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2.4.4 The Path Category of a Simplicial Set

Let $G$ be a directed graph, which we identify with a simplicial set $G_{\bullet }$ of dimension $\leq 1$ (Proposition 1.1.5.9). In §1.2.6, we introduced a category $\operatorname{Path}[G]$ called the path category of $G$ (Construction 1.2.6.1). The category $\operatorname{Path}[G]$ is characterized (up to isomorphism) by a universal property: for any category $\operatorname{\mathcal{C}}$, Proposition 1.2.6.5 supplies a bijection

\[ \{ \text{Functors $F: \operatorname{Path}[G] \rightarrow \operatorname{\mathcal{C}}$} \} \simeq \operatorname{Hom}_{\operatorname{Set_{\Delta }}}(G_{\bullet }, \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})). \]

In this section, we introduce a generalization of the construction $G \mapsto \operatorname{Path}[G]$, where we replace directed graphs by arbitrary simplicial sets (not necessarily of dimension $\leq 1$) and categories by simplicial categories.

Definition 2.4.4.1. Let $S_{\bullet }$ be a simplicial set and let $\operatorname{\mathcal{C}}_{\bullet }$ be a simplicial category. We will say that a map of simplicial sets $u: S_{\bullet } \rightarrow \operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}})$ exhibits $\operatorname{\mathcal{C}}_{\bullet }$ as a path category of $S_{\bullet }$ if, for every simplicial category $\operatorname{\mathcal{D}}_{\bullet }$, composition with $u$ induces a bijection

\[ \{ \text{Simplicial functors $F: \operatorname{\mathcal{C}}_{\bullet } \rightarrow \operatorname{\mathcal{D}}_{\bullet }$} \} \rightarrow \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( S_{\bullet }, \operatorname{N}_{\bullet }^{\operatorname{hc}}( \operatorname{\mathcal{D}}) ). \]

Notation 2.4.4.2 (The Path Category of a Simplicial Set). Let $S_{\bullet }$ be a simplicial set. It follows immediately from the definitions that if there exists a map of simplicial sets $u: S_{\bullet } \rightarrow \operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}})$ which exhibits $\operatorname{\mathcal{C}}_{\bullet }$ as the path category of $S_{\bullet }$, then the simplicial category $\operatorname{\mathcal{C}}_{\bullet }$ (and the morphism $u$) are unique up to isomorphism and depend functorially on $S_{\bullet }$. We will emphasize this dependence by denoting $\operatorname{\mathcal{C}}_{\bullet }$ by $\operatorname{Path}[ S ]_{\bullet }$ and referring to it as the path category of the simplicial set $S_{\bullet }$.

Proposition 2.4.4.3. Let $S_{\bullet }$ be a simplicial set. Then there exists simplicial category $\operatorname{\mathcal{C}}_{\bullet }$ and a morphism of simplicial sets $u: S_{\bullet } \rightarrow \operatorname{N}^{\operatorname{hc}}_{\bullet }(\operatorname{\mathcal{C}})$ which exhibits $\operatorname{\mathcal{C}}_{\bullet }$ as a path category of $S_{\bullet }$.

Proof. This is a special case of Proposition 1.1.8.20, since the category $\operatorname{Cat_{\Delta }}$ admits small colimits (Proposition 2.4.1.13). Explicitly, the simplicial path category of a simplicial set $S_{\bullet }$ is given by the generalized geometric realization

\[ | S_{\bullet } |^{ \operatorname{Path}[-]_{\bullet } } = \varinjlim _{ \Delta ^ n \rightarrow S_{\bullet } } \operatorname{Path}[n]_{\bullet }, \]

where $\operatorname{Path}[-]_{\bullet }$ denotes the cosimplicial object of $\operatorname{Cat_{\Delta }}$ defined in Notation 2.4.3.1. $\square$

Corollary 2.4.4.4. The homotopy coherent nerve functor $\operatorname{N}_{\bullet }^{\operatorname{hc}}: \operatorname{Cat_{\Delta }}\rightarrow \operatorname{Set_{\Delta }}$ admits a left adjoint

\[ \operatorname{Path}[ - ]_{\bullet }: \operatorname{Set_{\Delta }}\rightarrow \operatorname{Cat_{\Delta }}, \]

which associates to each simplicial set $S_{\bullet }$ the path category $\operatorname{Path}[ S ]_{\bullet }$ of Notation 2.4.4.2.

