The category of graphs

We define the category of simple undirected graphs without loops as a reflective subcategory of a presheaf category.

Let $\mathsf {G}$ be the category generated by the diagram

subject to the identities

\begin{align*} \begin{array}{ll} rs = rt = \operatorname{id}_{V}, & \sigma^2 = \operatorname{id}_{E}, \\ \sigma s = t, & \sigma t = s, \\ r \sigma = r. \end{array} \end{align*}

We write $\widehat {\mathsf {G}}$ for the functor category $\mathsf {Set}^{\mathsf {G}^{\mathrm {op}}}$.

For $G \in \widehat {\mathsf {G}}$, we write $G_V$ and $G_E$ for the sets $G(V)$ and $G(E)$, respectively. Explicitly, such a functor consists of sets $G_V$ and $G_E$ together with the following functions between them:

subject to the dual versions of identities in $\mathsf {G}$.

Let $\mathsf {Graph}$ denote the full subcategory of $\widehat {\mathsf {G}}$ spanned by graphs.

Figure 1. An example depiction of a graph.

In more concrete terms, a graph $G$ consists of a set $G_V$ of vertices and a set $G_E$ of ‘half-edges’. A half-edge $e \in G_E$ has source and target, and these are given by the maps $Gs$ and $Gt$, respectively. Each half-edge is paired with its other half via the map $G\sigma \colon G_E \to G_E$. Note that the edges paired by $G \sigma$ have swapped source and target, making the pair (i.e. whole edge) undirected. The map $Gr \colon G_V \to G_E$ takes a vertex to an edge whose source and target is that vertex (i.e. a loop). Finally, the condition that $(Gs, Gt)$ is a monomorphism ensures that there is at most one (whole) edge between any two vertices. This is equivalent to specifying a binary ‘incidence’ relation on $G_V$ which is reflexive and symmetric.

A map of graphs $f \colon G \to H$ is a natural transformation between such functors. However, since $(Hs, Ht)$ is a monomorphism, such a map is completely determined by a function $f_V \colon G_V \to H_V$ that preserves the incidence relation.

We may therefore assume that our graphs have no loops, but the maps between them, rather than merely preserving edges, are allowed to contract them to a single vertex. That is, a graph map $f \colon G \to H$ is a function $f \colon G_V \to H_V$ such that if $v, w \in G_V$ are connected by an edge, then either $f(v)$ and $f(w)$ are connected by an edge or $f(v) = f(w)$.

Proof. (i) The left adjoint is $\mathsf {Im} \colon \widehat {\mathsf {G}} \to \mathsf {Graph}$ given by

\begin{align*} (\mathsf{Im}\,G)_V = G_V, \quad (\mathsf{Im}\, G)_E = (Gs, Gt) (G_E), \end{align*}

where $(Gs, Gt) (G_E)$ is the image of $G_E$ under the map $(Gs, Gt) \colon G_E \to G_V \times G_V$.

(ii) The category $\widehat {\mathsf {G}}$ is (co)complete as a presheaf category. The conclusion follows from (i) as $\mathsf {Graph}$ is a reflective subcategory of a (co)complete category.

(iii) The left adjoint $\mathsf {Set} \to \mathsf {Graph}$ takes a set $A$ to the discrete graph with vertex set $A$. The right adjoint takes a set $A$ to the complete graph with vertex set $A$.

