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4.6.8 Composition of Morphisms

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

\[ \circ : \operatorname{Hom}_{\operatorname{\mathcal{C}}}(Y,Z) \times \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y) \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Z). \]

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 4.6.8.1. 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 4.6.8.2. 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 4.6.8.1 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 4.6.1.10).

Example 4.6.8.3. 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 4.6.8.1 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 4.6.8.4. 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

\[ \xymatrix@R =50pt@C=50pt{ \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X_0, \cdots , X_ n) \ar [d]^{\theta } \\ \operatorname{Hom}_{\operatorname{\mathcal{D}}}( \overline{X}_0, \cdots , \overline{X}_ n) \times _{ \prod _{i=1}^{n} \operatorname{Hom}_{\operatorname{\mathcal{D}}}( \overline{X}_{i-1}, \overline{X}_{i})} \prod _{i=1}^{n} \operatorname{Hom}_{\operatorname{\mathcal{C}}}( X_{i-1}, X_{i} ) } \]

is a trivial Kan fibration of simplicial sets.

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

\[ \theta ': \operatorname{Fun}( \Delta ^ n, \operatorname{\mathcal{C}}) \rightarrow \operatorname{Fun}( \operatorname{Spine}[n], \operatorname{\mathcal{C}}) \times _{ \operatorname{Fun}( \operatorname{Spine}[n], \operatorname{\mathcal{D}}) } \operatorname{Fun}( \Delta ^ n, \operatorname{\mathcal{D}}). \]

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 4.6.8.5. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category containing objects $X_0, X_1, \ldots , X_ n$. Then the restriction map

\[ \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X_0, X_1, \cdots , X_ n) \rightarrow \prod _{i=1}^{n} \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X_{i-1}, X_ i) \]

is a trivial Kan fibration of simplicial sets.

Example 4.6.8.6. 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

\[ \theta : \operatorname{Hom}_{\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})}(X_0, X_1, \cdots , X_ n) \rightarrow \prod _{i=1}^{n} \operatorname{Hom}_{\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})}(X_{i-1}, X_ i) \]

is an isomorphism of (discrete) simplicial sets.

Remark 4.6.8.7. It follows from Corollary 4.6.8.5 that the construction

\[ (X_0, X_1, \cdots , X_ n) \mapsto \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X_0, X_1, \cdots , X_ n) \]

endows the collection of objects of $\operatorname{\mathcal{C}}$ with the structure of a Segal category (see Definition ). We will return to this point in §.

Corollary 4.6.8.8. 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.

Construction 4.6.8.9. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category containing objects $X$, $Y$, and $Z$. By virtue of Corollary 4.6.8.5, the restriction map

\[ \theta : \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y,Z) \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{C}}}(Y,Z) \times \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y) \]

is a trivial Kan fibration, so its homotopy class $[\theta ]$ is an isomorphism in the homotopy category $\mathrm{h} \mathit{\operatorname{Kan}}$. We let

\[ \circ : \operatorname{Hom}_{\operatorname{\mathcal{C}}}(Y,Z) \times \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y) \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Z) \]

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 $\operatorname{\mathcal{C}}$.

Remark 4.6.8.10. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category containing objects $X$, $Y$, and $Z$. Then the composition law

\[ \circ : \operatorname{Hom}_{\operatorname{\mathcal{C}}}(Y,Z) \times \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y) \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Z) \]

of Construction 4.6.8.9 induces a map of sets

\[ \pi _0( \operatorname{Hom}_{\operatorname{\mathcal{C}}}(Y,Z) ) \times \pi _0( \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y) ) \rightarrow \pi _0( \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Z) ). \]

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.

Proposition 4.6.8.11 (Unitality). Let $\operatorname{\mathcal{C}}$ be an $\infty $-category containing a pair of objects $X$ and $Y$. Then:

$(1)$

The composition

\[ \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y) \simeq \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y) \times \{ \operatorname{id}_{X} \} \hookrightarrow \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y) \times \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,X) \xrightarrow {\circ } \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y) \]

is equal to the identity (in the homotopy category of Kan complexes $\mathrm{h} \mathit{\operatorname{Kan}}$).