Warning 2.4.4.5. We have now introduced several different notions of path category:

$(a)$

To every directed graph $G$, Construction 1.2.6.1 associates an ordinary category $\operatorname{Path}[G]$.

$(b)$

To every partially ordered set $Q$, Notation 2.4.3.1 associates a simplicial category $\operatorname{Path}[Q]_{\bullet }$.

$(c)$

To every simplicial set $S_{\bullet }$, Proposition 2.4.4.3 associates a simplicial category $\operatorname{Path}[ S ]_{\bullet }$.

We will show below that these constructions are closely related:

$(1)$

If $G$ is a directed graph and $S_{\bullet }$ denotes the associated simplicial set of dimension $\leq 1$ (Proposition 1.1.5.9), then the simplicial category $\operatorname{Path}[ S ]_{\bullet }$ of $(c)$ is constant, associated to the ordinary category $\operatorname{Path}[G]$ of $(a)$ (Proposition 2.4.4.7).

$(2)$

If $Q$ is a partially ordered set, then the simplicial category $\operatorname{Path}[Q]_{\bullet }$ of $(b)$ can be identified with the simplicial category $\operatorname{Path}[ \operatorname{N}(Q) ]_{\bullet }$ of $(c)$, where $\operatorname{N}_{\bullet }(Q)$ denotes the nerve of $Q$ (Proposition 2.4.4.14).

$(3)$

For any simplicial set $S_{\bullet }$, the simplicial category $\operatorname{Path}[ S ]_{\bullet }$ of $(c)$ has an underlying ordinary category $\operatorname{Path}[ S ]_0$, which can be described as the category $\operatorname{Path}[G]$ associated by $(a)$ to the underlying directed graph $G = \mathrm{Gr}( S_{\bullet } )$ of $S_{\bullet }$ (Proposition 2.4.4.12).

Assertions $(1)$ and $(2)$ imply that the path category constructions of §1.2.6 and §2.4.3 can be regarded as special cases of the construction $S_{\bullet } \mapsto \operatorname{Path}[ S ]_{\bullet }$. Assertion $(3)$ is a partial converse, which guarantees that the simplicial path category $\operatorname{Path}[S]_{\bullet }$ can be regarded as a simplicially enriched version of the classical path category studied in §1.2.6.

In the special case where $Q$ is a linearly ordered set of the form $[n] = \{ 0 < 1 < \cdots < n \} $, assertion $(2)$ of Warning 2.4.4.5 is immediate from the definitions:

Example 2.4.4.6 (The Path Category of a Simplex). Let $n \geq 0$ be a nonnegative integer and let $\operatorname{Path}[n]_{\bullet }$ denote the simplicial category of Notation 2.4.3.1. For any simplicial category $\operatorname{\mathcal{C}}_{\bullet }$, we have canonical bijections

\[ \operatorname{Hom}_{\operatorname{Cat_{\Delta }}}( \operatorname{Path}[n]_{\bullet }, \operatorname{\mathcal{C}}_{\bullet } ) \simeq \operatorname{N}^{\operatorname{hc}}_{n}(\operatorname{\mathcal{C}}) \simeq \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( \Delta ^ n, \operatorname{N}^{\operatorname{hc}}_{\bullet }(\operatorname{\mathcal{C}}) ). \]

It follows that $\operatorname{Path}[n]_{\bullet }$ is a path category for the standard simplex $\Delta ^ n$, in the sense of Definition 2.4.4.1.

Proposition 2.4.4.7. Let $G$ be a directed graph, let $\operatorname{Path}[G]$ denote the path category of Construction 1.2.6.1, and let $\underline{\operatorname{Path}[G]}_{\bullet }$ denote the associated constant simplicial category (Example 2.4.2.3). Then the comparison map $u: G_{\bullet } \rightarrow \operatorname{N}_{\bullet }( \operatorname{Path}[G] ) \simeq \operatorname{N}_{\bullet }^{\operatorname{hc}}( \underline{\operatorname{Path}[G]} )$ exhibits $\underline{\operatorname{Path}[G]}_{\bullet }$ as a path category of the simplicial set $G_{\bullet }$.

