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1.4.7 Universality of Path Categories

Let $G$ be a directed graph, let $G_{\bullet }$ denote the associated $1$-dimensional simplicial set (see Proposition 1.1.5.9), and let $\operatorname{Path}[G]$ denote the path category of $G$ (Construction 1.2.6.1). There is an evident map of simplicial sets $u: G_{\bullet } \rightarrow \operatorname{N}_{\bullet }( \operatorname{Path}[G] )$. By virtue of Proposition 1.2.6.5, this map exhibits $\operatorname{Path}[G]$ as the homotopy category of the simplicial set $G_{\bullet }$. In other words, the path category $\operatorname{Path}[G]$ is universal among categories $\operatorname{\mathcal{C}}$ which are equipped with a $G_{\bullet }$-indexed diagram (see Definition 1.4.2.1). Our goal in this section is to establish a variant of this statement in the setting of $\infty $-categories:

Theorem 1.4.7.1. Let $G$ be a directed graph and let $\operatorname{\mathcal{C}}$ be an $\infty $-category. Then composition with the map of simplicial sets $u: G_{\bullet } \rightarrow \operatorname{N}_{\bullet }( \operatorname{Path}[G] )$ induces a trivial Kan fibration of simplicial sets $\operatorname{Fun}( \operatorname{N}_{\bullet }( \operatorname{Path}[G] ), \operatorname{\mathcal{C}}) \rightarrow \operatorname{Fun}( G_{\bullet }, \operatorname{\mathcal{C}})$.

More informally, Theorem 1.4.7.1 asserts that any $G$-indexed diagram in an $\infty $-category $\operatorname{\mathcal{C}}$ admits an essentially unique extension to a functor of $\infty $-categories $\operatorname{N}_{\bullet }( \operatorname{Path}[G] ) \rightarrow \operatorname{\mathcal{C}}$.

Example 1.4.7.2. Let $G$ be the directed graph depicted in the diagram

\[ \xymatrix { \bullet \ar [r] & \bullet \ar [r] & \bullet . } \]

Then the map $u: G_{\bullet } \rightarrow \operatorname{N}_{\bullet }( \operatorname{Path}[G] )$ can be identified with the inclusion of simplicial sets $\Lambda ^{2}_{1} \hookrightarrow \Delta ^2$. In this case, Theorem 1.4.7.1 reduces to the statement that the map

\[ \operatorname{Fun}( \Delta ^2, \operatorname{\mathcal{C}}) \rightarrow \operatorname{Fun}( \Lambda ^{2}_{1}, \operatorname{\mathcal{C}}) \]

is a trivial Kan fibration, which is equivalent to the assumption that $\operatorname{\mathcal{C}}$ is an $\infty $-category by virtue of Theorem 1.4.6.1.

We will deduce Theorem 1.4.7.1 from the following more precise assertion.

Proposition 1.4.7.3. Let $G$ be a directed graph. Then the map of simplicial sets $u: G_{\bullet } \hookrightarrow \operatorname{N}_{\bullet }( \operatorname{Path}[G] )$ is inner anodyne (Definition 1.4.6.4).

Remark 1.4.7.4. Let $G$ be a directed graph and let $\operatorname{\mathcal{C}}$ be an ordinary category. Combining Proposition 1.4.7.3 with Variant 1.4.6.7, we deduce that the canonical map

\[ \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( \operatorname{N}_{\bullet }( \operatorname{Path}[G] ), \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) ) \rightarrow \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( G_{\bullet }, \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) ) \]

is bijective. Combining this observation with Proposition 1.2.2.1, we obtain a bijection

\[ \operatorname{Hom}_{\operatorname{Cat}}( \operatorname{Path}[G], \operatorname{\mathcal{C}}) \rightarrow \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( G_{\bullet }, \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) ). \]

Allowing $\operatorname{\mathcal{C}}$ to vary, we recover the assertion that $u: G_{\bullet } \rightarrow \operatorname{N}_{\bullet }(\operatorname{Path}[G] )$ exhibits $\operatorname{Path}[G]$ as the homotopy category of $G_{\bullet }$ (Proposition 1.2.6.5).

Let us first show that Proposition 1.4.7.3 implies Theorem 1.4.7.1.