$(2)$

The composition

\[ \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y) \simeq \{ \operatorname{id}_{Y} \} \times \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y) \hookrightarrow \operatorname{Hom}_{\operatorname{\mathcal{C}}}(Y,Y) \times \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y) \xrightarrow {\circ } \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y) \]

is equal to the identity (in the homotopy category of Kan complexes $\mathrm{h} \mathit{\operatorname{Kan}}$).

Proof. There is a diagram of Kan complexes

\[ \xymatrix@R =50pt@C=50pt{ & \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,X,Y) \ar [dr] \ar [d] & \\ \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y) \ar [r]^-{ \operatorname{id}\times \{ \operatorname{id}_{X} \} } \ar [ur] & \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y) \times \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,X) \ar@ {-->}[r]^-{\circ } & \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y), } \]

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 4.6.8.12 (Associativity). Let $\operatorname{\mathcal{C}}$ be an $\infty $-category containing objects $W$, $X$, $Y$, and $Z$. Then the diagram

4.68
\begin{equation} \begin{gathered}\label{equation:homotopy-composition-diagram} \xymatrix@R =50pt@C=30pt{ \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}

commutes (in the homotopy category of Kan complexes $\mathrm{h} \mathit{\operatorname{Kan}}$).

Proof. By virtue of Corollary 4.6.8.5, (4.68) is isomorphic to the diagram of restriction maps

\[ \xymatrix@R =50pt@C=50pt{ \operatorname{Hom}_{\operatorname{\mathcal{C}}}(W,X,Y,Z) \ar [r] \ar [d] & \operatorname{Hom}_{\operatorname{\mathcal{C}}}(W,X,Z) \ar [d] \\ \operatorname{Hom}_{\operatorname{\mathcal{C}}}(W,Y,Z) \ar [r] & \operatorname{Hom}_{\operatorname{\mathcal{C}}}(W,Z), } \]

which commutes in the category of simplicial sets (and therefore also in the homotopy category $\mathrm{h} \mathit{\operatorname{Kan}}$). $\square$

Construction 4.6.8.13 (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

    \[ \circ : \underline{\operatorname{Hom}}_{\mathrm{h} \mathit{\operatorname{\mathcal{C}}}}(Y,Z) \times \underline{\operatorname{Hom}}_{\mathrm{h} \mathit{\operatorname{\mathcal{C}}}}(X,Y) \rightarrow \underline{\operatorname{Hom}}_{\mathrm{h} \mathit{\operatorname{\mathcal{C}}}}(X,Z) \]

    is given by Construction 4.6.8.9.

Note that this definition satisfies the axiomatics of Definition 2.1.7.1 by virtue of Propositions 4.6.8.11 and 4.6.8.12 We will refer to $\mathrm{h} \mathit{\operatorname{\mathcal{C}}}$ as the enriched homotopy category of the $\infty $-category $\operatorname{\mathcal{C}}$.

Remark 4.6.8.14. 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

    \[ \operatorname{Hom}_{\mathrm{h} \mathit{\operatorname{\mathcal{C}}}}(X,Y) = \operatorname{Hom}_{ \mathrm{h} \mathit{\operatorname{Kan}} }( \Delta ^0, \underline{\operatorname{Hom}}_{\mathrm{h} \mathit{\operatorname{\mathcal{C}}}}( X,Y) ) \simeq \pi _0( \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y) ). \]

    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 $\operatorname{\mathcal{C}}$.

By virtue of Remark 4.6.8.10, 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.