Proof. Unwinding the definitions, we must show that for every simplicial category $\operatorname{\mathcal{D}}_{\bullet }$, the composite map

\begin{eqnarray*} \operatorname{Hom}_{\operatorname{Cat_{\Delta }}}( \underline{\operatorname{Path}[G]}_{\bullet }, \operatorname{\mathcal{D}}_{\bullet } ) & \rightarrow & \operatorname{Hom}_{\operatorname{Cat}}( \operatorname{Path}[G], \operatorname{\mathcal{D}}) \\ & \rightarrow & \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( G_{\bullet }, \operatorname{N}_{\bullet }(\operatorname{\mathcal{D}}) ) \\ & \rightarrow & \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( G_{\bullet }, \operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{D}}) ) \end{eqnarray*}

is a bijection. Here the first map is bijective because the simplicial category $\underline{\operatorname{Path}[G]}_{\bullet }$ is constant (Remark 2.4.2.5), the second by virtue of Proposition 1.2.6.5, and the third because $G_{\bullet }$ has dimension $\leq 1$ and the comparison map $\operatorname{N}_{\bullet }(\operatorname{\mathcal{D}}) \rightarrow \operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{D}})$ is an isomorphism on simplices of dimension $\leq 1$ (Example 2.4.3.9). $\square$

Warning 2.4.4.8. It follows from Proposition 2.4.4.7 that if $S_{\bullet }$ is a simplicial set of dimension $\leq 1$, then the simplicial category $\operatorname{Path}[ S ]_{\bullet }$ is constant. Beware that this is never true for simplicial sets of dimension $> 1$ (see Theorem 2.4.4.10 below).

The proof of Proposition 2.4.4.3 given above is somewhat unsatisfying: it constructs the path category of a simplicial set $S_{\bullet }$ abstractly, as the colimit of a certain diagram in $\operatorname{Cat_{\Delta }}$. In general, colimits in $\operatorname{Cat_{\Delta }}$ (like colimits in $\operatorname{Cat}$) can be difficult to describe. However, the (simplicial) path category $\operatorname{Path}[S]_{\bullet }$ actually has a relatively simple structure. For each nonnegative integer $m$, the category $\operatorname{Path}[ S ]_{m}$ is free in the sense of Definition 1.2.6.7: that is, it can be realized as the (ordinary) path category of a directed graph. To formulate a more precise statement, we need a bit of (temporary) notation.

Notation 2.4.4.9. Let $S_{\bullet }$ be a simplicial set. For each nonnegative integer $m$, we let $E(S,m)$ denote the collection of pairs $(\sigma , \overrightarrow {I} )$, where $\sigma : \Delta ^{n} \rightarrow S_{\bullet }$ is a nondegenerate simplex of $S_{\bullet }$ of dimension $n > 0$ and $\overrightarrow {I} = (I_0 \subseteq I_1 \subseteq \cdots \subseteq I_{m-1} \subseteq I_ m )$ is a chain of subsets of $[n]$ satisfying $I_0 = \{ 0, n\} $ and $I_ m = [n]$. Here we will view $\overrightarrow {I}$ as a $m$-simplex of the simplicial set $\operatorname{Hom}_{ \operatorname{Path}[n] }(0, n)_{\bullet }$.

Let $\operatorname{\mathcal{C}}_{\bullet }$ be a simplicial category and let $u: S_{\bullet } \rightarrow \operatorname{N}_{\bullet }^{\operatorname{hc}}( \operatorname{\mathcal{C}})$ be a morphism of simplicial sets. For each element $(\sigma , \overrightarrow {I}) \in E(S,m)$, the composite map

\[ \Delta ^{n} \xrightarrow {\sigma } S_{\bullet } \xrightarrow {u} \operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}}) \]

can be identified with a simplicial functor $u(\sigma ): \operatorname{Path}[n] \rightarrow \operatorname{\mathcal{C}}$. This functor carries $\overrightarrow {I}$ to a morphism in the ordinary category $\operatorname{\mathcal{C}}_{m}$, which we will denote by $u(\sigma , \overrightarrow {I} )$.