Examples of graphs

For $m \geq 0$, we have a map $l \colon I_{m+1} \to I_{m}$ defined by

\begin{align*} l(v) = \begin{cases} 0 & \text{if } v = 0 \\ v-1 & \text{otherwise.} \end{cases} \end{align*}

As well, we have a map $r \colon I_{m+1} \to I_m$ defined by

\begin{align*} r(v) = \begin{cases} m & \text{if } v = m + 1 \\ v & \text{otherwise.} \end{cases} \end{align*}

We write $c \colon I_{m+2} \to I_m$ for the composite $lr = rl$. Explicitly, this map is defined by

\begin{align*} c(v) = \begin{cases} 0 & \text{if } v = 0 \\ m & \text{if } v = m + 2 \\ v - 1 & \text{otherwise.} \end{cases} \end{align*}

We show the inclusion $\mathsf {Graph} \hookrightarrow \widehat {\mathsf {G}}$ preserves filtered colimits and use this to show all finite graphs are compact, i.e. if $G$ is a finite graph then the functor $\mathsf {Graph}(G, \mathord {- }) \colon \mathsf {Graph} \to \mathsf {Set}$ preserves filtered colimits.

Proof. Fix a filtered category $J$ and a diagram $D \colon J \to \mathsf {Graph}$. Let $i$ denote the inclusion $\mathsf {Graph} \hookrightarrow \widehat {\mathsf {G}}$. Recall that $\operatorname {colim} D$ is computed by $\mathsf {Im} (\operatorname {colim} (iD))$. It suffices to show $\operatorname {colim} (iD) \in \widehat {\mathsf {G}}$ is a graph, since then the unit map $\operatorname {colim} (iD) \to i \mathsf {Im} (\operatorname {colim} (iD))$ is an isomorphism $\operatorname {colim} (iD) \cong i (\operatorname {colim} D)$ natural in $D$.

Figure 2. The graphs $I_3$ and $C_3$, respectively.

Let $\lambda \colon iD \to \operatorname {colim}(iD)$ denote the colimit cone. Suppose two edges $e, e' \in \operatorname {colim}(iD)_E$ have the same source and target. Regarding $e, e'$ as maps $I_1 \to \operatorname {colim}(iD)$, these maps factor as

for some $x \in J$ since $I_1 \in \widehat {\mathsf {G}}$ is representable. Let $s, s' \in (iDx)_V$ denote the sources of $\overline {e}, \overline {e}'$ and let $t, t' \in (iDx)_V$ denote the targets, respectively. Using an explicit description of the colimit (Remark 1.3), since $\lambda _x(s) = \lambda _x(s')$ and $\lambda _x(t) = \lambda _x(t')$, there exist arrows $f, g \colon y \to x$ and $h, k \colon z \to x$ with vertices $v \in iDy$ and $w \in iDz$ such that

\begin{align*} \begin{array}{cccc} iDf(v) = s, & iDg(v) = s', & iDh(w) = t, & iDk(w) = t'. \end{array} \end{align*}

As $J$ is filtered, there exists an arrow $l \colon x \to w$ in $J$ such that $lf = lg$ and $lh = lk$. This implies the edges $iDl(\overline {e}), iDl(\overline {e}') \in (iDw)_E$ have the same source and target. As $iDw$ is a graph, it follows that $\overline {e} = \overline {e}'$, thus $e = e'$.

Proof. Given a filtered category $J$ and a diagram $D \colon J \to \mathsf {Graph}$, we have a natural isomorphism

\begin{align*} \mathsf{Graph}(G, \operatorname{colim} D) \cong \widehat{\mathsf{G}}(G, \operatorname{colim} D) \end{align*}

by Proposition 1.6 since $\mathsf {Graph}$ is a full subcategory. This then follows since $G \in \widehat {\mathsf {G}}$ is a finite colimit of representable presheaves.

Monoidal structure on the category of graphs

Define a functor $\otimes \colon \mathsf {G} \times \mathsf {G} \to \widehat {\mathsf {G}}$ by

\begin{align*} \begin{array}{ll} (V, V) \mapsto I_0, & (V, E) \mapsto I_1, \\ (E, V) \mapsto I_1, & (E, E) \mapsto C_4. \end{array} \end{align*}

Left Kan extension along the Yoneda embedding yields a monoidal product $\otimes \colon \widehat {\mathsf {G}} \times \widehat {\mathsf {G}} \to \widehat {\mathsf {G}}$:

Note that $\mathsf {Graph}$ is closed with respect to this product, i.e. if $G, H \in \mathsf {Graph}$ then $G \otimes H \in \widehat {\mathsf {G}}$ is a graph. Thus, this product descends to a monoidal product $\otimes \colon \mathsf {Graph} \times \mathsf {Graph} \to \mathsf {Graph}$, called the cartesian product.

