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
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:
The simplicial set $S$ is an $\infty $-category.
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 is a contractible Kan complex.
\[ \operatorname{Fun}( \Delta ^2, \operatorname{\mathcal{C}}) \times _{ \operatorname{Fun}( \Lambda ^2_1, \operatorname{\mathcal{C}}) } \{ (g, \bullet , f) \} \]
Proof. Combine Theorem 1.5.6.1 with Remark 1.5.5.10. $\square$
Remark 1.5.6.3. In the situation of Corollary 1.5.6.2, one can think of the simplicial set 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 $Z$).
\[ Z = \operatorname{Fun}( \Delta ^2, \operatorname{\mathcal{C}}) \underset {\operatorname{Fun}( \Lambda ^2_1, \operatorname{\mathcal{C}}) }{\times } \{ (g, \bullet , f) \} \]
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
is a trivial Kan fibration. Note that we can identify $r$ with the canonical map
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:
The simplicial set $S$ is an $\infty $-category.
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:
The simplicial set $S$ is isomorphic to the nerve of a category.
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:
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:
For every monomorphism of simplicial sets $i: A \hookrightarrow B$, the induced map
is inner anodyne.
The collection of inner anodyne morphisms is generated (as a weakly saturated class) by the inclusion maps
for $m \geq 0$.
\[ (B \times \Lambda ^2_1) \coprod _{A \times \Lambda ^2_1 } (A \times \Delta ^2) \subseteq B \times \Delta ^2 \]
\[ (\Delta ^ m \times \Lambda ^2_1) \coprod _{ \operatorname{\partial \Delta }^ m \times \Lambda ^2_1 } (\operatorname{\partial }\Delta ^ m \times \Delta ^2) \subseteq \Delta ^ m \times \Delta ^2 \]
Proof. Let $T$ be the weakly saturated class of morphisms generated by all inclusions of the form
and let $S$ be the collection of all morphisms of simplicial sets $A \rightarrow B$ for which the map
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
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
where the maps $s$ and $r$ are given on vertices by the formulae
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
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
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
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
We have a chain of inclusions
Each of these inclusions fits into a pushout diagram
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
each of which fits into a pushout diagram
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
admits a solution. Unwinding the definitions, we see that this is equivalent to the requirement that every lifting problem of the form
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
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). $\square$