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1.4.6 Uniqueness of Composition

Let $\operatorname{\mathcal{C}}$ be an $\infty $-category. Given a composable pair of morphisms $f: X \rightarrow Y$ and $g: Y \rightarrow Z$ in $\operatorname{\mathcal{C}}$, one can form a composition $g \circ f$ by choosing a $2$-simplex $\sigma $ with $d_0(\sigma ) = g$ and $d_2(\sigma ) = f$, as indicated in the diagram

\[ \xymatrix { & Y \ar [dr]^{g} & \\ X \ar [ur]^{f} \ar@ {-->}[rr]^{g \circ f} & & Z. } \]

In general, neither the $2$-simplex $\sigma $ nor the resulting morphism $g \circ f = d_1(\sigma )$ is uniquely determined. However, we saw in §1.3.4 that the composition $g \circ f$ is unique up to homotopy (Proposition 1.3.4.2). We now prove a stronger result, which asserts that the $2$-simplex $\sigma $ (hence also the composite morphism $g \circ f = d_1(\sigma )$) is unique up to a contractible space of choices.

Theorem 1.4.6.1 (Joyal). Let $S_{\bullet }$ be a simplicial set. The following conditions are equivalent:

$(1)$

The simplicial set $S_{\bullet }$ is an $\infty $-category.

$(2)$

The inclusion of simplicial sets $\Lambda ^2_1 \hookrightarrow \Delta ^2$ induces a trivial Kan fibration

\[ \operatorname{Fun}( \Delta ^2, S_{\bullet } ) \rightarrow \operatorname{Fun}( \Lambda ^2_1, S_{\bullet } ). \]

Corollary 1.4.6.2. Let $f: X \rightarrow Y$ and $g: Y \rightarrow Z$ be a composable pair of morphisms in an $\infty $-category $\operatorname{\mathcal{C}}$, so that the tuple $(g, \bullet , f)$ determines a map of simplicial sets $\Lambda ^2_1 \rightarrow \operatorname{\mathcal{C}}$ (see Exercise 1.1.2.14). Then the fiber product

\[ \operatorname{Fun}( \Delta ^2, \operatorname{\mathcal{C}}) \times _{ \operatorname{Fun}( \Lambda ^2_1, \operatorname{\mathcal{C}}) } \{ (g, \bullet , f) \} \]

is a contractible Kan complex.

Remark 1.4.6.3. In the situation of Corollary 1.4.6.2, one can think of the simplicial set

\[ Z_{\bullet } = \operatorname{Fun}( \Delta ^2, \operatorname{\mathcal{C}}) \underset {\operatorname{Fun}( \Lambda ^2_1, \operatorname{\mathcal{C}}) }{\times } \{ (g, \bullet , f) \} \]

as a “parameter space” for all choices of $2$-simplex $\sigma $ satisfying $d_0(\sigma ) = g$ and $d_2(\sigma ) = f$ (note that such $2$-simplices can be identified with the vertices of $Z_{\bullet }$). Consequently, we can summarize Corollary 1.4.6.2 informally by saying that this parameter space is contractible.

We will give the proof of Theorem 1.4.6.1 at the end of this section. First, let us study its consequences.

Proof of Theorem 1.4.3.7. Let $S_{\bullet }$ be a simplicial set and let $\operatorname{\mathcal{D}}$ be an $\infty $-category. We wish to show that the simplicial set $\operatorname{Fun}( S_{\bullet }, \operatorname{\mathcal{D}})$ is an $\infty $-category. By virtue of Theorem 1.4.6.1, it will suffice to show that the restriction map

\[ r: \operatorname{Fun}( \Delta ^2, \operatorname{Fun}( S_{\bullet } , \operatorname{\mathcal{D}}) ) \rightarrow \operatorname{Fun}( \Lambda ^2_1, \operatorname{Fun}(S_{\bullet }, \operatorname{\mathcal{D}}) ) \]

is a trivial Kan fibration. Note that we can identify $r$ with the canonical map

\[ \operatorname{Fun}( S_{\bullet }, \operatorname{Fun}( \Delta ^2, \operatorname{\mathcal{D}}) ) \rightarrow \operatorname{Fun}( S_{\bullet }, \operatorname{Fun}( \Lambda ^2_1, \operatorname{\mathcal{D}}) ), \]

which is a trivial Kan fibration by virtue of Corollary 1.4.5.6 and Theorem 1.4.6.1. $\square$

We now introduce some terminology which will be useful for the proof of Theorem 1.4.6.1.