Explicitly, the graph $G \otimes H$ has:

• – as vertices, pairs $(v, w)$ where $v \in G$ is a vertex of $G$ and $w \in H$ is a vertex of $H$;

• – an edge from $(v, w)$ to $(v', w')$ if either $v = v'$ and $w$ is connected to $w'$ in $H$ or $w = w'$ and $v$ is connected to $v'$ in $G$.

This structure makes the category of graphs into a closed symmetric monoidal category.

Homotopy theory of graphs

We now review the basics of discrete homotopy theory. Our treatment is brief and more categorically oriented, but the reader can find full details in any of the references [Reference MalleMal83, Reference Barcelo, Kramer, Laubenbacher and WeaverBKLW01, Reference Barcelo and LaubenbacherBL05, Reference Babson, Barcelo, de Longueville and LaubenbacherBBdLL06].

Note that one needs to allow the parameter $m$ appearing above to vary, as otherwise the notion of homotopy is not transitive. This is reminiscent of the notion of Moore path in topological spaces, which is a path parameterized by an interval of arbitrary length [Reference MayMay75, Reference van den Berg and GarnervdBG12, Reference Barthel and RiehlBR13].

We also define the A-homotopy groups of a graph. As in topological spaces, this requires the definition of based graph maps and based homotopies between them.

Explicitly, this data consists of maps $G \to H$ and $A \to B$ such that the following square commutes:

That is, $A$ is a subgraph of $G$ and the map $A \to B$ is the restriction of the map $G \to H$ to $A$, whose image is contained in the subgraph $B$.

In the context of relative graph maps, we denote a monomorphism $A \hookrightarrow G$ by $(G, A)$, suppressing the data of the map itself. An exception to this is for monomorphisms of the form $I_0 \to G$. This is exactly the data of a pointed graph, which we denote by $(G, v)$, where $v$ is the unique vertex in the image of the map $I_0 \to G$.

For a relative graph map $(f, g) \colon (G, A) \to (H, B)$, the map $g$ is uniquely determined by $f$. If a map $h \colon A \to B$ also forms a commutative square with $f$ then $g = h$ as the map $B \hookrightarrow Y$ is monic. Thus, we denote a relative graph map by $f \colon (G, A) \to (H, B)$. We additionally write $f \colon G \to H$ for the bottom map in the square and ${f}|_{A} \colon A \to B$ for the top map.

We define relative homotopies between relative graph maps as follows.

Explicitly, a relative homotopy from $f$ to $g$ consists of

• – a homotopy $\eta \colon G \otimes I_m \to H$ from $f$ to $g$,

• – a homotopy ${\eta }|_{A} \colon A \otimes I_m \to B$ from ${f}|_{A}$ to ${g}|_{A}$,

such that the following square commutes:

Remark 1.14 For graph maps $f, g \colon G \to H$, a path of length $m$ in $\operatorname {hom}^{\otimes }(G, H)$ from the vertex $f$ to the vertex $g$ is exactly a homotopy $G \otimes I_m \to H$ from $f$ to $g$. For relative graph maps $f, g \colon (G, A) \to (H, B)$, a path of length $m$ from $f$ to $g$ in the pullback graph

is exactly a relative homotopy from $f$ to $g$.

It follows that relative homotopy is an equivalence relation on relative graph maps.

Given a relative graph map $f \colon (G, A) \to (H, B)$, if the subgraph $B$ consists of a single vertex $v$ then we refer to $f$ as a graph map based at $v$ or a based graph map. We refer to a homotopy between two such maps as a based homotopy.

