# Kerodon

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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 morphism 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 uniquely determined 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 a 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 1.1.8.22, 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 2.4.4.15).

$(3)$

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 2.4.4.13).

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 2.4.2.4). 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 2.4.2.6), 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 $\overrightarrow {I} = (I_0 \supseteq I_{1} \supseteq \cdots \supseteq I_{m-1} \supseteq I_ m )$ is a chain of subsets of $[n]$ satisfying $I_0 = [n]$ and $I_ m = \{ 0, n \}$. 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 morphism 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 universal property 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.

Remark 2.4.4.12. Let $u: S_{\bullet } \hookrightarrow S'_{\bullet }$ be a monomorphism of simplicial sets. Then, for each $m \geq 0$, $u$ induces a monomorphism of sets $E(S,m) \hookrightarrow E(S',m)$ (see Notation 2.4.4.9). It follows from Theorem 2.4.4.10 that if $x$ and $y$ are vertices of $S_{\bullet }$, then the induced map of simplicial sets $\operatorname{Hom}_{ \operatorname{Path}[S]}(x,y)_{\bullet } \rightarrow \operatorname{Hom}_{ \operatorname{Path}[S'] }( u(x), u(y) )_{\bullet }$ is a monomorphism.

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 2.4.4.13. 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 $\overrightarrow {I} = (I_0 \supseteq \cdots \supseteq I_ m)$ is a chain of subsets of $[n]$ satisfying $I_0 = [n]$ and $I_ m = \{ 0, n \}$. In the case $m =0$, the equality $I_0 = I_ m$ forces $n = 1$, so that $E(S,0)$ can be identified (via the morphism $u_0$) 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 2.4.4.14. 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 2.4.4.15. Let $Q$ be a partially ordered set. Then the comparison map $u: \operatorname{N}_{\bullet }(Q) \rightarrow \operatorname{N}_{\bullet }^{\operatorname{hc}}( \operatorname{Path}[Q] )$ 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 2.4.4.15 follows immediately from Remark 2.4.4.11 together with the following:

Lemma 2.4.4.16. 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 morphism $u$ 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

$\overrightarrow {J} = (J_0 \supseteq J_1 \supseteq \cdots \supseteq J_ m)$

of finite linearly ordered subsets of $Q$, where each $J_ i$ 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 $J_ m = \{ x,y\}$. Moreover, an arbitrary morphism $\overrightarrow {J}$ from $x$ to $y$ with $J_ m = \{ x = x_0 < x_1 < \cdots < x_ k = y \}$ 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 $J_0 = \{ x = x_0 < \cdots < x_ n = y \}$, 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 $( \sigma ^{-1}(J_0) \supseteq \sigma ^{-1}(J_{1}) \supseteq \cdots \supseteq \sigma ^{-1}(J_ m) )$ 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. For each simplicial set $S$, let $G( S )$ denote the directed graph given by

$\operatorname{Vert}( G(S ) ) = \{ \textnormal{Vertices of S} \} \quad \quad \operatorname{Edge}( G(S ) ) = E( S, m),$

where we regard each element

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

as an edge of $G(S)$ having source $\sigma (0) \in \operatorname{Vert}( G(S) )$ and target $\sigma (n) \in \operatorname{Vert}( G(S))$. Let $u_{S}: S \rightarrow \operatorname{N}_{\bullet }^{\operatorname{hc}}( \operatorname{Path}[ S ] )$ exhibit the simplicial category $\operatorname{Path}_{\bullet }[S]$ as a path category of $S$. Then $u_{S}$ 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 $F_{S}: \operatorname{Path}[ G(S) ] \rightarrow \operatorname{Path}[ S ]_{m}$. Let us say that the simplicial set $S$ is good if $F_{S}$ 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 $\sigma :$

$\xymatrix@R =50pt@C=50pt{ S \ar [r] \ar [d] & T \ar [d] \\ S' \ar [r] & T', }$

where the horizontal maps are monomorphisms. Suppose that $S$, $T$, and $S'$ 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