Theorem 2.4.4.10. Let $S_{\bullet }$ be a simplicial set and let $u: S_{\bullet } \rightarrow \operatorname{N}_{\bullet }^{\operatorname{hc}}( \operatorname{Path}[S ] )$ be a map of simplicial sets which exhibits $\operatorname{Path}[S]_{\bullet }$ as a path category of $S_{\bullet }$. Then:

$(1)$

The map $u$ induces a bijection from the set of vertices of $S_{\bullet }$ to the set of objects of $\operatorname{Path}[S ]_{\bullet }$.

$(2)$

For each nonnegative integer $m \geq 0$, the category $\operatorname{Path}[ S ]_{m}$ is free (in the sense of Definition 1.2.6.7).

$(3)$

For each nonnegative integer $m \geq 0$, the construction $(\sigma , \overrightarrow {I} ) \mapsto u(\sigma , \overrightarrow {I} )$ of Notation 2.4.4.9 induces a bijection from $E(S,m)$ to the set of indecomposable morphisms of the category $\operatorname{Path}[S]_{m}$.

Remark 2.4.4.11. Let $S_{\bullet }$ be a simplicial set. Then the path category $\operatorname{Path}[S]_{\bullet }$ is characterized (up to isomorphism) by properties $(1)$, $(2)$, and $(3)$ of Theorem 2.4.4.10. More precisely, suppose that $\operatorname{\mathcal{C}}_{\bullet }$ is a simplicial category and that we are given a comparison map $u': S_{\bullet } \rightarrow \operatorname{N}_{\bullet }^{\operatorname{hc}}( \operatorname{\mathcal{C}})$ satisfying the following three conditions:

$(1')$

The map $u'$ induces a bijection from the set of vertices of $S_{\bullet }$ to the set of objects of $\operatorname{\mathcal{C}}_{\bullet }$.

$(2')$

For each nonnegative integer $m \geq 0$, the category $\operatorname{\mathcal{C}}_{m}$ is free.

$(3')$

For each nonnegative integer $m \geq 0$, the construction $(\sigma , \overrightarrow {I} ) \mapsto u'(\sigma , \overrightarrow {I} )$ induces a bijection from $E(S,m)$ to the set of indecomposable morphisms of the category $\operatorname{\mathcal{C}}_ m$.

Then $u'$ exhibits $\operatorname{\mathcal{C}}_{\bullet }$ as a path category of $S_{\bullet }$, in the sense of Definition 2.4.4.1. To prove this, we invoke the definition of $\operatorname{Path}[S]_{\bullet }$ to deduce that there is a unique simplicial functor $F: \operatorname{Path}[S]_{\bullet } \rightarrow \operatorname{\mathcal{C}}_{\bullet }$ for which the composite map

\[ S_{\bullet } \xrightarrow {u} \operatorname{N}_{\bullet }^{\operatorname{hc}}( \operatorname{Path}[S] ) \xrightarrow { \operatorname{N}_{\bullet }^{\operatorname{hc}}(F)} \operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}}) \]

is equal to $u'$. Combining Theorem 2.4.4.10 with assumptions $(1')$, $(2')$, and $(3')$, we deduce that for each $m \geq 0$, the induced functor $\operatorname{Path}[S]_{m} \rightarrow \operatorname{\mathcal{C}}_{m}$ is a map between free categories which is bijective on objects and indecomposable morphisms, and is therefore an isomorphism of categories.

Before giving the proof of Theorem 2.4.4.10, let us use it to deduce assertions $(2)$ and $(3)$ of Warning 2.4.4.5.

Proposition 2.4.4.12. Let $S_{\bullet }$ be a simplicial set and let $G$ be its underlying directed graph (Example 1.1.5.4), so that $G_{\bullet }$ can be identified with the $1$-skeleton of $S_{\bullet }$. Let $u: S_{\bullet } \rightarrow \operatorname{N}_{\bullet }^{\operatorname{hc}}( \operatorname{Path}[ S])$ denote the unit map. Then:

  • The restriction $u|_{ G_{\bullet } }$ factors uniquely as a composition

    \[ G_{\bullet } \xrightarrow {u_0} \operatorname{N}_{\bullet }( \operatorname{Path}[S ]_0 ) \rightarrow \operatorname{N}_{\bullet }^{\operatorname{hc}}( \operatorname{Path}[ S] ). \]
  • The map $u_0$ induces an isomorphism of categories $\operatorname{Path}[G] \xrightarrow {\sim } \operatorname{Path}[S]_0$.

