4.4 Isomorphisms and Isofibrations
Let $\operatorname{\mathcal{C}}$ be an $\infty $-category. Recall that a morphism $u: X \rightarrow Y$ in $\operatorname{\mathcal{C}}$ is an isomorphism if the homotopy class $[u]$ is an isomorphism in the homotopy category $\mathrm{h} \mathit{\operatorname{\mathcal{C}}}$ (Definition 1.4.6.1). Our goal in this section is to study the notion of isomorphism in more detail.
Our first goal is to show that the class of isomorphisms can be characterized by a lifting property. Let $u: X \rightarrow Y$ be an isomorphism in an $\infty $-category $\operatorname{\mathcal{C}}$, and let $f: X \rightarrow Z$ be any other morphism in $\operatorname{\mathcal{C}}$. Then the composition $[f] \circ [u]^{-1} \in \operatorname{Hom}_{\mathrm{h} \mathit{\operatorname{\mathcal{C}}}}(Y,Z)$ can be written as the homotopy class of some morphism $g: Y \rightarrow Z$ in $\operatorname{\mathcal{C}}$. The equality of homotopy classes $[f] = [g] \circ [u]$ is witnessed by some $2$-simplex $\sigma $ which we depict as a diagram
Phrased differently, $u$ and $f$ determine a morphism of simplicial sets $\sigma _0: \Lambda ^2_0 \rightarrow \operatorname{\mathcal{C}}$, and the preceding argument shows that $\sigma _0$ can be extended to a $2$-simplex of $\operatorname{\mathcal{C}}$. In §4.4.2, we extend this argument to simplices of higher dimension. Suppose that we are given an integer $n \geq 2$ and a morphism of simplicial sets $\sigma _0: \Lambda ^ n_{i} \rightarrow \operatorname{\mathcal{C}}$. If $0 < i < n$, then $\sigma _0$ can be extended to an $n$-simplex of $\operatorname{\mathcal{C}}$ by virtue of our assumption that $\operatorname{\mathcal{C}}$ is an $\infty $-category. In the extreme cases $i=0$ and $i=n$, such an extension need not exist. However, we will show that it exists in the case $i=0$ when $\sigma _0$ carries the initial edge $\operatorname{N}_{\bullet }( \{ 0 < 1 \} ) \subseteq \Lambda ^ n_ i$ to an isomorphism in $\operatorname{\mathcal{C}}$, or in the case $i=n$ when $\sigma _0$ carries the final edge $\operatorname{N}_{\bullet }( \{ n-1 < n \} ) \subseteq \Lambda ^ n_{i}$ to an isomorphism in $\operatorname{\mathcal{C}}$ (Theorem 4.4.2.6).
Theorem 4.4.2.6 has a number of useful consequences. For example, it implies that an $\infty $-category $\operatorname{\mathcal{C}}$ is a Kan complex if and only if every morphism of $\operatorname{\mathcal{C}}$ is an isomorphism (Proposition 4.4.2.1). More generally, it implies that any $\infty $-category $\operatorname{\mathcal{C}}$ contains a largest Kan complex, which we will denote by $\operatorname{\mathcal{C}}^{\simeq }$ and refer to as the core of $\operatorname{\mathcal{C}}$ (Construction 1.3.5.4). The construction $\operatorname{\mathcal{C}}\mapsto \operatorname{\mathcal{C}}^{\simeq }$ supplies a link between the theory of $\infty $-categories and the classical homotopy theory of Kan complexes, which will play an important role throughout this book.
Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be an inner fibration of $\infty $-categories. Then, for every object $D \in \operatorname{\mathcal{D}}$, the fiber $\operatorname{\mathcal{C}}_{D} = \{ D\} \times _{\operatorname{\mathcal{D}}} \operatorname{\mathcal{C}}$ is an $\infty $-category (Remark 4.1.1.6). Beware that, in general, this construction behaves poorly with respect to isomorphisms. For example, if the fiber $\operatorname{\mathcal{C}}_{D}$ is nonempty and $D' \in \operatorname{\mathcal{D}}$ is an object which is isomorphic to $D$, then the fiber $\operatorname{\mathcal{C}}_{D'}$ could be empty. One can rule out this sort of behavior by imposing an additional assumption on the functor $F$. In §4.4.1, we introduce the notion of an isofibration of $\infty $-categories (Definition 4.4.1.4). Roughly speaking, an isofibration between $\infty $-categories is an inner fibration which also satisfies a path lifting property for isomorphisms. This condition guarantees that passage to the fiber is a homotopy invariant operation. For example, if $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ is an isofibration of $\infty $-categories, then it restricts to a Kan fibration of cores $F^{\simeq }: \operatorname{\mathcal{C}}^{\simeq } \rightarrow \operatorname{\mathcal{D}}^{\simeq }$ (Proposotion 4.4.3.7).
Let $B$ be a simplicial set containing a simplicial subset $A$. Recall that, for every $\infty $-category $\operatorname{\mathcal{C}}$, the restriction functor $\theta : \operatorname{Fun}(B,\operatorname{\mathcal{C}}) \rightarrow \operatorname{Fun}(A,\operatorname{\mathcal{C}})$ is an inner fibration (Corollary 4.1.4.2). In §4.4.5, we prove that $\theta $ is an isofibration (Corollary 4.4.5.3; see Proposition 4.4.5.1 for a stronger relative statement). The proof is based on the following recognition principle, which we establish in §4.4.4: if $\operatorname{\mathcal{C}}$ is an $\infty $-category and $u: F \rightarrow G$ is a morphism in an $\infty $-category of the form $\operatorname{Fun}(X,\operatorname{\mathcal{C}})$, then $u$ is an isomorphism in $\operatorname{Fun}(X,\operatorname{\mathcal{C}})$ if and only if, for every vertex $x \in X$, the induced map $u_ x: F(x) \rightarrow G(x)$ is an isomorphism in the $\infty $-category $\operatorname{\mathcal{C}}$ (Theorem 4.4.4.4). In other words, if each $u_ x$ admits a homotopy inverse $v_ x: G(x) \rightarrow F(x)$, then we can choose the morphisms $\{ v_ x \} _{x \in X}$ (and homotopies witnessing the identifications $v_ x \circ u_ x \simeq \operatorname{id}_{ F(x)}$ and $u_ x \circ v_ x \simeq \operatorname{id}_{ G(x) }$) to depend functorially on $x \in X$.