Definition 1.16 Let $G$ be a graph and $n \geq 1$. For $i = 0, \ldots, n$ and $\varepsilon = 0, 1$, a map $f \colon I_\infty ^{\otimes {n}} \to G$ is stable in direction $(i,\varepsilon )$ if there exists $M \geq 0$ so that for $v_i > M$ we have

\begin{align*} f(v_1, \ldots, v_{i-1}, (2\varepsilon - 1)v_i, v_{i+1} \ldots, v_n) = f(v_1, \ldots, v_{i-1}, (2\varepsilon - 1)M, v_{i+1}, \ldots, v_n). \end{align*}

In words, a map $f \colon I_\infty ^{\otimes {n}} \to G$ is stable in direction $(i, 0)$ if it becomes constant (with respect to change in the $i$th coordinate) once the $i$th coordinate is sufficiently large in the negative direction. It is stable in direction $(i, 1)$ if it becomes constant (again, with respect to change in the $i$th coordinate) once the $i$th coordinate is sufficiently large in the positive direction.

For $n, M \geq 0$, let $I_{\geq M}^{\otimes {n}}$ denote the subgraph of $I_\infty ^{\otimes {n}}$ consisting of vertices $(v_1, \ldots, v_n)$ such that $|v_i| \geq M$ for some $i = 1, \ldots, n$. Given a based graph map $f \colon (I_\infty ^{\otimes {n}}, I_{\geq M}^{\otimes {n}}) \to (G,v)$ we may also regard $f$ as a based graph map $(I_\infty ^{\otimes {n}}, I_{\geq K}^{\otimes {n}}) \to (G,v)$ for any $K \geq M$. This gives a notion of based homotopy between maps $(I_\infty ^{\otimes {n}}, I_{\geq M}^{\otimes {n}}) \to (G, v)$ which, for some $M \geq 0$, are based at $v$.

Proposition 1.17 [Reference MalleMal83, Proposition 3.2]

Based homotopy is an equivalence relation on the set of based maps

\begin{align*} \{ (I_\infty^{\otimes{n}}, I_{\geq M}^{\otimes{n}}) \to (G, v) \mid M \geq 0 \}. \end{align*}

Let $n \geq 1$ and $i = 1, \ldots, n$. Given $f \colon (I_\infty ^{\otimes {n}}, I_{\geq M}^{\otimes {n}}) \to (G, v)$ and $g \colon (I_\infty ^{\otimes {n}}, I_{\geq M'}^{\otimes {n}}) \to (G, v)$, we define a binary operation $f \cdot _i g \colon (I_\infty ^{\otimes {n}}, I_{\geq M + M'}^{\otimes {n}}) \to (G, v)$ by

\begin{align*} (f \cdot_i g)(v_1, \ldots, v_n) := \begin{cases} f(v_1, \ldots, v_n) & v_i \leq M \\ g(v_1, \ldots, v_i - M - M', \ldots, v_n) & v_i > M. \end{cases} \end{align*}

This induces a group operation on homotopy groups $\cdot _i \colon A_n(G, v) \times A_n(G, v) \to A_n(G, v)$. For $n \geq 2$ and $1 \leq i < j \leq n$, it is straightforward to construct a homotopy witnessing that

\begin{align*} [[f] \cdot_i [g]] \cdot_j [[f] \cdot_i [g]] = [[f] \cdot_j [g]] \cdot_i [[f] \cdot_j [g]] \end{align*}

for any $[f], [g] \in A_n(G,v)$. The Eckmann–Hilton argument gives $[f] \cdot _i [g] = [f] \cdot _j [g]$ and that this operation is abelian.

Path and loop graphs

As vertices of $PG$ are paths that stabilize, we have graph maps $\partial ^{}_{1,0}, \partial ^{}_{1,1} \colon PG \to G$ which send a vertex $v \colon I_\infty \to G$ to its left and right endpoints, respectively.

