The contents of this tag are now at Corollary None.

Corollary 8.5.0.5. The contents of this tag are now at Proposition 8.5.0.1.

Remark 8.5.0.6. The contents of this tag are now at Corollary 7.1.4.2.

Corollary 8.5.0.7. The contents of this tag are now at Variant 7.1.5.6.

Example 8.5.0.8. The contents of this tag are now at Remark 7.1.2.3.

Example 8.5.0.9. The contents of this tag are now at Proposition 7.1.3.12.

Corollary 8.5.0.10. The contents of this tag are now at Proposition 8.5.0.3.

Lemma 8.5.0.11. The contents of this tag are now at Example 5.2.4.18.

Proposition 8.5.0.12. The contents of this tag are now at Corollary 4.3.3.24.

Definition 8.5.0.13. See Definition 5.6.2.1.

Corollary 8.5.0.14 (The Universal Cocartesian Fibration). The contents of this tag are now at Theorem 5.6.0.2.

Theorem 8.5.0.15 (Universality Theorem: Preliminary Version). See Theorem 5.6.0.2.

Corollary 8.5.0.16. The contents of this tag are now at Proposition 5.5.4.19.

Example 8.5.0.17 (Parametrized Covariant Transport). The contents of this tag are now at Definition 5.2.7.1.

Corollary 8.5.0.18. The contents of this tag are now at Proposition 5.2.7.8.

Proposition 8.5.0.19. The contents of this tag are now at Corollary 4.3.7.13.

Corollary 8.5.0.20. This tag is now at Lemma 8.8.2.6.

Example 8.5.0.21. The contents of this tag are now at Notation 5.2.2.5.

Example 8.5.0.22. The contents of this tag are now at Notation 5.2.2.13.

Example 8.5.0.23 (Parametrized Contravariant Transport). The contents of this tag are now at Variant 5.2.7.6.

Proposition 8.5.0.24. The contents of this tag are now at Corollary 4.6.4.18.

Example 8.5.0.25. The contents of this Example are now contained in Corollary 2.3.4.6.

Remark 8.5.0.26. The contents of this tag are now at Exercise None.

Remark 8.5.0.27. The contents of this tag are now at Construction 2.2.8.12.

Proposition 8.5.0.28. The contents of this tag are now at Corollary 5.5.2.13.

Variant 8.5.0.29. The contents of this tag are now contained in Definition 5.5.2.1.

Proposition 8.5.0.30. The contents of this tag are now at Exercise 5.0.0.6.

Variant 8.5.0.31. The contents of this tag are now part of Definition 5.0.0.1.

Proposition 8.5.0.32. The contents of this tag are now at Proposition 5.2.3.1.

Proposition 8.5.0.33. The contents of this tag are now at Corollary 4.4.5.9.

Lemma 8.5.0.34. The contents of this tag are now at Corollary None.

Lemma 8.5.0.35. The contents of this tag are now at Proposition 4.4.2.14.

Exercise 8.5.0.36. The contents of this tag are now mostly at Example None.

Proposition 8.5.0.37. The contents of this tag are essentially contained in Proposition 5.2.2.12.

Lemma 8.5.0.38. The contents of this tag are now at Corollary 4.4.3.8.

Proposition 8.5.0.39. The contents of this tag are now at Corollary 4.4.3.11.

Corollary 8.5.0.40. The contents of this tag are now contained in Proposition 4.4.2.13.

Lemma 8.5.0.41. The contents of this tag are now contained in Example 4.4.1.10.

Proposition 8.5.0.42. 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 8.5.0.43 (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

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 8.5.0.47).

Example 8.5.0.44. The contents of this tag can now be found in Remark 4.2.1.4.

Example 8.5.0.45. 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 8.5.0.43). 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 8.5.0.46. 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 8.5.0.47. 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

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 8.5.0.43), 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 8.5.0.48. 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

is a trivial Kan fibration.

Lemma 8.5.0.49. Let $f: A_{} \hookrightarrow B_{}$ and $f': A'_{} \hookrightarrow B'_{}$ be monomorphisms of simplicial sets. If either $f$ is anodyne, then the induced map

is anodyne.

Proposition 8.5.0.50. 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 8.5.0.51. 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 8.5.0.52. Assuming Theorem 5.2.2.15, 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 5.2.2.15) 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 1.3.6.10).

Remark 8.5.0.53. Assuming Theorem 5.2.2.15, 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 5.2.2.15, we deduce that $q$ is a Kan fibration. Since the fibers of $q$ are contractible, Proposition 3.3.7.4 guarantees that $q$ is a trivial Kan fibration.