$\xymatrix@R =50pt@C=50pt{ \operatorname{Path}[ G_0(S)] \ar [r] \ar [d] & \operatorname{Path}[ G_0(T) ] \ar [d] \\ \operatorname{Path}[ G( S' ) ] \ar [r] \ar [d]^{ F_{S'} } & \operatorname{Path}[ G( T' )] \ar [d]^{ F_{T'} } \\ \operatorname{Path}[ S' ]_{m} \ar [r] & \operatorname{Path}[ T' ]_{m}. }$

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

$\xymatrix@R =50pt@C=50pt{ \operatorname{Path}[ G_0(S)] \ar [r] \ar [d] & \operatorname{Path}[ G_0(T) ] \ar [d] \\ \operatorname{Path}[ G( S ) ] \ar [r] \ar [d]^{ F_{S} } & \operatorname{Path}[ G(T )] \ar [d]^{ F_{T} } \\ \operatorname{Path}[ S ]_{m} \ar [r] \ar [d] & \operatorname{Path}[ T ]_{m} \ar [d] \\ \operatorname{Path}[ S' ]_{m} \ar [r] & \operatorname{Path}[ T' ]_{m}. }$

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

$S(0) \hookrightarrow S(1) \hookrightarrow S(2) \hookrightarrow \cdots$

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

• Let $S$ be a simplicial set which can be written as a coproduct $\coprod _{i \in I} \Delta ^ n$; we must show that $S$ is good. Without loss of generality, we may assume that $I$ is a singleton, so that $S = \Delta ^ n$. 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 2.4.4.16.

$\square$

Remark 2.4.4.17. 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

$(\sigma , I_0 \supseteq \cdots \supseteq I_{m} ) \mapsto (\sigma , I_{0} \supseteq \cdots \supseteq I_{i-1} \supseteq I_{i} \supseteq I_{i} \supseteq I_{i+1} \supseteq \cdots \supseteq I_ m ).$
• 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

$(\sigma , I_0 \supseteq \cdots \supseteq I_{m} ) \mapsto (\sigma , I_{0} \supseteq \cdots \supseteq I_{i-1} \supseteq I_{i+1} \supseteq \cdots \supseteq I_ m ).$
• Each of the face maps $d_{0}: \operatorname{Path}[G_ m] \rightarrow \operatorname{Path}[G_{m-1}]$ is induced by a morphism 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 $\overrightarrow {I} = (I_0 \supseteq \cdots \supseteq I_{m})$ of $[n]$. Then the subset $I_{1} \subseteq I_{0} = [n]$ 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 $I_{i+1} \hookrightarrow I_{1} \simeq [n'] \twoheadrightarrow [n'']$, and set $\overrightarrow {J} = (J_{0} \supseteq J_1 \supseteq \cdots \supseteq J_{m-1} )$. In the case $n'' = 0$, the morphism $f$ carries $(\sigma , \overrightarrow {I} )$ to the vertex $\tau \in \operatorname{Vert}(G_{m-1} )$. 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 $d_ m: \operatorname{Path}[G_ m] \rightarrow \operatorname{Path}[G_{m-1} ]$ 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 $I_{m-1} = \{ 0 = i_0 < i_1 < \cdots < i_ k = m \}$, then $d_0$ carries $(\sigma , \overrightarrow {I} )$ to a path of length $k$ in the category $\operatorname{Path}[ G_{m-1} ]$.

Let us record a consequence of Remark 2.4.4.12 which will be useful later.

Corollary 2.4.4.18. Let $Q$ be a partially ordered set, let $q \in Q$ be an element, and set $Q_{-} = \{ q_{-} \in Q: q_{-} \leq q \}$ and $Q_{+} = \{ q_+ \in Q: q \leq q_{+} \}$. Let $\operatorname{\mathcal{C}}$ be the smallest simplicial subcategory of $\operatorname{Path}[Q]_{\bullet }$ which contains $\operatorname{Path}[ Q_{-} ]_{\bullet }$ and $\operatorname{Path}[ Q_{+} ]_{\bullet }$. Then the diagram

$\xymatrix@R =50pt@C=50pt{ \{ q\} \ar [r] \ar [d] & \operatorname{Path}[ Q_{-} ]_{\bullet } \ar [d] \\ \operatorname{Path}[ Q_{+} ]_{\bullet } \ar [r] & \operatorname{\mathcal{C}}}$

is a pushout square of simplicial categories.