Lemma 1.4.7.5. Let $f: X_{\bullet } \hookrightarrow Y_{\bullet }$ and $f': X'_{\bullet } \hookrightarrow Y'_{\bullet }$ be monomorphisms of simplicial sets. If $f$ is inner anodyne, then the induced map

\[ u_{f,f'}: (Y_{\bullet } \times X'_{\bullet } ) \coprod _{ (X_{\bullet } \times X'_{\bullet })} (X_{\bullet } \times Y'_{\bullet } ) \hookrightarrow Y_{\bullet } \times Y'_{\bullet } \]

is inner anodyne.

Proof. Let us regard the morphism $f': X'_{\bullet } \hookrightarrow Y'_{\bullet }$ as fixed. Let $T$ be the collection of all morphisms $f: X_{\bullet } \rightarrow Y_{\bullet }$ for which the map $u_{f,f'}$ is inner anodyne. Then $T$ is weakly saturated. To prove Lemma 1.4.7.5, we must show that $T$ contains all inner anodyne morphisms of simplicial sets. By virtue of Lemma 1.4.6.8, it will suffice to show that $T$ contains every morphism of the form

\[ u_{i,j}: (B_{\bullet } \times \Lambda ^2_1) \coprod _{A_{\bullet } \times \Lambda ^2_1 } (A_{\bullet } \times \Delta ^2) \subseteq B_{\bullet } \times \Delta ^2, \]

where $i: A_{\bullet } \hookrightarrow B_{\bullet }$ is a monomorphism of simplicial sets and $j: \Lambda ^2_1 \hookrightarrow \Delta ^2$ is the inclusion. Setting

\[ A'_{\bullet } = (B_{\bullet } \times X'_{\bullet }) \coprod _{ (A_{\bullet } \times X'_{\bullet } )} ( A_{\bullet } \times Y'_{\bullet }) \quad \quad B'_{\bullet } = B_{\bullet } \times Y'_{\bullet }, \]

we are reduced to the problem of showing that the map

\[ u_{i',j}: (B'_{\bullet } \times \Lambda ^2_1) \coprod _{A'_{\bullet } \times \Lambda ^2_1 } (A'_{\bullet } \times \Delta ^2) \subseteq B'_{\bullet } \times \Delta ^2, \]

is inner anodyne, which follows from Lemma 1.4.6.8. $\square$

Proposition 1.4.7.6. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $f: X_{\bullet } \hookrightarrow Y_{\bullet }$ be an inner anodyne morphism of simplicial sets. Then the induced map $p: \operatorname{Fun}( Y_{\bullet }, \operatorname{\mathcal{C}}) \rightarrow \operatorname{Fun}( X_{\bullet }, \operatorname{\mathcal{C}})$ is a trivial Kan fibration.

Proof. To show that $p$ is a trivial Kan fibration, it will suffice to show that it has the right lifting property with respect to respect to every monomorphism of simplicial sets $f': X'_{\bullet } \hookrightarrow Y'_{\bullet }$. This is equivalent to the assertion that every map of simplicial sets

\[ g_0: (Y_{\bullet } \times X'_{\bullet } ) \coprod _{ (X_{\bullet } \times X'_{\bullet })} (X_{\bullet } \times Y'_{\bullet } ) \rightarrow \operatorname{\mathcal{C}} \]

can be extended to a map $g: Y_{\bullet } \times Y'_{\bullet } \rightarrow \operatorname{\mathcal{C}}$. This follows from Proposition 1.4.6.6, since $\operatorname{\mathcal{C}}$ is an $\infty $-category and the map

\[ u_{f,f'}: (Y_{\bullet } \times X'_{\bullet } ) \coprod _{ (X_{\bullet } \times X'_{\bullet })} (X_{\bullet } \times Y'_{\bullet } ) \hookrightarrow Y_{\bullet } \times Y'_{\bullet } \]

is inner anodyne (Lemma 1.4.7.5). $\square$

Proof of Theorem 1.4.7.1. Let $G$ be a graph and let $\operatorname{\mathcal{C}}$ be an $\infty $-category; we wish to show that the canonical map

\[ \operatorname{Fun}( \operatorname{N}_{\bullet }(\operatorname{Path}[G] ), \operatorname{\mathcal{C}}) \rightarrow \operatorname{Fun}( G_{\bullet }, \operatorname{\mathcal{C}}) \]

is a trivial Kan fibration. This follows from Proposition 1.4.7.6, since the inclusion $G_{\bullet } \hookrightarrow \operatorname{N}_{\bullet }( \operatorname{Path}[G] )$ is inner anodyne (Proposition 1.4.7.3). $\square$

Before giving the proof of Proposition 1.4.7.3, let us illustrate its contents with some examples.

