# Kerodon

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### 1.5.6 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 $d^{2}_0(\sigma ) = g$ and $d^{2}_2(\sigma ) = f$, as indicated in the diagram

$\xymatrix@R =50pt@C=50pt{ & Y \ar [dr]^{g} & \\ X \ar [ur]^{f} \ar@ {-->}[rr]^{g \circ f} & & Z. }$

In general, neither the $2$-simplex $\sigma$ nor the resulting morphism $g \circ f = d^{2}_1(\sigma )$ is uniquely determined. However, we saw in §1.4.4 that the composition $g \circ f$ is unique up to homotopy (Proposition 1.4.4.2). We now prove a stronger result, which asserts that the $2$-simplex $\sigma$ (hence also the composite morphism $g \circ f = d^{2}_1(\sigma )$) is unique up to a contractible space of choices.

Theorem 1.5.6.1 (Joyal). Let $S$ be a simplicial set. The following conditions are equivalent:

$(1)$

The simplicial set $S$ is an $\infty$-category.

$(2)$

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

$\operatorname{Fun}( \Delta ^2, S ) \rightarrow \operatorname{Fun}( \Lambda ^2_1, S ).$

Corollary 1.5.6.2. 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 Proposition 1.2.4.7). 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 1.5.6.3. In the situation of Corollary 1.5.6.2, one can think of the simplicial set

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

as a “parameter space” for all choices of $2$-simplex $\sigma$ satisfying $d^{2}_0(\sigma ) = g$ and $d^{2}_2(\sigma ) = f$ (note that such $2$-simplices can be identified with the vertices of $Z$).

We will give the proof of Theorem 1.5.6.1 at the end of this section. First, let us note one of its consequences.

Proof of Theorem 1.5.3.7. Let $S$ be a simplicial set and let $\operatorname{\mathcal{D}}$ be an $\infty$-category. We wish to show that the simplicial set $\operatorname{Fun}( S, \operatorname{\mathcal{D}})$ is an $\infty$-category. By virtue of Theorem 1.5.6.1, it will suffice to show that the restriction map

$r: \operatorname{Fun}( \Delta ^2, \operatorname{Fun}( S , \operatorname{\mathcal{D}}) ) \rightarrow \operatorname{Fun}( \Lambda ^2_1, \operatorname{Fun}(S, \operatorname{\mathcal{D}}) )$

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

$\operatorname{Fun}( S, \operatorname{Fun}( \Delta ^2, \operatorname{\mathcal{D}}) ) \rightarrow \operatorname{Fun}( S, \operatorname{Fun}( \Lambda ^2_1, \operatorname{\mathcal{D}}) ),$

which is a trivial Kan fibration by virtue of Corollary 1.5.5.7 and Theorem 1.5.6.1. $\square$

We now introduce some terminology which will be useful for the proof of Theorem 1.5.6.1.

Definition 1.5.6.4. Let $f: A \rightarrow B$ 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 1.5.6.5. Let $f: A \rightarrow B$ 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 1.5.5.14), since every inner horn inclusion $\Lambda ^ n_ i \hookrightarrow \Delta ^ n$ is a monomorphism.

Exercise 1.5.6.6. Let $f: A \hookrightarrow B$ be an inner anodyne morphism of simplicial sets. Show that the underlying map on vertices $A_0 \rightarrow B_0$ is a bijection.

Proposition 1.5.6.7. Let $S$ be a simplicial set. The following conditions are equivalent:

$(1)$

The simplicial set $S$ is an $\infty$-category.

$(2)$

For every inner anodyne map of simplicial sets $i: A \hookrightarrow B$ and every map $f_0: A \rightarrow S$, there exists a map $f: B \rightarrow S$ 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 $\Lambda ^ n_ i \hookrightarrow \Delta ^ n$ is weakly left orthogonal to the projection map $p: S \rightarrow \Delta ^0$. It then follows from Remark 1.5.4.14 that every inner anodyne map is weakly left orthogonal to $p$. $\square$

Variant 1.5.6.8. Let $S$ be a simplicial set. The following conditions are equivalent:

$(1)$

The simplicial set $S$ is isomorphic to the nerve of a category.