Proof. The first assertion follows immediately from Example 2.4.3.9, since $G_{\bullet }$ is a simplicial set of dimension $\leq 1$. To prove the second assertion, we note that Theorem 2.4.4.10 guarantees that $\operatorname{Path}[S]_0$ is a free category, whose objects can be identified with the vertices of $S_{\bullet }$ and whose indecomposable morphisms can be identified with elements of the set $E(S,0)$ of Notation 2.4.4.9. By definition, $E(S,m)$ consists of pairs $(\sigma , \overrightarrow {I} )$, where $\sigma $ is a nondegenerate $n$-simplex of $S_{\bullet }$ for $n > 0$ and $\overrightarrow {I} = (I_0 \subseteq \cdots \subseteq I_ m)$ is a chain of subsets of $[n]$ satisfying $I_0 = \{ 0, n \} $ and $I_ m = [n]$. In the case $m =0$, the equality $I_0 = I_ m$ forces $n = 1$, so that $E(S,0)$ can be identified (via the map $u_0$) with the collection of nondegenerate $1$-simplices of $S_{\bullet }$: that is, with the collection of edges of the graph $G$. The freeness of $\operatorname{Path}[ S ]_0$ now guarantees that the induced map $\operatorname{Path}[G] \simeq \operatorname{Path}[S]_0$ is an isomorphism of categories (see Proposition 1.2.6.11). $\square$

Exercise 2.4.4.13. Use Theorem 2.4.4.10 to give a different proof of Proposition 2.4.4.7 (show that if $S_{\bullet }$ is a simplicial set of dimension $\leq 1$, then the sets $E(S,m)$ appearing in Notation 2.4.4.9 do not depend on $m$).

Let $Q$ be a partially ordered set. Note that every $n$-simplex $\sigma \in \operatorname{N}_{\bullet }(Q)$ can be identified with a map of partially ordered sets $[n] \rightarrow Q$, and therefore induces a simplicial functor $\operatorname{Path}[n]_{\bullet } \rightarrow \operatorname{Path}[Q]_{\bullet }$ which we can view as an $n$-simplex of the homotopy coherent nerve $\operatorname{N}_{\bullet }^{\operatorname{hc}}( \operatorname{Path}[Q] )$. This construction determines a map of simplicial sets $u: \operatorname{N}_{\bullet }(Q) \rightarrow \operatorname{N}_{\bullet }^{\operatorname{hc}}( \operatorname{Path}[Q] )$.

Proposition 2.4.4.14. Let $Q$ be a partially ordered set. Then the comparison map $u: \operatorname{N}_{\bullet }(Q) \rightarrow \operatorname{N}_{\bullet }^{\operatorname{hc}}( \operatorname{Path}[Q] )$ described above exhibits $\operatorname{Path}[Q]_{\bullet }$ as a path category for the simplicial set $\operatorname{N}_{\bullet }(Q)$ (in the sense of Definition 2.4.4.1).

Proposition 2.4.4.14 follows immediately from Remark 2.4.4.11 together with the following:

Lemma 2.4.4.15. Let $Q$ be a partially ordered set. Then the comparison map $u: \operatorname{N}_{\bullet }(Q) \rightarrow \operatorname{N}_{\bullet }^{\operatorname{hc}}( \operatorname{Path}[Q] )$ satisfies conditions $(1')$, $(2')$, and $(3')$ of Remark 2.4.4.11.

Proof. Assertion $(1')$ is immediate (the map $u$ is bijective on vertices by construction). For each $m \geq 0$, the category $\operatorname{Path}[Q]_ m$ can be described concretely as follows:

  • The objects of $\operatorname{Path}[Q]_ m$ are the elements of $Q$.

  • If $x$ and $y$ are elements of $Q$, then a morphism from $x$ to $y$ in $\operatorname{Path}[Q]_ m$ is a chain

    \[ \overrightarrow {J} = (J_0 \subseteq J_1 \subseteq \cdots \subseteq J_ m) \]

    of finite linearly ordered subsets of $Q$, where each $S_ i$ has least element $x$ and greatest element $y$.

