Let $\operatorname{\mathcal{C}}$ be an ordinary category. For every triple of objects $X, Y, Z \in \operatorname{Ob}(\operatorname{\mathcal{C}})$, the composition of morphisms in $\operatorname{\mathcal{C}}$ determines a map
Our goal in this section is to construct an analogous operation in the $\infty $-categorical setting. Here the situation is more subtle: as we saw in §1.3.4, a pair of morphisms $f: X \rightarrow Y$ and $g: Y \rightarrow Z$ in an $\infty $-category $\operatorname{\mathcal{C}}$ generally do not have a unique composition. Nevertheless, we will show that the mapping spaces of Construction 4.6.1.1 can be endowed with a composition law which is well-defined up to homotopy (and even up to a contractible space of choices). To describe this composition law, it will be convenient to introduce a generalization of Construction 4.6.1.1.
Notation 4.6.3.1. Let $\operatorname{\mathcal{C}}$ be a simplicial set containing a (nonempty) finite sequence of vertices $X_0, X_1, \ldots , X_ n$. We let $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(X_{0}, X_1, \cdots , X_ n)$ denote the simplicial set given by the fiber product
\[ \operatorname{Fun}( \Delta ^ n, \operatorname{\mathcal{C}}) \times _{ \operatorname{Fun}( \{ 0, 1, \cdots , n \} , \operatorname{\mathcal{C}}) } \{ (X_0, X_1, \cdots , X_ n) \} . \]
Example 4.6.3.2. Let $\operatorname{\mathcal{C}}$ be a simplicial set containing vertices $X_0$ and $X_1$. Then the simplicial set $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(X_0, X_1)$ of Notation 4.6.3.1 agrees with the morphism space $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(X_0, X_1)$ of Construction 4.6.1.1. In particular, if $\operatorname{\mathcal{C}}$ is an $\infty $-category, then $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(X_0, X_1)$ is a Kan complex (Proposition 4.6.1.8).
Example 4.6.3.3. Let $\operatorname{\mathcal{C}}$ be a simplicial set and let $X_0$ be a vertex of $\operatorname{\mathcal{C}}$. Then the simplicial set $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(X_0)$ of Notation 4.6.3.1 is isomorphic to $\Delta ^0$.
Let $\operatorname{\mathcal{C}}$ be a simplicial set containing a sequence of vertices $X_0, X_1, \ldots , X_ n$. For every pair of integers $0 \leq i < j \leq n$, precomposition with the edge $\Delta ^1 \simeq \operatorname{N}_{\bullet }( \{ i < j \} ) \hookrightarrow \Delta ^ n$ determines a restriction map $\operatorname{Hom}_{\operatorname{\mathcal{C}}}( X_0, X_1, \cdots , X_ n) \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{C}}}( X_ i, X_ j)$.
Proposition 4.6.3.4. Let $q: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be an inner fibration of simplicial sets, and let $X_0, X_1, \ldots , X_ n$ be vertices of $\operatorname{\mathcal{C}}$ having images $\overline{X}_0, \overline{X}_1, \ldots , \overline{X}_ n \in \operatorname{\mathcal{D}}$. Then the restriction map is a trivial Kan fibration of simplicial sets.
\[ \xymatrix { \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X_0, \cdots , X_ n) \ar [d]^{\theta } \\ \operatorname{Hom}_{\operatorname{\mathcal{D}}}( \overline{X}_0, \cdots , \overline{X}_ n) \times _{ \prod _{i=1}^{n} \operatorname{Hom}_{\operatorname{\mathcal{D}}}( \overline{X}_{i-1}, \overline{X}_{i})} \prod _{i=1}^{n} \operatorname{Hom}_{\operatorname{\mathcal{C}}}( X_{i-1}, X_{i} ) } \]
Proof. Let $\operatorname{Spine}[n]$ denote the spine of the standard $n$-simplex $\Delta ^ n$ (see Example 1.4.7.7). Unwinding the definitions, we see that $\theta $ is a pullback of the restriction map
Since $q$ is an inner fibration and the inclusion $\operatorname{Spine}[n] \hookrightarrow \Delta ^ n$ is inner anodyne (Example 1.4.7.7), the morphism $\theta '$ is a trivial Kan fibration (Proposition 4.1.4.4). $\square$
Corollary 4.6.3.5. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category containing objects $X_0, X_1, \ldots , X_ n$. Then the restriction map is a trivial Kan fibration of simplicial sets.
\[ \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X_0, X_1, \cdots , X_ n) \rightarrow \prod _{i=1}^{n} \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X_{i-1}, X_ i) \]
Example 4.6.3.6. Let $\operatorname{\mathcal{C}}$ be an ordinary category containing objects $X_0, X_1, \ldots , X_ n$, which we also regard as objects of the $\infty $-category $\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$. Then the restriction map is an isomorphism of (discrete) simplicial sets.
\[ \theta : \operatorname{Hom}_{\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})}(X_0, X_1, \cdots , X_ n) \rightarrow \prod _{i=1}^{n} \operatorname{Hom}_{\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})}(X_{i-1}, X_ i) \]
Remark 4.6.3.7. It follows from Corollary 4.6.3.5 that the construction endows the collection of objects of $\operatorname{\mathcal{C}}$ with the structure of a Segal category (see Definition
\[ (X_0, X_1, \cdots , X_ n) \mapsto \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X_0, X_1, \cdots , X_ n) \]
). We will return to this point in §.