Example 1.4.7.7 (The Spine of a Simplex). Let $n \geq 0$ and let $\Delta ^{n}$ be the standard $n$-simplex (Construction 1.1.2.1). We let $\operatorname{Spine}[n]$ denote the simplicial subset of $\Delta ^{n}$ whose $k$-simplices are monotone maps $\sigma : [k] \rightarrow [n]$ satisfying $\sigma (k) \leq \sigma (0) + 1$. We will refer to $\operatorname{Spine}[n]$ as the spine of the simplex $\Delta ^{n}$. More informally, it is comprised of all vertices of $\Delta ^{n}$, together with those edges which join adjacent vertices. The spine $\operatorname{Spine}[n]$ is a simplicial set of dimension $\leq 1$, which we can identify with the directed graph $G$ depicted in the diagram

\[ \xymatrix { 0 \ar [r] & 1 \ar [r] & 2 \ar [r] & \cdots \ar [r] & n. } \]

Under this identification, the map $u: G_{\bullet } \rightarrow \operatorname{N}_{\bullet }( \operatorname{Path}[G] )$ corresponds to the inclusion $\operatorname{Spine}[n] \hookrightarrow \Delta ^ n$ (see Example 1.2.6.2). Invoking Proposition 1.4.7.3 and Theorem 1.4.7.1, we obtain the following:

$(a)$

The inclusion $\operatorname{Spine}[n] \hookrightarrow \Delta ^ n$ is inner anodyne.

$(b)$

For any $\infty $-category $\operatorname{\mathcal{C}}$, the restriction map $\operatorname{Fun}( \Delta ^ n, \operatorname{\mathcal{C}}) \rightarrow \operatorname{Fun}( \operatorname{Spine}[n], \operatorname{\mathcal{C}})$ is a trivial Kan fibration.

Remark 1.4.7.8 (The Generalized Associative Law). Let $\operatorname{\mathcal{C}}$ be an ordinary category and let $n \geq 0$ be an integer. Applying Remark 1.4.7.4 to the inner anodyne inclusion $\operatorname{Spine}[n] \hookrightarrow \Delta ^ n$ of Example 1.4.7.7, we deduce that every diagram

\[ X_0 \xrightarrow {f_1} X_1 \xrightarrow { f_2} X_2 \rightarrow \cdots \xrightarrow {f_ n} X_ n \]

can be extended uniquely to a functor $[n] \rightarrow \operatorname{\mathcal{C}}$. In particular, it shows that $\operatorname{\mathcal{C}}$ satisfies the “generalized associative law”: the iterated composition $f_{n} \circ f_{n-1} \circ \cdots \circ f_{2} \circ f_{1}$ is well-defined (that is, it does not depend on a choice of parenthesization). In essence, Proposition 1.4.7.3 can be regarded as an extension of this generalized associative law to the setting of $\infty $-categories.

Example 1.4.7.9 (The Simplicial Circle). Let $S^{1}_{\bullet }$ denote the simplicial set obtained from $\Delta ^1$ by collapsing the boundary $\partial \Delta ^1$ to a point, so that we have a pushout diagram of simplicial sets

\[ \xymatrix { \partial \Delta ^1 \ar [r] \ar [d] & \Delta ^1 \ar [d] \\ \Delta ^0 \ar [r] & S^{1}_{\bullet }. } \]

We will refer to $S^{1}_{\bullet }$ as the simplicial circle; note that the geometric realization $| S^{1}_{\bullet } |$ is isomorphic to the standard circle $S^1$ as a topological space. The simplicial set $S^{1}_{\bullet }$ has dimension $\leq 1$, and can therefore be identified with the directed graph $G$ depicted in the diagram

\[ \xymatrix { \bullet \ar@ (ur,ul)[] } \]

Note that the path category $\operatorname{Path}[G]$ can be identified with the category $B\operatorname{\mathbf{Z}}_{\geq 0}$ associated to the monoid $\operatorname{\mathbf{Z}}_{\geq 0}$ of nonnegative numbers under addition (Example 1.2.6.4) whose nerve is the simplicial set $B_{\bullet } \operatorname{\mathbf{Z}}_{\geq 0}$ of Example 1.2.4.3. Invoking Proposition 1.4.7.3 and Theorem 1.4.7.1, we obtain the following:

$(a)$

The inclusion of simplicial sets $S^{1}_{\bullet } \hookrightarrow B_{\bullet } \operatorname{\mathbf{Z}}_{\geq 0}$ is inner anodyne.