Notation 4.6.8.15. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category containing objects $X$, $Y$, and $Z$. For every morphism $f: X \rightarrow Y$ in $\operatorname{\mathcal{C}}$, the composition law of Construction 4.6.8.9 restricts to a morphism of Kan complexes

\[ \operatorname{Hom}_{\operatorname{\mathcal{C}}}(Y,Z) \simeq \operatorname{Hom}_{\operatorname{\mathcal{C}}}(Y,Z) \times \{ f\} \hookrightarrow \operatorname{Hom}_{\operatorname{\mathcal{C}}}(Y,Z) \times \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y) \xrightarrow {\circ } \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Z), \]

which is well-defined up to homotopy. Note that this morphism depends only on the homotopy class $[f]$ of the morphism $f$. We will denote this map by $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(Y,Z) \xrightarrow { \circ [f]} \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Z)$ and refer to it as precomposition with $f$. Similarly, for every morphism $g: Y \rightarrow Z$, the composition law of Remark 4.6.8.10 determines a homotopy class of morphisms $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y) \xrightarrow { [g] \circ } \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Z)$, which we will refer to as postcomposition with $g$.

To describe the precomposition morphism of Notation 4.6.8.15 concretely, it is convenient to replace the morphism spaces $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Z)$ and $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(Y,Z)$ by their right-pinched variants $\operatorname{Hom}_{\operatorname{\mathcal{C}}}^{\mathrm{R}}(X,Z) = \operatorname{\mathcal{C}}_{X/} \times _{\operatorname{\mathcal{C}}} \{ Z\} $ and $\operatorname{Hom}_{\operatorname{\mathcal{C}}}^{\mathrm{R}}(Y,Z) = \operatorname{\mathcal{C}}_{Y/} \times _{\operatorname{\mathcal{C}}} \{ Z\} $, respectively (see Construction 4.6.5.1).

Proposition 4.6.8.16. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $f: X \rightarrow Y$ be a morphism of $\operatorname{\mathcal{C}}$. For every object $Z \in \operatorname{\mathcal{C}}$, the diagram of Kan complexes

\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{C}}_{Y/} \times _{\operatorname{\mathcal{C}}} \{ Z\} \ar [d]^{\iota ^{\mathrm{R}}_{Y,Z}}_{\sim } & \operatorname{\mathcal{C}}_{f/} \times _{\operatorname{\mathcal{C}}} \{ Z\} \ar [l]^{\sim } \ar [r] & \operatorname{\mathcal{C}}_{X/} \times _{\operatorname{\mathcal{C}}} \{ Z\} \ar [d]^{\iota ^{\mathrm{R}}_{X,Z}}_{\sim } \\ \operatorname{Hom}_{\operatorname{\mathcal{C}}}(Y,Z) \ar [rr]^{ \circ [f] } & & \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Z) } \]

commutes up to homotopy, where the vertical maps are the right-pinch inclusion morphisms of Construction 4.6.5.6.

Remark 4.6.8.17. In the situation of Proposition 4.6.8.16, the morphisms

\[ \iota ^{\mathrm{R}}_{Y,Z}: \operatorname{\mathcal{C}}_{Y/} \times _{\operatorname{\mathcal{C}}} \{ Z\} \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{C}}}(Y,Z) \quad \quad \iota ^{\mathrm{R}}_{X,Z}: \operatorname{\mathcal{C}}_{X/} \times _{\operatorname{\mathcal{C}}} \{ Z\} \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Z) \]

are homotopy equivalences, by virtue of Proposition 4.6.5.9. Moreover, the restriction map $\operatorname{\mathcal{C}}_{f/} \times _{\operatorname{\mathcal{C}}} \{ Z\} \rightarrow \operatorname{\mathcal{C}}_{Y/} \times _{\operatorname{\mathcal{C}}} \{ Z\} $ is a trivial Kan fibration (Corollary 4.3.6.13). Consequently, the precomposition map $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(Y,Z) \xrightarrow { \circ [f]} \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Z)$ is characterized (up to homotopy) by the conclusion of Proposition 4.6.8.16.