Note that a morphism $\overrightarrow {J}$ from $x$ to $y$ is indecomposable (in the sense of Definition 1.2.6.8) if and only if $x < y$ and $J_0 = \{ x,y\} $. Moreover, an arbitrary morphism $\overrightarrow {J}$ from $x$ to $y$ with $J_0 = \{ x = x_0 < x_1 < \cdots < x_ k = y \} $ decomposes uniquely as a composition of indecomposable morphisms

\[ x_0 \xrightarrow { \overrightarrow {J(1)}} x_1 \xrightarrow { \overrightarrow {J(2)}} x_2 \rightarrow \cdots \xrightarrow { \overrightarrow {J(k)}} x_ k \]

where $J(a)_{b} = \{ z \in J_ b: x_{a-1} \leq z \leq x_ a \} $. Applying Proposition 1.2.6.11, we deduce that the category $\operatorname{Path}[Q]_ m$ is free, which proves $(2')$. To prove $(3')$, we observe that every indecomposable morphism $\overrightarrow {J}$ can be written uniquely in the form $u( \sigma , \overrightarrow {I} )$, where $(\sigma , \overrightarrow {I} )$ is an element of the set $E(S,m)$ of Notation 2.4.4.9. Writing $J_ m = \{ x = x_0 < \cdots < x_ n = y \} $, we see that $\sigma $ must be the nondegenerate $n$-simplex of $\operatorname{N}_{\bullet }(Q)$ given by the map

\[ [n] \rightarrow Q \quad \quad i \mapsto x_ i, \]

and $\overrightarrow {I}$ must be the chain $( \sigma ^{-1}(J_0) \subseteq \sigma ^{-1}(J_1) \subseteq \cdots \subseteq \sigma ^{-1}(J_ m) )$ of subsets of $[n]$. $\square$

Proof of Theorem 2.4.4.10. Let $m$ be a nonnegative integer, which we regard as fixed throughout the proof. Let $G( S )$ denote the directed graph given by

\[ \operatorname{Vert}( G(S ) ) = S_0 \quad \quad \operatorname{Edge}( G(S ) ) = E( S, m), \]

where we regard each element

\[ (\sigma : \Delta ^ n \rightarrow S_{\bullet }, \vec{I} \in \operatorname{Hom}_{ \operatorname{Path}[n] }( 0, n)_{m} ) \in \operatorname{Edge}( G(S ) ) \]

as having source $\sigma (0) \in \operatorname{Vert}( G(S) )$ and target $\sigma (n) \in \operatorname{Vert}( G(S))$. Let $u_{S_{\bullet }}: S_{\bullet } \rightarrow \operatorname{N}_{\bullet }^{\operatorname{hc}}( \operatorname{Path}[ S ] )$ denote the unit map. Then $u_{S_{\bullet } }$ induces a map of simplicial sets $G(S)_{\bullet } \rightarrow \operatorname{N}_{\bullet }( \operatorname{Path}[S]_{m} )$, which we can identify with a functor of ordinary categories

\[ F_{S_{\bullet }}: \operatorname{Path}[ G ] \rightarrow \operatorname{Path}[ S ]_{m}. \]

Let us say that the simplicial set $S_{\bullet }$ is good if $F_{ S_{\bullet } }$ is an isomorphism of categories. Theorem 2.4.4.10 is equivalent to the assertion that every simplicial set is good (for every choice of nonnegative integer $m$). We will prove this by verifying that the collection of good simplicial sets satisfies the hypotheses of Lemma 1.1.8.15:

  • Suppose we are given a pushout diagram of simplicial sets $\sigma :$

    \[ \xymatrix { S_{\bullet } \ar [r] \ar [d] & T_{\bullet } \ar [d] \\ S'_{\bullet } \ar [r] & T'_{\bullet }, } \]

    where the horizontal maps are monomorphisms. Suppose that $S_{\bullet }$, $T_{\bullet }$, and $S'_{\bullet }$ are good; we wish to show that $T'_{\bullet }$ is good. Note that the horizonal maps induce monomorphisms of directed graphs

    \[ G(S) \hookrightarrow G(T ) \quad \quad G(S' ) \hookrightarrow G(T'). \]