$(b)$

For any $\infty $-category $\operatorname{\mathcal{C}}$, the restriction map $\operatorname{Fun}( B_{\bullet } \operatorname{\mathbf{Z}}_{\geq 0}, \operatorname{\mathcal{C}}) \rightarrow \operatorname{Fun}( S^1_{\bullet }, \operatorname{\mathcal{C}})$ is a trivial Kan fibration.

Example 1.4.7.10 (Free Monoids). Let $M$ be the free monoid generated by a set $E$. Then we can identify $BM$ with the path category $\operatorname{Path}[G]$ of a directed graph $G$ satisfying

\[ \operatorname{Vert}(G) = \{ x\} \quad \quad \operatorname{Edge}(G) = E; \]

see Example 1.2.6.3. Invoking Proposition 1.4.7.3 and Theorem 1.4.7.1, we obtain the following:

$(a)$

The inclusion of simplicial sets $G_{\bullet } \hookrightarrow B_{\bullet }M$ is inner anodyne.

$(b)$

For any $\infty $-category $\operatorname{\mathcal{C}}$, the restriction map $\operatorname{Fun}( B_{\bullet }M , \operatorname{\mathcal{C}}) \rightarrow \operatorname{Fun}( G_{\bullet }, \operatorname{\mathcal{C}})$ is a trivial Kan fibration.

Note that if $\operatorname{\mathcal{C}}$ is an $\infty $-category, then a map of simplicial sets $\sigma _0: G_{\bullet } \rightarrow \operatorname{\mathcal{C}}$ can be identified with a choice of object $X \in \operatorname{\mathcal{C}}$ together with a collection of morphisms $\{ f_{e}: X \rightarrow X \} _{e \in E}$ indexed by $E$. It follows from $(b)$ that any such map admits an (essentially unique) extension to a functor $\sigma : B_{\bullet } M \rightarrow \operatorname{\mathcal{C}}$, which we can interpret as an action of the monoid $M$ on the object $X \in \operatorname{\mathcal{C}}$.

Proof of Proposition 1.4.7.3. Let $G$ be a directed graph and let $\operatorname{Path}[G]$ denote its path category. By definition, a morphism from $x \in \operatorname{Vert}(G)$ to $y \in \operatorname{Vert}(G)$ in the category $\operatorname{Path}[G]$ is given by a sequence of edges $\vec{e} = (e_ m, e_{m-1}, \ldots , e_1)$ satisfying

\[ s(e_1) = x \quad \quad t(e_{i}) = s(e_{i+1} ) \quad \quad t(e_ m) = y. \]

In this case, we will refer to $m$ as the length of the morphism $\vec{e}$ and write $m = \ell (\vec{e})$. If $\sigma : \Delta ^ n \rightarrow \operatorname{N}_{\bullet }(\operatorname{Path}[G])$ is an $n$-simplex given by a diagram

\[ x_0 \xrightarrow { \vec{e}_1 } x_1 \xrightarrow { \vec{e}_2 } \cdots \xrightarrow { \vec{e}_ n} x_ n \]

in $\operatorname{Path}[G]$, we define the length $\ell (\sigma )$ to be the sum $\ell ( \vec{e}_1 ) + \cdots + \ell ( \vec{e}_ n ) = \ell ( \vec{e}_ n \circ \cdots \circ \vec{e}_1 )$. For each positive integer $k$, let $\operatorname{N}_{\bullet }^{\leq k}( \operatorname{Path}[G] )$ denote the simplicial subset of $\operatorname{N}_{\bullet }( \operatorname{Path}[G] )$ consisting of those simplices having length $\leq k$. We then have inclusions

\[ \operatorname{N}_{\bullet }^{\leq 1}( \operatorname{Path}[G] ) \subseteq \operatorname{N}_{\bullet }^{\leq 2}( \operatorname{Path}[G] ) \subseteq \operatorname{N}_{\bullet }^{\leq 3}( \operatorname{Path}[G] ) \subseteq \operatorname{N}_{\bullet }^{\leq 4}( \operatorname{Path}[G] ) \subseteq \cdots , \]

where $\operatorname{N}_{\bullet }^{\leq 1}( \operatorname{Path}[G] ) = G_{\bullet }$ and $\operatorname{N}_{\bullet }( \operatorname{Path}[G] ) = \bigcup \operatorname{N}_{\bullet }^{\leq k}( \operatorname{Path}[G] )$. Consequently, to show that the inclusion $G_{\bullet } \hookrightarrow \operatorname{N}_{\bullet }( \operatorname{Path}[G] )$ is inner anodyne, it will suffice to show that each of the inclusion maps $\operatorname{N}_{\bullet }^{\leq k}(\operatorname{Path}[G]) \hookrightarrow \operatorname{N}_{\bullet }^{\leq k+1}(\operatorname{Path}[G] )$ is inner anodyne.