Proof of Proposition 4.6.8.16. It will suffice to show that there exists a morphism of Kan complexes

\[ \iota ^{\mathrm{R}}_{X,Y,Z}: \operatorname{\mathcal{C}}_{f/} \times _{\operatorname{\mathcal{C}}} \{ Z\} \rightarrow \{ f\} \times _{\operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y) } \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y,Z) \]

for which the diagram

\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{C}}_{Y/} \times _{\operatorname{\mathcal{C}}} \{ Z\} \ar [d]^{\iota ^{\mathrm{R}}_{Y,Z}} & \operatorname{\mathcal{C}}_{f/} \times _{\operatorname{\mathcal{C}}} \{ Z\} \ar [l] \ar [d]^{ \iota ^{\mathrm{R}}_{X,Y,Z} } \ar [r] & \operatorname{\mathcal{C}}_{X/} \times _{\operatorname{\mathcal{C}}} \{ Z\} \ar [d]^{\iota ^{\mathrm{R}}_{X,Z}} \\ \operatorname{Hom}_{\operatorname{\mathcal{C}}}(Y,Z) & \{ f\} \times _{\operatorname{Hom}_{\operatorname{\mathcal{C}}}(Y,Z) } \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y,Z) \ar [l] \ar [r] & \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Z) } \]

commutes (in the category of simplicial sets).

We first observe that there is a unique morphism of simplicial sets $e: \Delta ^2 \times \operatorname{\mathcal{C}}_{f/} \rightarrow \Delta ^1 \star \operatorname{\mathcal{C}}_{f/}$ with the property that $e|_{ \Delta ^1 \times \operatorname{\mathcal{C}}_{f/} }$ is given by projection onto the first factor, and $e|_{ \{ 2\} \times \operatorname{\mathcal{C}}_{f/} }$ is given by projection onto the second factor. Note that the composite map

\[ \Delta ^2 \times \operatorname{\mathcal{C}}_{f/} \xrightarrow {e} \Delta ^1 \star \operatorname{\mathcal{C}}_{f/} \rightarrow \operatorname{\mathcal{C}} \]

can be identified with a morphism of simplicial sets $e': \operatorname{\mathcal{C}}_{f/} \rightarrow \operatorname{Fun}( \Delta ^2, \operatorname{\mathcal{C}})$. Unwinding the definition, we see that $e'$ restricts to a morphism of simplicial subsets

\[ \iota _{X,Y,Z}^{\mathrm{R}}: \operatorname{\mathcal{C}}_{f/} \times _{\operatorname{\mathcal{C}}} \{ Z\} \rightarrow \{ f\} \times _{\operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y) } \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y,Z) \subseteq \operatorname{Fun}(\Delta ^2, \operatorname{\mathcal{C}}) \]

having the desired properties. $\square$

Corollary 4.6.8.18. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $f: X \rightarrow Y$ and $g: X \rightarrow Z$ be morphisms of $\operatorname{\mathcal{C}}$, which we identify with objects of the coslice $\infty $-category $\operatorname{\mathcal{C}}_{X/}$. Then the morphism space $\operatorname{Hom}_{ \operatorname{\mathcal{C}}_{X/} }( f, g)$ can be identified with the homotopy fiber of the composition map $\operatorname{Hom}_{ \operatorname{\mathcal{C}}}(Y, Z ) \xrightarrow { \circ [f] } \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Z)$ over the vertex $g \in \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Z)$.

Proof. Using Proposition 4.6.8.16, we can replace the composition map $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(Y,Z) \xrightarrow { \circ [f] } \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Z)$ with the restriction map $\theta : \operatorname{\mathcal{C}}_{f/} \times _{\operatorname{\mathcal{C}}} \{ Z\} \rightarrow \operatorname{\mathcal{C}}_{X/} \times _{\operatorname{\mathcal{C}}} \{ Z\} $. The morphism $\theta $ is a left fibration (Corollary 4.3.6.11). Since the left-pinched morphism space $\operatorname{\mathcal{C}}_{X/} \times _{\operatorname{\mathcal{C}}} \{ Z\} = \operatorname{Hom}_{\operatorname{\mathcal{C}}}^{\mathrm{L}}(X,Z)$ is a Kan complex (Proposition 4.6.5.4), it follows that $\theta $ is a Kan fibration (Corollary 4.4.3.8). In particular, the homotopy fiber of the composition map $\operatorname{Hom}_{ \operatorname{\mathcal{C}}}(Y, Z ) \xrightarrow { \circ [f] } \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Z)$ over the vertex $g$ can be identified with the fiber

\[ \theta ^{-1} \{ g\} \simeq \operatorname{\mathcal{C}}_{f/} \times _{\operatorname{\mathcal{C}}_{X/} } \{ g\} = \operatorname{Hom}_{\operatorname{\mathcal{C}}_{X/}}^{\mathrm{L}}( f, g ), \]

which is homotopy equivalent to $\operatorname{Hom}_{\operatorname{\mathcal{C}}_{X/}}(f,g)$ by virtue of Proposition 4.6.5.9. $\square$

