In §1.1 and §1.2, we considered two closely related conditions on a simplicial set $S_{\bullet }$:
For $n > 0$ and $0 \leq i \leq n$, every map of simplicial sets $\sigma _0: \Lambda ^{n}_{i} \rightarrow S_{\bullet }$ can be extended to a map $\sigma : \Delta ^{n} \rightarrow S_{\bullet }$.
For $0 < i < n$, every map of simplicial sets $\sigma _0: \Lambda ^{n}_{i} \rightarrow S_{\bullet }$ can be extended uniquely to a map $\sigma : \Delta ^{n} \rightarrow S_{\bullet }$.
Simplicial sets satisfying $(\ast )$ are called Kan complexes and form the basis for a combinatorial approach to homotopy theory, while simplicial sets satisfying $(\ast ')$ can be identified with categories (Propositions 1.2.2.1 and 1.2.3.1). These notions admit a common generalization:
Definition 1.3.0.1. An $\infty $-category is a simplicial set $S_{\bullet }$ which satisfies the following condition:
For $0 < i < n$, every map of simplicial sets $\sigma _0: \Lambda ^{n}_{i} \rightarrow S_{\bullet }$ can be extended to a map $\sigma : \Delta ^{n} \rightarrow S_{\bullet }$.
Remark 1.3.0.2. Condition $(\ast '')$ is commonly known as the weak Kan extension condition. It was introduced by Boardman and Vogt in [MR0420609], who refer to $\infty $-categories as weak Kan complexes. The theory was developed further by Joyal ([MR1935979] and [joyal]), who refers to $\infty $-categories as quasicategories.
Example 1.3.0.3. Every Kan complex is an $\infty $-category. In particular, if $X$ is a topological space, then the singular simplicial set $\operatorname{Sing}_{\bullet }(X)$ is an $\infty $-category.
Example 1.3.0.4. For every category $\operatorname{\mathcal{C}}$, the nerve $\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$ is an $\infty $-category.
Remark 1.3.0.5. We will often abuse terminology by identifying a category $\operatorname{\mathcal{C}}$ with its nerve $\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$ (this abuse is essentially harmless by virtue of Proposition 1.2.2.1). Adopting this convention, we can state Example 1.3.0.4 more simply: every category is an $\infty $-category. To minimize the possibility of confusion, we will sometimes refer to categories as ordinary categories.
Example 1.3.0.6 (Products of $\infty $-Categories). Let $\{ S_{\alpha \bullet } \} _{\alpha \in A}$ be a collection of simplicial sets parametrized by a set $A$, and let $S_{\bullet } = \prod _{\alpha \in A} S_{\alpha \bullet }$ denote their product. If each $S_{\alpha \bullet }$ is an $\infty $-category, then $S_{\bullet }$ is an $\infty $-category. The converse holds provided that each $S_{\alpha \bullet }$ is nonempty.
Example 1.3.0.7 (Coproducts of $\infty $-Categories). Let $\{ S_{\alpha \bullet } \} _{\alpha \in A}$ be a collection of simplicial sets parametrized by a set $A$, and let $S_{\bullet } = \coprod _{\alpha \in A} S_{\alpha \bullet }$ denote their coproduct. For each $0 < i < n$, the restriction map can be identified with the coproduct (formed in the arrow category $\operatorname{Fun}( [1], \operatorname{Set})$) of restriction maps $\theta _{\alpha }: \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( \Delta ^ n, S_{\alpha \bullet }) \rightarrow \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( \Lambda ^ n_ i, S_{\alpha \bullet } )$. It follows that $\theta $ is a surjection if and only if each $\theta _{\alpha }$ is a surjection. Allowing $n$ and $i$ to vary, we conclude that $S_{\bullet }$ is an $\infty $-category if and only if each summand $S_{\alpha \bullet }$ is an $\infty $-category.
\[ \theta : \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( \Delta ^ n, S_{\bullet } ) \rightarrow \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( \Lambda ^ n_ i, S_{\bullet } ) \]
Remark 1.3.0.8. Let $S_{\bullet }$ be a simplicial set. Combining Example 1.3.0.7 with Proposition 1.1.6.13, we deduce that $S_{\bullet }$ is an $\infty $-category if and only if each connected component of $S_{\bullet }$ is an $\infty $-category.
Remark 1.3.0.9. Suppose we are given a filtered diagram of simplicial sets $\{ S(\alpha )_{\bullet } \} $ having colimit $S_{\bullet } = \varinjlim S(\alpha )_{\bullet }$. If each $S(\alpha )_{\bullet }$ is an $\infty $-category, then $S_{\bullet }$ is an $\infty $-category.
Throughout this book, we will generally use calligraphic letters (like $\operatorname{\mathcal{C}}$, $\operatorname{\mathcal{D}}$, and $\operatorname{\mathcal{E}}$) to denote $\infty $-categories, and we will generally describe them using terminology borrowed from category theory. For example, if $\operatorname{\mathcal{C}}= S_{\bullet }$ is an $\infty $-category, then we will refer to vertices of the simplicial set $S_{\bullet }$ as objects of the $\infty $-category $\operatorname{\mathcal{C}}$, and to edges of the simplicial set $S_{\bullet }$ as morphisms of the $\infty $-category $\operatorname{\mathcal{C}}$ (see §1.3.1). One of the central themes of this book is that $\infty $-categories behave much like ordinary categories. In particular, for any $\infty $-category $\operatorname{\mathcal{C}}$, there is a notion of composition for morphisms of $\operatorname{\mathcal{C}}$, which we study in §1.3.4. Given a pair of morphisms $f: X \rightarrow Y$ and $g: Y \rightarrow Z$ in $\operatorname{\mathcal{C}}$ (corresponding to edges $f,g \in S_{1}$ satisfying $d_0(f) = d_1(g)$), the pair $(f,g)$ defines a map of simplicial sets $\sigma _0: \Lambda ^{2}_{1} \rightarrow \operatorname{\mathcal{C}}$. Applying condition $(\ast '')$, we can extend $\sigma _0$ to a $2$-simplex $\sigma $ of $\operatorname{\mathcal{C}}$, which we can think of heuristically as a commutative diagram
In this case, we will refer to the morphism $h = d_1(\sigma )$ as a composition of $f$ and $g$. However, this comes with a caveat: the extension $\sigma $ is usually not unique, so the morphism $h$ is not completely determined by $f$ and $g$. However, we will show that it is unique up to a certain notion of homotopy which we study in §1.3.3. We apply this observation in §1.3.5 to give a concrete description of the homotopy category $\mathrm{h} \mathit{\operatorname{\mathcal{C}}}$ (in the sense of Definition 1.2.5.1) when $\operatorname{\mathcal{C}}$ is an $\infty $-category (see Definition 1.3.5.3 and Proposition 1.3.5.7).