$(2)$

For every inner anodyne map of simplicial sets $i: A \hookrightarrow B$ and every map $f_0: A \rightarrow S$, there exists a unique map $f: B \rightarrow S$ such that $f_0 = f \circ i$.

Proof. Let us regard the simplicial set $S$ as fixed, and let $T$ be the collection of all morphisms of simplicial sets $i: A \rightarrow B$ for which the induced map $\operatorname{Hom}_{\operatorname{Set_{\Delta }}}( B, S ) \rightarrow \operatorname{Hom}_{\operatorname{Set_{\Delta }}}(A, S )$ is bijective. Then $T$ is weakly saturated (in the sense of Definition 1.5.4.12). It follows that $(2)$ is equivalent to the following a priori weaker assertion:

$(2')$

For every pair of integers $0 < i < n$, the map $\operatorname{Hom}_{\operatorname{Set_{\Delta }}}( \Delta ^ n, S ) \rightarrow \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( \Lambda ^{n}_ i, S )$ is bijective.

The equivalence of $(1)$ and $(2')$ is the content of Proposition 1.3.4.1. $\square$

We will deduce Theorem 1.5.6.1 from the following technical result:

Lemma 1.5.6.9 (Joyal).

$(a)$

For every monomorphism of simplicial sets $i: A \hookrightarrow B$, the induced map

$(B \times \Lambda ^2_1) \coprod _{A \times \Lambda ^2_1 } (A \times \Delta ^2) \subseteq B \times \Delta ^2$

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 $A \rightarrow B$ for which the map

$(B \times \Lambda ^2_1) \coprod _{A \times \Lambda ^2_1 } (A \times \Delta ^2) \subseteq B \times \Delta ^2$

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 1.5.5.14). Consequently, to prove Lemma 1.5.6.9, 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

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

Each of these inclusions fits into a pushout diagram

$\xymatrix@R =50pt@C=50pt{ \Lambda ^{m+1}_{ i+1} \ar [r] \ar [d] & X(j) \cup \sigma _{0j} \cup \cdots \cup \sigma _{(i-1)j} \ar [d] \\ \sigma _{ij} \ar [r] & X(j) \cup \sigma _{0j} \cup \cdots \cup \sigma _{ij}, }$

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

$\xymatrix@R =50pt@C=50pt{ \Lambda ^{m+2}_{i+1} \ar [r] \ar [d] & Y(j) \cup \tau _{0j} \cup \cdots \cup \tau _{(i-1)j} \ar [d] \\ \tau _{ij} \ar [r] & Y(j) \cup \tau _{0j} \cup \cdots \cup \tau _{ij}, }$

and is therefore inner anodyne. 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 $X(0) \hookrightarrow Y(0) \hookrightarrow Y(1) \hookrightarrow \cdots Y(m+1) = \Delta ^ m \times \Delta ^2$ is inner anodyne, as desired. $\square$

Proof of Theorem 1.5.6.1. Let $S$ be a simplicial set and let $p: \operatorname{Fun}( \Delta ^2, S ) \rightarrow \operatorname{Fun}( \Lambda ^2_1, S )$ denote the restriction map. Then $p$ is a trivial Kan fibration if and only if every lifting problem

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

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

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

admits a solution. Let $T$ be the collection of all morphisms of simplicial sets which are weakly left orthogonal to the projection $S \rightarrow \Delta ^0$. 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 1.5.4.13), this is equivalent to the requirement that $T$ contains all inner anodyne morphisms (Lemma 1.5.6.9), which is in turn equivalent to the requirement that $S$ is an $\infty$-category (Proposition 1.5.6.7). $\square$