Let $\operatorname{\mathcal{C}}$ be a locally Kan simplicial category, so that the homotopy coherent nerve $\operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}})$ is an $\infty $-category (Theorem 2.4.5.1). In this case, the composition law of Construction 4.6.8.9 has a direct description:

Proposition 4.6.8.19. Let $\operatorname{\mathcal{C}}$ be a locally Kan simplicial category. For every pair of objects $X,Y \in \operatorname{\mathcal{C}}$, let $\theta _{X,Y}: \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)_{\bullet } \rightarrow \operatorname{Hom}_{ \operatorname{N}_{\bullet }^{\operatorname{hc}} }(X,Y)$ denote the homotopy equivalence of Kan complexes supplied by Remark 4.6.7.6. Then, for every triple of objects $X,Y,Z \in \operatorname{\mathcal{C}}$, the diagram

\[ \xymatrix@R =50pt@C=50pt{ \operatorname{Hom}_{\operatorname{\mathcal{C}}}(Y,Z)_{\bullet } \times \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)_{\bullet } \ar [d]^{ [\theta _{Y,Z} \times \theta _{X,Y}] }_{\sim } \ar [r]^-{\circ } & \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Z)_{\bullet } \ar [d]^{ [ \theta _{X,Z} ]}_{\sim } \\ \operatorname{Hom}_{\operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}})}( Y,Z) \times \operatorname{Hom}_{\operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}})}( X,Y) \ar [r]^-{\circ } & \operatorname{Hom}_{\operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}})}( X,Z) } \]

commutes in the homotopy category $\mathrm{h} \mathit{\operatorname{Kan}}$; here the lower horizontal map is the composition law of Construction 4.6.8.9.

Proof. We will show that there exists a morphism of Kan complexes

\[ \theta _{X,Y,Z}: \operatorname{Hom}_{\operatorname{\mathcal{C}}}(Y,Z)_{\bullet } \times \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)_{\bullet } \rightarrow \operatorname{Hom}_{ \operatorname{N}_{\bullet }^{\operatorname{hc}}( \operatorname{\mathcal{C}})}(X,Y,Z) \]

for which the diagram

\[ \xymatrix@R =50pt@C=20pt{ \operatorname{Hom}_{\operatorname{\mathcal{C}}}(Y,Z)_{\bullet } \times \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)_{\bullet } \ar [d]^{\theta _{Y,Z} \times \theta _{X,Y}} \ar [rr]^-{\circ } \ar [dr]^{\theta _{X,Y,Z}} & & \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Z)_{\bullet } \ar [d]^{ \theta _{X,Z} } \\ \operatorname{Hom}_{\operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}})}( Y,Z) \times \operatorname{Hom}_{\operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}})}( X,Y) & \operatorname{Hom}_{\operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}})}(X,Y,Z) \ar [l] \ar [r] & \operatorname{Hom}_{\operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}})}( X,Z) } \]

is commutative.

Fix an integer $n \geq 0$. Let $\operatorname{\mathcal{E}}$ denote the simplicial category with object set $\operatorname{Ob}(\operatorname{\mathcal{E}}) = \{ x,y,z \} $ and morphism spaces given by

\[ \operatorname{Hom}_{\operatorname{\mathcal{E}}}( x,x )_{\bullet } = \{ \operatorname{id}_ x \} \quad \quad \operatorname{Hom}_{\operatorname{\mathcal{E}}}( y,y )_{\bullet } = \{ \operatorname{id}_ y \} \quad \quad \operatorname{Hom}_{\operatorname{\mathcal{E}}}(z,z)_{\bullet } = \{ \operatorname{id}_ z \} \]