Proof. Using Proposition 2.4.4.15, we can identify the pushout $\operatorname{Path}[ Q_{-} ]_{\bullet } \coprod _{ \{ q\} } \operatorname{Path}[ Q_{+} ]_{\bullet }$ with the simplicial path category of the simplicial set $S = \operatorname{N}_{\bullet }(Q_{-} ) \coprod _{ \{ q\} } \operatorname{N}_{\bullet }( Q_{+} )$. The tautological map $S \rightarrow \operatorname{N}_{\bullet }(Q)$ is a monomorphism of simplicial sets, and therefore induces an equivalence from $\operatorname{Path}[S]_{\bullet }$ to a simplicial subcategory $\operatorname{\mathcal{C}}\subseteq \operatorname{Path}[Q]_{\bullet }$ (Remark 2.4.4.12). It is clear that this subcategory contains both $\operatorname{Path}[ Q_{-} ]_{\bullet }$ and $\operatorname{Path}[ Q_{+} ]_{\bullet }$. To complete the proof, it will suffice to show that if $\operatorname{\mathcal{D}}$ is any other simplicial subcategory of $\operatorname{Path}[Q]_{\bullet }$ which contains $\operatorname{Path}[Q_{-} ]_{\bullet }$ and $\operatorname{Path}[ Q_{+} ]_{\bullet }$, then $\operatorname{\mathcal{D}}$ contains $\operatorname{\mathcal{C}}$. This is clear: the universal property of $\operatorname{\mathcal{C}}$ guarantees that there is a unique simplicial functor $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ which is the identity on both $\operatorname{Path}[ Q_{-} ]_{\bullet }$ and $\operatorname{Path}[ Q_{+} ]_{\bullet }$. Invoking the universal property of $\operatorname{\mathcal{C}}$ again, we deduce that the composite functor $\operatorname{\mathcal{C}}\xrightarrow {F} \operatorname{\mathcal{D}}\hookrightarrow \operatorname{Path}[Q]_{\bullet }$ coincides with the inclusion map, so that $\operatorname{\mathcal{C}}\subseteq \operatorname{\mathcal{D}}$. $\square$

Remark 2.4.4.19. In the situation of Corollary 2.4.4.18, the simplicial subcategory $\operatorname{\mathcal{C}}\subseteq \operatorname{Path}[Q]_{\bullet }$ can be described more concretely:

• The objects of $\operatorname{\mathcal{C}}$ are elements of the subset $Q_{-} \cup Q_{+} \subseteq Q$.

• Let $a$ and $b$ be objects of $\operatorname{\mathcal{C}}$, and write $\operatorname{Hom}_{ \operatorname{Path}[Q]}(a,b)_{\bullet } = \operatorname{N}_{\bullet }(P_{a,b} )$, where $P_{a,b}$ is the collection finite linearly ordered $J \subseteq Q$ having smallest element $a$ and largest element $b$, ordered by reverse inclusion. Then $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(a,b)_{\bullet }$ can be identified with the nerve of the partially ordered subset $P'_{a,b} \subseteq P_{a,b}$ given by

$P'_{a,b} = \begin{cases} \{ J \in P_{a,b}: q \in J \} & \text{ if } a \leq q \leq b \\ P_{a,b} & \text{ otherwise. } \end{cases}$

Stated more informally, $\operatorname{\mathcal{C}}$ is a simplicial subcategory of $\operatorname{Path}[Q]_{\bullet }$ whose morphisms are paths which, when possible, contain the element $q$.

Corollary 2.4.4.20. Let $Q$ be a partially ordered set, let $q \in Q$ be an element, and suppose that $Q = Q_{-} \cup Q_{+}$ for $Q_{-} = \{ q_{-} \in Q: q_{-} \leq q \}$ and $Q_{+} = \{ q_+ \in Q: q \leq q_{+} \}$ (this condition is automatically satisfied, for example, if $Q$ is linearly ordered). Then the simplicial functor

$\operatorname{Path}[ Q_{-} ]_{\bullet } \coprod _{ \{ q\} } \operatorname{Path}[ Q_{+} ]_{\bullet } \rightarrow \operatorname{Path}[ Q ]_{\bullet }$

has a unique left inverse $R: \operatorname{Path}[ Q ]_{\bullet } \rightarrow \operatorname{Path}[ Q_{-} ]_{\bullet } \coprod _{ \{ q\} } \operatorname{Path}[ Q_{+} ]_{\bullet }$.