We henceforth regard the integer $k \geq 1$ as fixed. Let $\sigma : \Delta ^ n \rightarrow \operatorname{N}_{\bullet }(\operatorname{Path}[G])$ be an $n$-simplex of $\operatorname{N}_{\bullet }( \operatorname{Path}[G] )$ having length $k+1$, corresponding to a diagram

\[ x_0 \xrightarrow { \vec{e}_1 } x_1 \xrightarrow { \vec{e}_2 } \cdots \xrightarrow { \vec{e}_ n} x_ n \]

as above. Note that $\sigma $ is nondegenerate if and only if each $\vec{e}_ i$ has positive length. We will say that $\sigma $ is normalized if it is nondegenerate and $\ell ( \vec{e}_1 ) = 1$. Let $S(n)$ be the collection of all normalized $n$-simplices of $\operatorname{N}_{\bullet }^{\leq k+1}( \operatorname{Path}[G] )$ having length $k+1$. We make the following observations:

$(i)$

If $\sigma $ belongs to $S(n)$, then the faces $d_0(\sigma )$ and $d_ n(\sigma )$ have length $\leq k$, and are therefore contained in $\operatorname{N}_{\bullet }^{\leq k}( \operatorname{Path}[G] )$.

$(ii)$

If $\sigma $ belongs to $S(n)$ and $1 < i < n$, then the face $d_ i(\sigma )$ is a normalized $(n-1)$-simplex of $\operatorname{N}_{\bullet }^{\leq k+1}( \operatorname{Path}[G] )$ of length $k+1$, and therefore belongs to $S(n-1)$.

$(iii)$

If $\sigma $ belongs to $S(n)$, then the face $d_1(\sigma )$ is not normalized. Moreover, the construction $\sigma \mapsto d_1(\sigma )$ induces a bijection from $S(n)$ to the collection of $(n-1)$-simplices of $\operatorname{N}_{\bullet }^{\leq k+1}( \operatorname{Path}[G] )$ which are nondegenerate, of length $k+1$, and not normalized.

For each $n \geq 1$, let $X(n)_{\bullet }$ denote the simplicial subset of $\operatorname{N}_{\bullet }^{\leq k+1}( \operatorname{Path}[G] )$ given by the union of the $(n-1)$-skeleton $\operatorname{sk}_{n-1}( \operatorname{N}_{\bullet }^{\leq k+1}( \operatorname{Path}[G] ))$, the simplicial set $\operatorname{N}_{\bullet }^{\leq k}( \operatorname{Path}[G] )$, and the collection of normalized $n$-simplices of $\operatorname{N}_{\bullet }^{\leq k+1}( \operatorname{Path}[G] )$. We have inclusions

\[ X(1)_{\bullet } \subseteq X(2)_{\bullet } \subseteq X(3)_{\bullet } \subseteq X(4)_{\bullet } \subseteq \cdots , \]

where $\operatorname{N}_{\bullet }^{\leq k}( \operatorname{Path}[G] ) = X(1)_{\bullet }$ and $\operatorname{N}_{\bullet }^{\leq k+1}( \operatorname{Path}[G] ) = \bigcup _{n} X(n)_{\bullet }$. It will therefore suffice to show that the inclusion maps $X(n-1)_{\bullet } \hookrightarrow X(n)_{\bullet }$ are inner anodyne for $n \geq 2$. We conclude by observing that $(i)$, $(ii)$, and $(iii)$ guarantee the existence of a pushout diagram of simplicial sets

\[ \xymatrix { \coprod _{\sigma \in S(n)} \Lambda ^{n}_{1} \ar [r] \ar [d] & \coprod _{\sigma \in S(n)} \Delta ^ n \ar [d] \\ X(n-1)_{\bullet } \ar [r] & X(n)_{\bullet }. } \]
$\square$