\[ \operatorname{Hom}_{\operatorname{\mathcal{E}}}( y,x )_{\bullet } = \emptyset \quad \quad \operatorname{Hom}_{\operatorname{\mathcal{E}}}(z,x)_{\bullet } = \emptyset \quad \quad \operatorname{Hom}_{\operatorname{\mathcal{E}}}( z,y)_{\bullet } = \emptyset \]

\[ \operatorname{Hom}_{\operatorname{\mathcal{E}}}(x,y)_{\bullet } = \Delta ^ n \quad \quad \operatorname{Hom}_{\operatorname{\mathcal{E}}}(y,z)_{\bullet } = \Delta ^ n, \]

where the composition law $\operatorname{Hom}_{\operatorname{\mathcal{E}}}(y,z)_{\bullet } \times \operatorname{Hom}_{\operatorname{\mathcal{E}}}(x,y)_{\bullet } \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{E}}}(x,z)_{\bullet }$ is an isomorphism (so that $\operatorname{Hom}_{\operatorname{\mathcal{E}}}(x,z)_{\bullet }$ can be identified with the product $\Delta ^ n \times \Delta ^ n$). Note that there is a unique simplicial functor $F: \operatorname{Path}[ \Delta ^2 \times \Delta ^ n ]_{\bullet } \rightarrow \operatorname{\mathcal{E}}$ satisfying the following conditions:

  • On objects, the functor $F$ is given by the formula

    \[ F(i,j) = \begin{cases} x & \text{ if $i=0$ } \\ y & \text{ if $i=1$} \\ z & \text{ if $i=2$.} \end{cases} \]
  • Let $(i,j)$ and $(i',j')$ be vertices of $\Delta ^2 \times \Delta ^ n$ satisfying $i < i'$ and $j \leq j'$, so that there is a unique indecomposable morphism $u$ from $(i,j)$ to $(i', j')$ in the path category $\operatorname{Path}[ \Delta ^2 \times \Delta ^ n ]$ (given by the chain $\{ (i,j) < (i',j') \} $). If $i=0$ and $i' = 1$, then $F(u)$ is the vertex $j'$ of $\Delta ^ n = \operatorname{Hom}_{\operatorname{\mathcal{E}}}(x,y)_{\bullet }$. If $i = 1$ and $i' =2$, then $F(u)$ is the vertex $j'$ of $\Delta ^ n = \operatorname{Hom}_{\operatorname{\mathcal{E}}}(y,z)_{\bullet }$. If $i= 0$ and $i' = 2$, then $F(u)$ is the vertex $(j',j')$ of $\Delta ^ n \times \Delta ^ n = \operatorname{Hom}_{\operatorname{\mathcal{E}}}(x,z)_{\bullet }$.

Let $\sigma $ and $\tau $ be $n$-simplices of the Kan complexes $\operatorname{Hom}_{\operatorname{\mathcal{C}}}( Y,Z)_{\bullet }$ and $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)_{\bullet }$, respectively. Then there is a unique simplicial functor $G_{\sigma ,\tau }: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ satisfying the following conditions:

  • On objects, the functor $G_{\sigma ,\tau }$ is given by $G_{\sigma ,\tau }(x) = X$, $G_{\sigma ,\tau }(y) = Y$, and $G_{\sigma ,\tau }(z) = Z$.

  • The induced map $\Delta ^ n = \operatorname{Hom}_{\operatorname{\mathcal{E}}}( x,y)_{\bullet } \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)_{\bullet }$ is the $n$-simplex $\tau $.

  • The induced map $\Delta ^ n = \operatorname{Hom}_{\operatorname{\mathcal{E}}}( y,z)_{\bullet } \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{C}}}(Y,Z)_{\bullet }$ is the $n$-simplex $\sigma $.