    Define subgraphs $G_0(S) \subseteq G(S )$ and $G_0( T) \subseteq G(T )$ by the formulae

    \[ \operatorname{Vert}( G_0(S ) ) = \operatorname{Vert}( G(S) ) = S_0 \quad \quad \operatorname{Vert}( G_0(T ) ) = \operatorname{Vert}( G(T) ) = T_0 \]
    \[ \operatorname{Edge}( G_0( S) ) = \emptyset \quad \quad \operatorname{Edge}( G_0( T ) ) = \operatorname{Edge}( G(T) ) \setminus \operatorname{Edge}( G( S) ). \]

    We then have a commutative diagram of categories

    \[ \xymatrix { \operatorname{Path}[ G_0(S)] \ar [r] \ar [d] & \operatorname{Path}[ G_0(T) ] \ar [d] \\ \operatorname{Path}[ G( S' ) ] \ar [r] \ar [d]^{ F_{S'} } & \operatorname{Path}[ G( T' )] \ar [d]^{ F_{T'} } \\ \operatorname{Path}[ S' ]_{m} \ar [r] & \operatorname{Path}[ T' ]_{m}. } \]

    We wish to show that the functor $F_{T'}$ is an isomorphism of categories, and the map $F_{S' }$ is an isomorphism by assumption. It will therefore suffice to show that the lower square in this diagram is a pushout. Note that the upper square is a pushout (since it is obtained from a pushout diagram in the category of directed graphs by passing to path categories). We are therefore reduced to showing that the outer rectangle is a pushout. We can rewrite this as the outer rectangle in another commutative diagram of categories

    \[ \xymatrix { \operatorname{Path}[ G_0(S)] \ar [r] \ar [d] & \operatorname{Path}[ G_0(T) ] \ar [d] \\ \operatorname{Path}[ G( S ) ] \ar [r] \ar [d]^{ F_{S} } & \operatorname{Path}[ G(T )] \ar [d]^{ F_{T} } \\ \operatorname{Path}[ S ]_{m} \ar [r] \ar [d] & \operatorname{Path}[ T ]_{m} \ar [d] \\ \operatorname{Path}[ S' ]_{m} \ar [r] & \operatorname{Path}[ T' ]_{m}. } \]

    We now conclude by observing that the upper square in this diagram is a pushout (because it is obtained from a pushout diagram of directed graphs by passing to path categories), the middle square is a pushout (since $F_{S}$ and $F_{T}$ are isomorphisms), and the lower square is a pushout (since the construction $X_{\bullet } \mapsto \operatorname{Path}[ X ]_{m}$ preserves colimits).

  • Suppose we are given a sequence of monomorphisms of simplicial sets

    \[ S(0)_{\bullet } \hookrightarrow S(1)_{\bullet } \hookrightarrow S(2)_{\bullet } \hookrightarrow \cdots \]

    with colimit $S_{\bullet }$. Then the functor $F_{S_{\bullet }}: \operatorname{Path}[ G(S_{\bullet })] \rightarrow \operatorname{Path}[ S ]_{m}$ can be written as a filtered colimit of functors $F_{S(i)_{\bullet }}: \operatorname{Path}[ G(S(i)_{\bullet })] \rightarrow \operatorname{Path}[ S(i) ]_{m}$. Consequently, if each $S(i)_{\bullet }$ is good, then $S_{\bullet }$ is good.

  • Let $S_{\bullet }$ be a simplicial set which can be written as a coproduct $\coprod _{i \in I} \Delta ^ n$; we must show that $S_{\bullet }$ is good. Without loss of generality, we may assume that $I$ is a singleton, so that $S_{\bullet } = \Delta ^ n$. In this case, Example 2.4.4.6 supplies an equivalence of simplicial categories $\operatorname{Path}[S]_{\bullet } \simeq \operatorname{Path}[n]_{\bullet }$. The desired result now follows from Lemma 2.4.4.15.

$\square$

Remark 2.4.4.16. Let $S_{\bullet }$ be a simplicial set. For each $m \geq 0$, Theorem 2.4.4.10 guarantees that $\operatorname{Path}[S]_ m$ can be realized as the path category of a directed graph $G_ m$ (Construction 1.2.6.1), which can be described explicitly as follows:

  • The vertices of $G_{m}$ are the vertices of the simplicial set $S_{\bullet }$.