Definition 1.4.6.4. Let $f: A_{\bullet } \rightarrow B_{\bullet }$ be a morphism of simplicial sets. We will say that $f$ is inner anodyne if it belongs to the weakly saturated class of morphisms generated by the collection of all inner horn inclusions $\Lambda ^{n}_{i} \hookrightarrow \Delta ^ n$ (so that $0 < i < n$).

Remark 1.4.6.5. Let $f: A_{\bullet } \rightarrow B_{\bullet }$ be an inner anodyne map of simplicial sets. Then $f$ is a monomorphism. This follows from the observation that the collection of monomorphisms is weakly saturated (Proposition 1.4.5.12), since every inner horn inclusion $\Lambda ^ n_ i \hookrightarrow \Delta ^ n$ is a monomorphism.

Proposition 1.4.6.6. Let $S_{\bullet }$ be a simplicial set. The following conditions are equivalent:

$(1)$

The simplicial set $S_{\bullet }$ is an $\infty $-category.

$(2)$

For every inner anodyne map of simplicial sets $i: A_{\bullet } \hookrightarrow B_{\bullet }$ and every map $f_0: A_{\bullet } \rightarrow S_{\bullet }$, there exists a map $f: B_{\bullet } \rightarrow S_{\bullet }$ such that $f_0 = f \circ i$.

Proof. The implication $(2) \Rightarrow (1)$ is immediate (since every inner horn inclusion $\Lambda ^ n_ i \hookrightarrow \Delta ^ n$ is inner anodyne). Conversely, if $(1)$ is satisfied, then every inner horn inclusion $\Lambda ^ n_ i \hookrightarrow \Delta ^ n$ has the left lifting property with respect to the projection map $p: S_{\bullet } \rightarrow \Delta ^0$. It then follows from Remark 1.4.4.17 that every inner anodyne map has the left lifting property with respect to $p$. $\square$

We will deduce Theorem 1.4.6.1 from the following technical result:

Lemma 1.4.6.7 (Joyal).

$(a)$

For every monomorphism of simplicial sets $i: A_{\bullet } \hookrightarrow B_{\bullet }$, the induced map

\[ (B_{\bullet } \times \Lambda ^2_1) \coprod _{A_{\bullet } \times \Lambda ^2_1 } (A_{\bullet } \times \Delta ^2) \subseteq A_{\bullet } \times \Delta ^2 \]

is inner anodyne.

$(b)$

The collection of inner anodyne morphisms is generated (as a weakly saturated class) by the inclusion maps

\[ (\Delta ^ m \times \Lambda ^2_1) \coprod _{ \operatorname{\partial \Delta }^ m \times \Lambda ^2_1 } (\operatorname{\partial }\Delta ^ m \times \Delta ^2) \subseteq \Delta ^ m \times \Delta ^2 \]

for $m \geq 0$.

Proof. Let $T$ be the weakly saturated class of morphisms generated by all inclusions of the form

\[ (\Delta ^ m \times \Lambda ^2_1) \coprod _{ \operatorname{\partial \Delta }^ m \times \Lambda ^2_1 } (\operatorname{\partial }\Delta ^ m \times \Delta ^2) \subseteq \Delta ^ m \times \Delta ^2, \]

and let $S$ be the collection of all morphisms of simplicial sets $A_{\bullet } \rightarrow B_{\bullet }$ for which the map

\[ (B_{\bullet } \times \Lambda ^2_1) \coprod _{A_{\bullet } \times \Lambda ^2_1 } (A_{\bullet } \times \Delta ^2) \subseteq A_{\bullet } \times \Delta ^2 \]