The composite simplicial functor

\[ \operatorname{Path}[ \Delta ^2 \times \Delta ^ n ]_{\bullet } \xrightarrow { F } \operatorname{\mathcal{E}}\xrightarrow { G_{\sigma ,\tau } } \operatorname{\mathcal{C}} \]

determines a morphism from $\Delta ^2 \times \Delta ^ n$ to the homotopy coherent nerve $\operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}})$, which can be identified with an $n$-simplex $\theta _{X,Y,Z}(\sigma ,\tau )$ of the Kan complex $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y,Z)_{\bullet }$. Allowing $n$ to vary, the construction $(\sigma ,\tau ) \mapsto \theta _{X,Y,Z}(\sigma ,\tau )$ determines a morphism of simplicial sets $\theta _{X,Y,Z}: \operatorname{Hom}_{\operatorname{\mathcal{C}}}(Y,Z)_{\bullet } \times \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)_{\bullet } \rightarrow \operatorname{Hom}_{ \operatorname{N}_{\bullet }^{\operatorname{hc}}( \operatorname{\mathcal{C}})}(X,Y,Z)$ having the desired properties. $\square$

Corollary 4.6.8.20. Let $\operatorname{\mathcal{C}}$ be a locally Kan simplicial category, and let $U: \mathrm{h} \mathit{\operatorname{\mathcal{C}}} \rightarrow \mathrm{h} \mathit{ \operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}}) }$ be the isomorphism of homotopy categories supplied by Proposition 2.4.6.9. Then the homotopy equivalences $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)_{\bullet } \rightarrow \operatorname{Hom}_{ \operatorname{N}_{\bullet }^{\operatorname{hc}} }(X,Y)$ of Remark 4.6.7.6 promote $U$ to an isomorphism of $\mathrm{h} \mathit{\operatorname{Kan}}$-enriched categories. Here $\mathrm{h} \mathit{\operatorname{\mathcal{C}}}$ is endowed with the $\mathrm{h} \mathit{\operatorname{Kan}}$-enrichment of Remark 3.1.5.12 and $\mathrm{h} \mathit{ \operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}}) }$ is endowed with the $\mathrm{h} \mathit{\operatorname{Kan}}$-enrichment of Construction 4.6.8.13.

Let $\operatorname{\mathcal{C}}$ be a differential graded category. For every pair of objects $X,Y \in \operatorname{\mathcal{C}}$, we let $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)_{\ast }$ denote the chain complex of morphisms from $X$ to $Y$ and $\mathrm{K}( \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)_{\ast } )$ the associated Eilenberg-MacLane space (Construction 2.5.6.3). In what follows, let us write

\[ \rho _{Y,X}: \mathrm{K}( \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)_{\ast } ) \hookrightarrow \operatorname{Hom}_{ \operatorname{N}_{\bullet }^{\operatorname{dg}}(\operatorname{\mathcal{C}}) }( X, Y) \]

for the composition of the isomorphism $\mathrm{K}( \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)_{\ast } ) \simeq \operatorname{Hom}^{\mathrm{L}}_{ \operatorname{N}_{\bullet }^{\operatorname{dg}}(\operatorname{\mathcal{C}}) }( X, Y)$ of Example 4.6.5.14 with the pinch inclusion morphism $\operatorname{Hom}^{\mathrm{L}}_{ \operatorname{N}_{\bullet }^{\operatorname{dg}}(\operatorname{\mathcal{C}}) }( X, Y) \hookrightarrow \operatorname{Hom}^{\mathrm{L}}_{ \operatorname{N}_{\bullet }^{\operatorname{dg}}(\operatorname{\mathcal{C}}) }( X, Y)$ of Construction 4.6.5.6. We then have the following:

Proposition 4.6.8.21. Let $\operatorname{\mathcal{C}}$ be a differential graded category containing objects $X$, $Y$, and $Z$, so that the composition law

\[ \circ : \operatorname{Hom}_{\operatorname{\mathcal{C}}}(Y,Z)_{\ast } \otimes \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)_{\ast } \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Z)_{\ast } \]

induces a bilinear map of simplicial abelian groups

\[ \mu : \mathrm{K}( \operatorname{Hom}_{\operatorname{\mathcal{C}}}(Y,Z)_{\ast } ) \times \mathrm{K}( \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)_{\ast }) \rightarrow \mathrm{K}( \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Z)_{\ast } ) \]