  • The edges of $G_{m}$ are the elements of the set $E(S,m)$ of Notation 2.4.4.9.

It follows that we can regard the construction $[m] \mapsto \operatorname{Path}[ G_ m ]$ as a simplicial object of $\operatorname{Cat}$. The face and degeneracy operators on this simplicial object can be described as follows:

  • For $0 \leq i \leq m$, the degeneracy operator $s_{i}: \operatorname{Path}[ G_ m ] \rightarrow \operatorname{Path}[ G_{m+1} ]$ is induced by a map of directed graphs from $G_ m$ to $G_{m+1}$, which is the identity on vertices and given on edges by the construction

    \[ (\sigma , I_0 \subseteq \cdots \subseteq I_{m} ) \mapsto (\sigma , I_{0} \subseteq \cdots \subseteq I_{i-1} \subseteq I_{i} \subseteq I_{i} \subseteq I_{i+1} \subseteq \cdots \subseteq I_ m ). \]
  • For $0 < i < m$, the face map $d_{i}: \operatorname{Path}[G_ m] \rightarrow \operatorname{Path}[G_{m-1} ]$ is induced by a map of directed graphs from $G_{m}$ to $G_{m-1}$, which is the identity on vertices and given on edges by the construction

    \[ (\sigma , I_0 \subseteq \cdots \subseteq I_{m} ) \mapsto (\sigma , I_{0} \subseteq \cdots \subseteq I_{i-1} \subseteq I_{i+1} \subseteq \cdots \subseteq I_ m ). \]
  • Each of the face maps $d_{m}: \operatorname{Path}[G_ m] \rightarrow \operatorname{Path}[G_{m-1}]$ is induced by a map directed graphs $f: G_ m \rightarrow G_{m-1}$ which is the identity on vertices. Let $(\sigma , \overrightarrow {I})$ be an edge of $G_{m}$, given by a nondegenerate simplex $\sigma : \Delta ^ n \rightarrow S_{\bullet }$ and a chain of subsets $\overrightarrow {I} = (I_0 \subseteq \cdots \subseteq I_{m})$ of $[n]$. Then the subset $I_{m-1} \subseteq I_{m} = [n]$ is the image of a unique monotone injection $\alpha : [n'] \hookrightarrow [n]$, and the composite map $\Delta ^{n'} \xrightarrow {\alpha } \Delta ^{n} \xrightarrow {\sigma } S_{\bullet }$ factors uniquely as a composition $\Delta ^{n'} \twoheadrightarrow \Delta ^{n''} \xrightarrow {\tau } S_{\bullet }$, where the first map is surjective on vertices and $\tau $ is a nondegenerate $n''$-simplex of $S_{\bullet }$. For $0 \leq i < m$, let $J_{i} \subseteq [n'']$ denote the image of the composite map $I_{i} \hookrightarrow I_{m-1} \xrightarrow {\alpha ^{-1}} [n'] \twoheadrightarrow [n'']$, and set $\overrightarrow {J} = (J_0 \subseteq J_1 \subseteq \cdots \subseteq J_{m-1} )$. In the case $n'' = 0$, the morphism $f$ carries $(\sigma , \overrightarrow {I} )$ to the vertex $\tau \in \operatorname{Vert}(G_{m-1} ) = S_0$. In the case $n'' > 0$ the morphism $f$ carries $(\sigma , \overrightarrow {I} )$ to the edge $( \tau , \overrightarrow {J}) \in \operatorname{Edge}( G_{m-1})$.

  • The face maps $d_0: \operatorname{Path}[G_ m] \rightarrow \operatorname{Path}[G_{m-1} ]$ are generally not induced by maps of directed graphs $G_{m} \rightarrow G_{m-1}$: that is, they do not carry indecomposable morphisms of $\operatorname{Path}[G_ m]$ to indecomposable morphisms of $\operatorname{Path}[G_{m-1}]$. More precisely, if $(\sigma , \overrightarrow {I} )$ is an edge of $G_{n}$ with $I_{1} = \{ 0 = i_0 < i_1 < \cdots < i_ k = m \} $, then $d_0$ carries $(\sigma , \overrightarrow {I} )$ to a path of length $k$ in the category $\operatorname{Path}[ G_{m-1} ]$.