Proof. By virtue of Corollary 2.4.4.18, we can identify the pushout $\operatorname{Path}[ Q_{-} ]_{\bullet } \coprod _{ \{ q\} } \operatorname{Path}[ Q_{+} ]_{\bullet }$ with a simplicial subcategory $\operatorname{\mathcal{C}}\subseteq \operatorname{Path}[Q]_{\bullet }$; we wish to show that there is a unique simplicial functor $R: \operatorname{Path}[Q]_{\bullet } \rightarrow \operatorname{\mathcal{C}}$ satisfying $R|_{\operatorname{\mathcal{C}}} = \operatorname{id}_{\operatorname{\mathcal{C}}}$. Our assumption that $Q = Q_{-} \cup Q_{+}$ guarantees that $\operatorname{\mathcal{C}}$ contains every object of $\operatorname{Path}[Q]_{\bullet }$. To prove existence, we take the simplicial functor $R$ to be the identity on objects and given on morphisms by the maps

$\operatorname{Hom}_{ \operatorname{Path}[Q] }(a,b)_{\bullet } = \operatorname{N}_{\bullet }(P_{a,b} ) \rightarrow \operatorname{N}_{\bullet }( P'_{a,b} ) = \operatorname{Hom}_{\operatorname{\mathcal{C}}}(a,b)_{\bullet }$

$(J \in P_{a,b}) \mapsto \begin{cases} J \cup \{ q\} & \text{ if a \leq q \leq b} \\ J & \text{ otherwise, } \end{cases}$

where $P_{a,b}$ and the subset $P'_{a,b} \subseteq P_{a,b}$ are defined as in Remark 2.4.4.19.

To prove uniqueness, let $R': \operatorname{Path}[Q]_{\bullet } \rightarrow \operatorname{\mathcal{C}}$ be another simplicial functor satisfying $R'|_{\operatorname{\mathcal{C}}} = \operatorname{id}_{\operatorname{\mathcal{C}}}$; we wish to show that $R' = R$. It is clear that $R$ and $R'$ agree at the level of objects. For every pair of elements $a,b \in Q$, the simplicial functors $R$ and $R'$ induce maps $\theta , \theta ': \operatorname{Hom}_{ \operatorname{Path}[Q] }(a,b)_{\bullet } \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{C}}}(a,b)_{\bullet }$; we wish to show that $\theta = \theta '$. Since $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(a,b)_{\bullet }$ can be identified with the nerve of the partially ordered set $P'_{a,b}$, it will suffice to show that $\theta$ and $\theta '$ agree on vertices. For every finite linearly ordered subset $J \subseteq Q$ having least element $a$ and greatest element $b$, let $f_{J}: a \rightarrow b$ denote the corresponding morphism in the path category $\operatorname{Path}[Q]$; we wish to show that $\theta ( f_{J} ) = \theta '( f_{J} )$. Without loss of generality, we may assume that the morphism $f_{J}$ is indecomposable: that is, that we have $a \neq b$ and that $J = \{ a < b \}$. We may further assume that $a < q < b$ (otherwise, $f_{J}$ is a morphism in the category $\operatorname{\mathcal{C}}$ and we have $\theta (f_ J) = f_{J} = \theta '(f_ J)$). Set $J^{+} = \{ a < q < b \}$, so that $\theta ( f_ J ) = f_{J^{+} }$. Write $\theta '( f_ J ) = f_{K}$ where $K \subseteq Q$ is a finite linearly ordered subset having least element $a$ and greatest element $b$. Since $f_{J^{+}}$ is a morphism of $\operatorname{\mathcal{C}}$, we have $\theta '( f_{J^{+}} ) = f_{J^{+} }$. The inclusion $J \subseteq J^{+}$ then implies that $K \subseteq J^{+}$. On the other hand, $f_{K}$ is also a morphism of $\operatorname{\mathcal{C}}$, so we must have $q \in K$. It follows that $K = J^{+}$, so that $\theta ( f_ J) = f_{J^{+}} = f_{K} = \theta '( f_ J )$ as desired. $\square$