4.1 Inner Fibrations
Recall that a simplicial set $X$ is an $\infty $-category if, for every pair of integers $0 < i < n$, every morphism of simplicial sets $\sigma _0: \Lambda ^{n}_{i} \rightarrow X$ can be extended to an $n$-simplex of $X$ (Definition 1.4.0.1). The goal of this section is to introduce and study a relative version of this condition. We say that a morphism of simplicial sets $q: X \rightarrow S$ is an inner fibration if it is weakly right orthogonal to the horn inclusion $\Lambda ^{n}_{i} \hookrightarrow \Delta ^ n$ for $0 < i < n$ (Definition 4.1.1.1). In the special case $S = \Delta ^0$, this reduces to the assumption that $X$ is an $\infty $-category (Example 4.1.1.2). More generally, we will see in §4.1.1 that a morphism $q: X \rightarrow S$ is an inner fibration if and only if the inverse image of every simplex of $S$ is an $\infty $-category (Remark 4.1.1.13).
Let $\operatorname{\mathcal{C}}$ be an $\infty $-category. We will say that a simplicial subset $\operatorname{\mathcal{C}}' \subseteq \operatorname{\mathcal{C}}$ is a subcategory of $\operatorname{\mathcal{C}}$ if the inclusion map $\operatorname{\mathcal{C}}' \hookrightarrow \operatorname{\mathcal{C}}$ is an inner fibration (Definition 4.1.2.2). In this case, $\operatorname{\mathcal{C}}'$ is also an $\infty $-category, whose homotopy category $\mathrm{h} \mathit{\operatorname{\mathcal{C}}'}$ can be identified with a subcategory of $\mathrm{h} \mathit{\operatorname{\mathcal{C}}}$ (in the sense of classical category theory). In §4.1.2, we show that every subcategory of $\mathrm{h} \mathit{\operatorname{\mathcal{C}}}$ can be obtained (uniquely) in this way: more precisely, the construction $\operatorname{\mathcal{C}}' \mapsto \mathrm{h} \mathit{\operatorname{\mathcal{C}}'}$ induces a bijection from the set of subcategories of $\operatorname{\mathcal{C}}$ to the set of subcategories of $\mathrm{h} \mathit{\operatorname{\mathcal{C}}}$ (Proposition 4.1.2.10).
Recall that a morphism of simplicial sets $i: A \hookrightarrow B$ is said to be inner anodyne if it can be constructed from inner horn inclusions $\Lambda ^ n_ i \hookrightarrow \Delta ^ n$ using pushouts, retracts, and transfinite composition (Definition 1.5.6.4). It follows immediately from the definitions that a morphism of simplicial sets $q: X \rightarrow S$ is an inner fibration if and only if it is weakly right orthogonal to all inner anodyne morphisms (Proposition 4.1.3.1). In §4.1.3, we use a version of Quillen's small object argument (Proposition 4.1.3.2) to show that, conversely, a morphism $i: A \hookrightarrow B$ is inner anodyne if and only if it is weakly left orthogonal to every inner fibration (Corollary 4.1.3.4).
If $\operatorname{\mathcal{C}}$ is an $\infty $-category and $K$ is an arbitrary simplicial set, then the simplicial set $\operatorname{Fun}(K, \operatorname{\mathcal{C}})$ is also an $\infty $-category (Theorem 1.5.3.7). In §4.1.4, we establish a relative form of this result: if $q: X \rightarrow S$ is an inner fibration of simplicial sets, then postcomposition with $q$ induces another inner fibration $\operatorname{Fun}(K,X) \rightarrow \operatorname{Fun}(K,S)$ (Corollary 4.1.4.3). This is a special case of a more general result (Proposition 4.1.4.1), which is essentially equivalent to the stability of inner anodyne morphisms under the formation of pushout-products (see Lemma 1.5.7.5).