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4.4.1 Isofibrations of $\infty$-Categories

Let us begin by reviewing a bit of classical category theory.

Definition 4.4.1.1. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor between categories. We say that $F$ is an isofibration if it satisfies the following condition:

$(\ast )$

For every object $C \in \operatorname{\mathcal{C}}$ and every isomorphism $u: D \rightarrow F(C)$ in the category $\operatorname{\mathcal{D}}$, there exists an isomorphism $\overline{u}: \overline{D} \rightarrow C$ in the category $\operatorname{\mathcal{C}}$ satisfying $F(\overline{u}) = u$.

Example 4.4.1.2. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor between categories. If $F$ is a fibration in groupoids (or an opfibration in groupoids), then $F$ is an isofibration. For a more general statement, see Example 4.4.1.10.

The notion of isofibration is self-dual:

Proposition 4.4.1.3. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor between categories. Then $F$ is an isofibration if and only if the opposite functor $F^{\operatorname{op}}: \operatorname{\mathcal{C}}^{\operatorname{op}} \rightarrow \operatorname{\mathcal{D}}^{\operatorname{op}}$ is an isofibration.

Proof. Assume that $F$ is an isofibration; we will show that $F^{\operatorname{op}}$ is also an isofibration (the reverse implication follows by the same argument). Fix an object $C \in \operatorname{\mathcal{C}}$ and an isomorphism $u: F(C) \rightarrow D$ in the category $\operatorname{\mathcal{D}}$. Since $F$ is an isofibration, the inverse isomorphism $u^{-1}: D \rightarrow F(C)$ can be lifted to an isomorphism $v: \overline{D} \rightarrow C$ in the category $\operatorname{\mathcal{C}}$. Then $v^{-1}: C \rightarrow \overline{D}$ satisfies $F( v^{-1} ) = u$. $\square$

We now introduce an $\infty$-categorical counterpart of Definition 4.4.1.1.

Definition 4.4.1.4. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor between $\infty$-categories. We say that $F$ is an isofibration if it is an inner fibration (Definition 4.1.1.1) which satisfies the following additional condition:

$(\ast )$

For every object $C \in \operatorname{\mathcal{C}}$ and every isomorphism $u: D \rightarrow F(C)$ in the category $\operatorname{\mathcal{D}}$, there exists an isomorphism $\overline{u}: \overline{D} \rightarrow C$ in the category $\operatorname{\mathcal{C}}$ satisfying $F(\overline{u}) = u$.

Example 4.4.1.5. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor between ordinary categories. Then $F$ is an isofibration (in the sense of Definition 4.4.1.1) if and only if the induced map of simplicial sets $\operatorname{N}_{\bullet }(F): \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) \rightarrow \operatorname{N}_{\bullet }(\operatorname{\mathcal{D}})$ is an isofibration of $\infty$-categories. This follows from the observation that $\operatorname{N}_{\bullet }(F)$ is automatically an inner fibration (see Proposition 4.1.1.10).

Example 4.4.1.6. Let $\operatorname{\mathcal{C}}$ be an $\infty$-category and let $\operatorname{\mathcal{D}}$ be an ordinary category. By virtue of Proposition 4.1.1.10, every functor $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{N}_{\bullet }(\operatorname{\mathcal{D}})$ is automatically an inner fibration. If every isomorphism in $\operatorname{\mathcal{D}}$ is an identity morphism, then $F$ is also an isofibration. In particular, every functor of $\infty$-categories $\operatorname{\mathcal{C}}\rightarrow \Delta ^ n$ is automatically an isofibration.

Proposition 4.4.1.7. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be an inner fibration between $\infty$-categories. Then $F$ is an isofibration of $\infty$-categories (in the sense of Definition 4.4.1.4) if and only if the induced functor of homotopy categories $f: \mathrm{h} \mathit{\operatorname{\mathcal{C}}} \rightarrow \mathrm{h} \mathit{\operatorname{\mathcal{D}}}$ is an isofibration of ordinary categories (in the sense of Definition 4.4.1.1).

Proof. Assume first that $F$ is an isofibration and let $C \in \operatorname{\mathcal{C}}$ be an object, and let $[u]: D \rightarrow F(C)$ be an isomorphism in the homotopy category $\mathrm{h} \mathit{\operatorname{\mathcal{D}}}$, given by the homotopy class of some morphism $u: D \rightarrow F(C)$ in the $\infty$-category $\operatorname{\mathcal{D}}$. Then $u$ is an isomorphism, so our assumption that $F$ is an isofibration guarantees that we can lift $u$ to an isomorphism $\overline{u}: \overline{D} \rightarrow C$ in the $\infty$-category $\operatorname{\mathcal{C}}$. The homotopy class $[ \overline{u} ]$ is then an isomorphism in the homotopy category $\mathrm{h} \mathit{\operatorname{\mathcal{C}}}$ satisfying $f( [ \overline{u}] ) = [u]$. Allowing $C$ and $[u]$ to vary, we conclude that $f$ is an isofibration of ordinary categories.