belongs to $T$. By construction, $S$ contains all inclusions of the form $\operatorname{\partial \Delta }^ m \hookrightarrow \Delta ^ m$. Moreover, since $T$ is weakly saturated, the class $S$ is also weakly saturated. It follows that every monomorphism of simplicial sets belongs to $S$ (Proposition 1.4.5.12). Consequently, to prove Lemma 1.4.6.7, it will suffice to show that $T$ coincides with the class of inner anodyne morphisms of $\operatorname{Set_{\Delta }}$. We first show that every inner anodyne morphism belongs to $T$. Since $T$ is weakly saturated, we are reduced to showing that every inner horn inclusion $f: \Lambda ^{n}_ i \hookrightarrow \Delta ^ n$ belongs to $T$. Since $f$ belongs to $S$, the monomorphism

\[ \overline{f}: (\Delta ^ n \times \Lambda ^2_1) \coprod _{ \Lambda ^ n_ i \times \Lambda ^2_1 } ( \Lambda ^ n_ i \times \Delta ^2) \subseteq \Delta ^ n \times \Delta ^2. \]

belongs to $T$. We conclude by observing that the morphism $f$ is a retract of $\overline{f}$. More precisely, we have a commutative diagram of simplicial sets

\[ \xymatrix@C =40pt@R=40pt{ \Lambda ^ n_ i \ar [r] \ar [d]^{f} & (\Delta ^ n \times \Lambda ^2_1) \coprod _{ \Lambda ^ n_ i \times \Lambda ^2_1 } ( \Lambda ^ n_ i \times \Delta ^2) \ar [r] \ar [d]^{\overline{f}} & \Lambda ^ n_ i \ar [d]^{f} \\ \Delta ^ n \ar [r]^{s} & \Delta ^ n \times \Delta ^2 \ar [r]^{r} & \Delta ^ n, } \]

where the maps $s$ and $r$ are given on vertices by the formulae

\[ s(j) = \begin{cases} (j,0) & \text{if } j < i \\ (j,1) & \text{if } j = i \\ (j,2) & \text{if } j > i \end{cases} \]

\[ r(j,k) = \begin{cases} j & \text{if } j < i, k=0 \\ j & \text{if } j > i, k = 2 \\ i & \text{otherwise.} \end{cases} \]

We now show that every morphism of $T$ is inner anodyne. Since the collection of inner anodyne morphisms is weakly saturated, it will suffice to show that the inclusion map

\[ (\Delta ^ m \times \Lambda ^2_1) \coprod _{ \operatorname{\partial \Delta }^ m \times \Lambda ^2_1 } (\operatorname{\partial }\Delta ^ m \times \Delta ^2) \subseteq \Delta ^ m \times \Delta ^2 \]

is inner anodyne for each $m \geq 0$. For each $0 \leq i \leq j < m$, we let $\sigma _{ij}$ denote the $(m+1)$-simplex of $\Delta ^ m \times \Delta ^2$ given by the map of partially ordered sets

\[ f_{ij}: [m+1] \rightarrow [m] \times [2] \]

\[ f_{ij}(k) = \begin{cases} (k,0) & \text{if } 0 \leq k \leq i \\ (k-1, 1) & \text{if } i+1 \leq k \leq j+1 \\ (k-1, 2) & \text{if } j+2 \leq k \leq m+1. \end{cases} \]

For each $0 \leq i \leq j \leq m$, we let $\tau _{ij}$ denote the $(m+2)$-simplex of $\Delta ^ m \times \Delta ^2$ given by the map of partially ordered sets

\[ g_{ij}: [m+2] \rightarrow [m] \times [2] \]

\[ g_{ij}(k) = \begin{cases} (k,0) & \text{if } 0 \leq k \leq i \\ (k-1, 1) & \text{if } i+1 \leq k \leq j+1 \\ (k-2, 2) & \text{if } j+2 \leq k \leq m+2. \end{cases} \]

We will regard each $\sigma _{ij}$ and $\tau _{ij}$ as a simplicial subset of $\Delta ^ m \times \Delta ^2$.