(see Proposition 2.5.9.1). Then the diagram of Kan complexes

4.72
\begin{equation} \begin{gathered}\label{equation:composition-law-on-dg-nerve} \xymatrix { \mathrm{K}( \operatorname{Hom}_{\operatorname{\mathcal{C}}}(Y,Z)_{\ast } ) \times \mathrm{K}( \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)_{\ast }) \ar [r]^-{\mu } \ar [d]^{ \rho _{Z,Y} \times \rho _{Y,X} } & \mathrm{K}(\operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Z)_{\ast }) \ar [d]^{ \rho _{Z,X} } \\ \operatorname{Hom}_{\operatorname{N}_{\bullet }^{\operatorname{dg}}(\operatorname{\mathcal{C}}) }(Y,Z) \ar [r] & \operatorname{Hom}_{ \operatorname{N}_{\bullet }^{\operatorname{dg}}(\operatorname{\mathcal{C}}) }( X,Y) \ar [r] & \operatorname{Hom}_{ \operatorname{N}_{\bullet }^{\operatorname{dg}}(\operatorname{\mathcal{C}})}(X,Z) } \end{gathered} \end{equation}

commutes up to homotopy, where the bottom horizontal map is the composition law of Construction 4.6.8.9.

Remark 4.6.8.22. In the situation of Proposition 4.6.8.21, the morphisms $\rho _{Y,X}$, $\rho _{Z,Y}$, and $\rho _{Z,X}$ are homotopy equivalences (Proposition 4.6.5.9). Consequently, Proposition 4.6.8.21 determines the composition law on the $\mathrm{h} \mathit{\operatorname{Kan}}$-enriched homotopy category of $\operatorname{N}_{\bullet }^{\operatorname{dg}}(\operatorname{\mathcal{C}})$.

Proof of Proposition 4.6.8.21. Let $\operatorname{\mathcal{C}}^{\Delta }$ denote the underlying simplicial category of the differential graded category $\operatorname{\mathcal{C}}$ (Construction 2.5.9.2). By virtue of Exercise 4.6.7.4, we can identify (4.72) with the outer rectangle of a larger diagram

\[ \xymatrix { \mathrm{K}( \operatorname{Hom}_{\operatorname{\mathcal{C}}}(Y,Z)_{\ast } ) \times \mathrm{K}( \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)_{\ast }) \ar [r]^-{\mu } \ar [d] & \mathrm{K}(\operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Z)_{\ast }) \ar [d] \\ \operatorname{Hom}_{ \operatorname{N}^{\operatorname{hc}}_{\bullet }( \operatorname{\mathcal{C}}^{\Delta } )}( Y,Z) \times \operatorname{Hom}_{ \operatorname{N}^{\operatorname{hc}}_{\bullet }( \operatorname{\mathcal{C}}^{\Delta } )}( X,Y) \ar [r] \ar [d] & \operatorname{Hom}_{ \operatorname{N}_{\bullet }^{\operatorname{hc}}( \operatorname{\mathcal{C}}^{\Delta })}(X,Z) \ar [d] \\ \operatorname{Hom}_{\operatorname{N}_{\bullet }^{\operatorname{dg}}(\operatorname{\mathcal{C}}) }(Y,Z) \ar [r] & \operatorname{Hom}_{ \operatorname{N}_{\bullet }^{\operatorname{dg}}(\operatorname{\mathcal{C}}) }( X,Y) \ar [r] & \operatorname{Hom}_{ \operatorname{N}_{\bullet }^{\operatorname{dg}}(\operatorname{\mathcal{C}})}(X,Z), } \]

where middle horizontal map is given by the composition law of the $\infty $-category $\operatorname{N}_{\bullet }^{\operatorname{hc}}( \operatorname{\mathcal{C}}^{\Delta } )$. We now observe that the upper square commutes up to homotopy by virtue of Proposition 4.6.8.19, and the lower square commutes up to homotopy by the functoriality of Construction 4.6.8.9. $\square$