Corollary 4.6.3.8. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category. For every sequence of objects $X_0, X_1, \cdots , X_ n \in \operatorname{\mathcal{C}}$, the simplicial set $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(X_0, X_1, \cdots , X_ n)$ is a Kan complex.
Proof. Combine Corollary 4.6.3.5 with Proposition 4.6.1.8. $\square$
Construction 4.6.3.9. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category containing objects $X$, $Y$, and $Z$. By virtue of Corollary 4.6.3.5, the restriction map is a trivial Kan fibration, so its homotopy class $[\theta ]$ is an isomorphism in the homotopy category $\mathrm{h} \mathit{\operatorname{Kan}}$. We let denote the morphism in $\mathrm{h} \mathit{\operatorname{Kan}}$ obtained by composing $[\theta ]^{-1}$ with (the homotopy class of) the restriction map $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y,Z) \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Z)$. We will refer to $\circ $ as the composition law on the $\infty $-category $\operatorname{\mathcal{C}}$.
\[ \theta : \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y,Z) \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{C}}}(Y,Z) \times \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y) \]
\[ \circ : \operatorname{Hom}_{\operatorname{\mathcal{C}}}(Y,Z) \times \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y) \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Z) \]
Remark 4.6.3.10. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category containing objects $X$, $Y$, and $Z$. Then the composition law of Construction 4.6.3.9 induces a map of sets Concretely, this map is given by the construction $([g], [f]) \mapsto [h]$, where $h$ is a composition of $f$ and $g$ in the sense of Definition 1.3.4.1.
\[ \circ : \operatorname{Hom}_{\operatorname{\mathcal{C}}}(Y,Z) \times \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y) \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Z) \]
\[ \pi _0( \operatorname{Hom}_{\operatorname{\mathcal{C}}}(Y,Z) ) \times \pi _0( \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y) ) \rightarrow \pi _0( \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Z) ). \]
Proposition 4.6.3.11 (Unitality). Let $\operatorname{\mathcal{C}}$ be an $\infty $-category containing a pair of objects $X$ and $Y$. Then:
The composition
is equal to the identity (in the homotopy category of Kan complexes $\mathrm{h} \mathit{\operatorname{Kan}}$).
The composition
is equal to the identity (in the homotopy category of Kan complexes $\mathrm{h} \mathit{\operatorname{Kan}}$).
\[ \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y) \simeq \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y) \times \{ \operatorname{id}_{X} \} \hookrightarrow \operatorname{Hom})_{\operatorname{\mathcal{C}}}(X,Y) \times \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,X) \xrightarrow {\circ } \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y) \]
\[ \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y) \simeq \{ \operatorname{id}_{Y} \} \times \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y) \hookrightarrow \operatorname{Hom})_{\operatorname{\mathcal{C}}}(Y,Y) \times \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y) \xrightarrow {\circ } \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y) \]
Proof. There is a diagram of Kan complexes
where the left diagonal arrow is induced by the map $\sigma ^0: [2] \rightarrow [1]$ of Notation 1.1.1.9 and the right diagonal arrow is induced by the map $\delta ^{1}: [1] \rightarrow [2]$ of Notation 1.1.1.8. Here the solid arrows are well-defined as morphisms of simplicial sets, while the dotted arrow is well-defined only as a morphism in the homotopy category $\mathrm{h} \mathit{\operatorname{Kan}}$. We now observe that the triangle on the left is strictly commutative, the triangle on the right commutes up to homotopy (by the construction of the composition law $\circ $). Assertion $(1)$ follows from the observation that the composition of the diagonal arrows is the identity on the Kan complex $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)$ (since $\sigma ^{0} \circ \delta ^{1}$ is the identity on the object $[1] \in \operatorname{{\bf \Delta }}$). Assertion $(2)$ follows by a similar argument. $\square$
Proposition 4.6.3.12 (Associativity). Let $\operatorname{\mathcal{C}}$ be an $\infty $-category containing objects $W$, $X$, $Y$, and $Z$. Then the diagram commutes (in the homotopy category of Kan complexes $\mathrm{h} \mathit{\operatorname{Kan}}$).