Set $X(0) = (\Delta ^ m \times \Lambda ^2_1) \coprod _{ \operatorname{\partial \Delta }^ m \times \Lambda ^2_1 } (\operatorname{\partial }\Delta ^ m \times \Delta ^2)$. For $0 \leq j < m$, we let

\[ X(j+1) = X(j) \cup \sigma _{0j} \cup \cdots \cup \sigma _{jj}. \]

We have a chain of inclusions

\[ X(j) \subseteq X(j) \cup \sigma _{0j} \subseteq \cdots \subset X(j) \cup \sigma _{0j} \cup \cdots \cup \sigma _{jj} = X(j+1). \]

Each of these inclusions fits into a pushout diagram

\[ \xymatrix { \Lambda ^{m+1}_{ i+1} \ar [r] \ar [d] & X(j) \cup \sigma _{0j} \cup \cdots \cup \sigma _{(i-1)j} \ar [d] \\ \sigma _{ij} \ar [r] & X(j) \cup \sigma _{0j} \cup \cdots \cup \sigma _{ij}, } \]

and is therefore inner anodyne. Set $Y(0) = X(m)$, so that the inclusion $X(0) \subseteq Y(0)$ is inner anodyne. We now set $Y(j+1) = Y(j) \cup \tau _{0j} \cup \cdots \cup \tau _{jj}$ for $0 \leq j \leq m$. As before, we have a chain of inclusions

\[ Y(j) \subseteq Y(j) \cup \tau _{0j} \subseteq \cdots \subseteq Y_{j} \cup \tau _{0j} \cup \cdots \cup \tau _{jj} = Y(j+1), \]

each of which fits into a pushout diagram

\[ \xymatrix { \Lambda ^{m+2}_{i+1} \ar [r] \ar [d] & Y_{j} \cup \tau _{0j} \cup \cdots \cup \tau _{(i-1)j} \ar [d] \\ \tau _{ij} \ar [r] & Y_{j} \cup \tau _{0j} \cup \cdots \cup \tau _{ij}, } \]

and is therefore inner andoyne. It follows that each inclusion $Y(j) \subseteq Y(j+1)$ is inner anodyne. Since the collection of inner anodyne morphisms is closed under composition, we conclude that the inclusion map $X(0) \hookrightarrow Y(0) \hookrightarrow Y(1) \hookrightarrow \cdots Y(m+2) = \Delta ^ m \times \Delta ^2$ is inner anodyne, as desired. $\square$

Proof of Theorem 1.4.6.1. Let $S_{\bullet }$ be a simplicial set and let $p: \operatorname{Fun}( \Delta ^2, S_{\bullet } ) \rightarrow \operatorname{Fun}( \Lambda ^2_1, S_{\bullet } )$ denote the restriction map. Then $p$ is a trivial Kan fibration if and only if every lifting problem

\[ \xymatrix@C =40pt@R=40pt{ \operatorname{\partial \Delta }^ m \ar [r] \ar [d] & \operatorname{Fun}( \Delta ^2, S_{\bullet }) \ar [d]^{p} \\ \Delta ^ m \ar@ {-->}[ur] \ar [r] & \operatorname{Fun}( \Lambda ^2_1, S_{\bullet }) } \]

admits a solution. Unwinding the definitions, we see that this is equivalent to the requirement that every lifting problem of the form

\[ \xymatrix@C =40pt@R=40pt{ (\Delta ^ m \times \Lambda ^2_1) \underset { \operatorname{\partial \Delta }^ m \times \Lambda ^2_1}{\coprod } (\operatorname{\partial }\Delta ^ m \times \Delta ^2) \ar [d]^{i} \ar [r] & S_{\bullet } \ar [d] \\ \Delta ^ m \times \Delta ^2 \ar [r] \ar@ {-->}[ur] & \Delta ^0, } \]

admits a solution. Let $T$ be the collection of all morphisms of simplicial sets which have the left lifting property with respect to the projection $S_{\bullet } \rightarrow \Delta ^0$. Then $p$ is a trivial Kan fibration if and only if $T$ contains each of the inclusion maps

\[ (\Delta ^ m \times \Lambda ^2_1) \coprod _{ \operatorname{\partial \Delta }^ m \times \Lambda ^2_1 } (\operatorname{\partial }\Delta ^ m \times \Delta ^2) \subseteq \Delta ^ m \times \Delta ^2. \]

Since $T$ is weakly saturated (Proposition 1.4.4.16), this is equivalent to the requirement that $T$ contains all inner anodyne morphisms (Lemma 1.4.6.7), which is in turn equivalent to the requirement that $S_{\bullet }$ is an $\infty $-category (Proposition 1.4.6.6). $\square$