Now suppose that $f$ is an isofibration, let $C \in \operatorname{\mathcal{C}}$ be an object, and let $u: D \rightarrow F(C)$ be an isomorphism in the $\infty$-category $\operatorname{\mathcal{D}}$. Then the homotopy class $[u]: D \rightarrow F(C)$ is an isomorphism in the homotopy category $\mathrm{h} \mathit{\operatorname{\mathcal{D}}}$. Invoking our assumption that $f$ is an isofibration, we conclude that there exists an isomorphism $[v]: \overline{D} \rightarrow C$ in the homotopy category $\mathrm{h} \mathit{\operatorname{\mathcal{C}}}$ satisfying $f( [v] ) = [u]$. Then $[v]$ can be realized as the homotopy class of some morphism $v: \overline{D} \rightarrow C$ in the $\infty$-category $\operatorname{\mathcal{C}}$, which is automatically an isomorphism. The equation $f([v]) = [u]$ guarantees that there exists a homotopy from $F(v)$ to $u$ in the $\infty$-category $\operatorname{\mathcal{D}}$, given by a $2$-simplex $\sigma :$

$\xymatrix@R =50pt@C=50pt{ & F(C) \ar [dr]^{ \operatorname{id}_{ F(C) }} & \\ D \ar [ur]^{ F(v) } \ar [rr]^{ u } & & F(C). }$

Since $F$ is an inner fibration, it has the right lifting property with respect to the inclusion $\Lambda ^{2}_{1} \hookrightarrow \Delta ^2$. We can therefore lift $\sigma$ to a $2$-simplex $\overline{\sigma }:$

$\xymatrix@R =50pt@C=50pt{ & C \ar [dr]^{ \operatorname{id}_{C }} & \\ \overline{D} \ar [ur]^{ v } \ar [rr]^{ \overline{u} } & & C. }$

in the $\infty$-category $\operatorname{\mathcal{C}}$. Since $v$ and $\operatorname{id}_{C}$ are isomorphisms, it follows that $\overline{u}$ is an isomorphism (Remark 1.3.6.3). Allowing $C$ and $u$ to vary, we conclude that $F$ is an isofibration of $\infty$-categories. $\square$

Corollary 4.4.1.8. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor between $\infty$-categories. Then $F$ is an isofibration if and only if the opposite functor $F^{\operatorname{op}}: \operatorname{\mathcal{C}}^{\operatorname{op}} \rightarrow \operatorname{\mathcal{D}}^{\operatorname{op}}$ is an isofibration.

Remark 4.4.1.9. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ and $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{E}}$ be isofibrations of $\infty$-categories. Then the composition $G \circ F$ is also an isofibration of $\infty$-categories (for a more general statement, see Remark 4.5.5.13).

Example 4.4.1.10. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a right fibration between $\infty$-categories. Then $F$ is an inner fibration (Remark 4.2.1.4), and any isomorphism $u: D \rightarrow F(C)$ can be lifted to a morphism $\overline{u}: \overline{D} \rightarrow C$ in $\operatorname{\mathcal{C}}$, which is automatically an isomorphism by virtue of Proposition 4.4.2.11. It follows that $F$ is an isofibration. Similarly, any left fibration of $\infty$-categories is an isofibration. For a more general statement, see Corollary 5.7.7.5.

Example 4.4.1.11 (Replete Subcategories). Let $\operatorname{\mathcal{C}}$ be an $\infty$-category and let $\operatorname{\mathcal{C}}' \subseteq \operatorname{\mathcal{C}}$ be a subcategory (Definition 4.1.2.2). The following conditions are equivalent:

$(1)$

The inclusion functor $\operatorname{\mathcal{C}}' \hookrightarrow \operatorname{\mathcal{C}}$ is an isofibration.

$(2)$

If $u: X \rightarrow Y$ is an isomorphism in $\operatorname{\mathcal{C}}$ and the object $Y$ belongs to the subcategory $\operatorname{\mathcal{C}}'$, then the isomorphism $u$ also belongs to the subcategory $\operatorname{\mathcal{C}}'$ (and, in particular, the object $X$ also belongs to $\operatorname{\mathcal{C}}'$).

$(3)$

If $u: X \rightarrow Y$ is an isomorphism in $\operatorname{\mathcal{C}}$ and the object $X$ belongs to the subcategory $\operatorname{\mathcal{C}}'$, then the isomorphism $u$ also belongs to the subcategory $\operatorname{\mathcal{C}}'$ (and, in particular, the object $Y$ also belongs to $\operatorname{\mathcal{C}}'$).

If these conditions are satisfied, then we say that the subcategory $\operatorname{\mathcal{C}}' \subseteq \operatorname{\mathcal{C}}$ is replete.

Exercise 4.4.1.12. Let $X$ be a Kan complex, and let $Y \subseteq X$ be a simplicial subset. Show that $Y$ is a summand of $X$ (Definition 1.1.6.1) if and only if it is a replete full subcategory of $X$.

Example 4.4.1.13. Let $\operatorname{\mathcal{C}}$ be an $\infty$-category, and let $\operatorname{Isom}(\operatorname{\mathcal{C}})$ denote the full subcategory of $\operatorname{Fun}(\Delta ^1, \operatorname{\mathcal{C}})$ spanned by the isomorphisms in $\operatorname{\mathcal{C}}$. Then the subcategory $\operatorname{Isom}(\operatorname{\mathcal{C}}) \subseteq \operatorname{Fun}(\Delta ^1, \operatorname{\mathcal{C}})$ is replete. Unwinding the definitions, this amounts to the observation that for every commutative diagram

$\xymatrix@R =50pt@C=50pt{ X \ar [r]^-{u} \ar [d]^{v} & Y \ar [d]^{v'} \\ X' \ar [r]^-{u'} & Y' }$

in the $\infty$-category $\operatorname{\mathcal{C}}$ where $u$, $v$, and $v'$ are isomorphisms, the morphism $u'$ is also an isomorphism. This follows immediately from the two-out-of-three property of Remark 1.3.6.3.