4.31
\begin{equation} \begin{gathered}\label{equation:homotopy-composition-diagram} \xymatrix { \operatorname{Hom}_{\operatorname{\mathcal{C}}}(Y,Z) \times \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y) \times \operatorname{Hom}_{\operatorname{\mathcal{C}}}(W,X) \ar [r]^{\circ } \ar [d]^{\circ } & \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Z) \times \operatorname{Hom}_{\operatorname{\mathcal{C}}}(W,X) \ar [d]^{\circ } \\ \operatorname{Hom}_{\operatorname{\mathcal{C}}}(Y,Z) \times \operatorname{Hom}_{\operatorname{\mathcal{C}}}(W,Y) \ar [r]^{\circ } & \operatorname{Hom}_{\operatorname{\mathcal{C}}}(W,Z) } \end{gathered} \end{equation}
Proof. By virtue of Corollary 4.6.3.5, (4.31) is isomorphic to the diagram of restriction maps
which commutes in the category of simplicial sets (and therefore also in the homotopy category $\mathrm{h} \mathit{\operatorname{Kan}}$). $\square$
Construction 4.6.3.13 (The Enriched Homotopy Category). Let $\mathrm{h} \mathit{\operatorname{Kan}}$ denote the homotopy category of Kan complexes, which we endow with the monoidal structure given by cartesian products. To every $\infty $-category $\operatorname{\mathcal{C}}$, we define an $\mathrm{h} \mathit{\operatorname{Kan}}$-enriched category $\mathrm{h} \mathit{\operatorname{\mathcal{C}}}$ as follows:
The objects of $\mathrm{h} \mathit{\operatorname{\mathcal{C}}}$ are the objects of $\operatorname{\mathcal{C}}$.
For every pair of objects $X,Y \in \operatorname{\mathcal{C}}$, the Kan complex $\underline{\operatorname{Hom}}_{ \mathrm{h} \mathit{\operatorname{\mathcal{C}}} }(X,Y)$ is the morphism space $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)$ of Construction 4.6.1.1.
For every object $X \in \operatorname{\mathcal{C}}$, the unit map $\Delta ^{0} \rightarrow \underline{\operatorname{Hom}}_{\mathrm{h} \mathit{\operatorname{\mathcal{C}}}}(X,X)$ is the homotopy class of the inclusion $\{ \operatorname{id}_{X} \} \hookrightarrow \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,X)$.
For every triple of objects $X,Y,Z \in \operatorname{\mathcal{C}}$, the composition law
is given by Construction 4.6.3.9.
\[ \circ : \underline{\operatorname{Hom}}_{\mathrm{h} \mathit{\operatorname{\mathcal{C}}}}(Y,Z) \times \underline{\operatorname{Hom}}_{\mathrm{h} \mathit{\operatorname{\mathcal{C}}}}(X,Y) \rightarrow \underline{\operatorname{Hom}}_{\mathrm{h} \mathit{\operatorname{\mathcal{C}}}}(X,Z) \]
Note that this definition satisfies the axiomatics of Definition 2.1.7.1 by virtue of Propositions 4.6.3.11 and 4.6.3.12 We will refer to $\mathrm{h} \mathit{\operatorname{\mathcal{C}}}$ as the enriched homotopy category of the $\infty $-category $\operatorname{\mathcal{C}}$.
Remark 4.6.3.14. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $\mathrm{h} \mathit{\operatorname{\mathcal{C}}}$ denote the enriched homotopy category of $\operatorname{\mathcal{C}}$. Then $\mathrm{h} \mathit{\operatorname{\mathcal{C}}}$ has an underlying category (Example 2.1.7.5), which we will also denote by $\mathrm{h} \mathit{\operatorname{\mathcal{C}}}$. Concretely, this category can be described as follows:
The objects of $\mathrm{h} \mathit{\operatorname{\mathcal{C}}}$ are the objects of $\operatorname{\mathcal{C}}$.
For every pair of objects $X,Y \in \operatorname{\mathcal{C}}$, we have
In other words, $\operatorname{Hom}_{\mathrm{h} \mathit{\operatorname{\mathcal{C}}}}(X,Y)$ can be identified with the set of homotopy classes of morphisms from $X$ to $Y$ in the $\infty $-category $\operatorname{\mathcal{C}}$.
\[ \operatorname{Hom}_{\mathrm{h} \mathit{\operatorname{\mathcal{C}}}}(X,Y) = \operatorname{Hom}_{ \mathrm{h} \mathit{\operatorname{Kan}} }( \Delta ^0, \underline{\operatorname{Hom}}_{\mathrm{h} \mathit{\operatorname{\mathcal{C}}}}( X,Y) ) \simeq \pi _0( \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y) ). \]
By virtue of Remark 4.6.3.10, the composition of morphisms in the category $\mathrm{h} \mathit{\operatorname{\mathcal{C}}}$ agrees with the composition law of Construction 1.3.5.1. In other words, we can identify $\mathrm{h} \mathit{\operatorname{\mathcal{C}}}$ with the homotopy category constructed in §1.3.5.