Proposition 1.20 For a graph $G$, we have an isomorphism

\begin{align*} PG \cong \operatorname{colim} \big( \operatorname{hom}^{\otimes}(I_1, G) \xrightarrow{l^*} \operatorname{hom}^{\otimes}(I_2, G) \xrightarrow{r^*} \operatorname{hom}^{\otimes}(I_3, G) \xrightarrow{l^*} \cdots \big) \end{align*}

natural in $G$.

Proof. For each $m \geq 1$, we write $m = 2k + b$, where $k \geq 0$ and $b \in \{0, 1\}$, and define a graph map $I_\infty \to I_m$ by

\begin{align*} v \mapsto \begin{cases} 0 & v \leq -k \\ k + v & -k \leq v \leq k+b \\ m & v \geq k+b. \end{cases} \end{align*}

Geometrically, this function maps the subinterval $[-k, k+b]$ surjectively onto $I_m$, and collapses all other vertices to the endpoints. By pre-composition, this induces a cone

which one verifies is a colimit cone.

From the definition, the following proposition is immediate.

Proposition 1.22 For a pointed graph $(G, v)$, the square

is a pullback.

The loop graph of a pointed graph $(G, v)$ has a distinguished vertex which is the constant path at $v$. This gives an endofunctor $\Omega \colon \mathsf {Graph}_* \to \mathsf {Graph}_*$. From this, we define the notion of $n$th loop graphs.

Definition 1.23 For $n \geq 0$, we define the $n$th loop graph to be

\begin{align*} \Omega^{n}(X,x) := \begin{cases} (X,x) & n = 0 \\ \Omega(\Omega^{n-1}(X,x)) & \text{otherwise.} \end{cases} \end{align*}

In [Reference Babson, Barcelo, de Longueville and LaubenbacherBBdLL06], it is shown that the $n$th homotopy groups of a graph correspond to the connected components of $\Omega ^{n}(G,v)$.

Proposition 1.24 [Reference Babson, Barcelo, de Longueville and LaubenbacherBBdLL06, Proposition 7.4]

For $n \geq 0$, we have an isomorphism

\begin{align*} A_n(G, v) \cong A_0(\Omega^{n}(G, v)). \end{align*}

Cubical sets

We begin by defining the box category $\Box$. As used in this paper, the box category will include both positive and negative connections. This variant of the box category was introduced in [Reference Brown and HigginsBH81] and was later used in [Reference Doherty, Kapulkin, Lindsey and SattlerDKLS24] to model $(\infty, 1)$-categories. The objects of $\Box$ are posets of the form $[1]^n = \{ 0 \leq 1 \}^n$ and the maps are generated (inside the category of posets) under composition by the following four special classes:

• – faces $\partial ^n_{i,\varepsilon } \colon [1]^{n-1} \to [1]^n$ for $i = 1, \ldots, n$ and $\varepsilon = 0, 1$ given by

  \begin{align*} \partial^n_{i,\varepsilon} (x_1, x_2, \ldots, x_{n-1}) = (x_1, x_2, \ldots, x_{i-1}, \varepsilon, x_i, \ldots, x_{n-1}); \end{align*}

• – degeneracies $\sigma ^n_i \colon [1]^n \to [1]^{n-1}$ for $i = 1, 2, \ldots, n$ given by

  \begin{align*} \sigma^n_i ( x_1, x_2, \ldots, x_n) = (x_1, x_2, \ldots, x_{i-1}, x_{i+1}, \ldots, x_n); \end{align*}

• – negative connections $\gamma ^n_{i,0} \colon [1]^n \to [1]^{n-1}$ for $i = 1, 2, \ldots, n-1$ given by

  \begin{align*} \gamma^n_{i,0} (x_1, x_2, \ldots, x_n) = (x_1, x_2, \ldots, x_{i-1}, \max\{ x_i , x_{i+1}\}, x_{i+2}, \ldots, x_n) . \end{align*}

• – positive connections $\gamma ^n_{i,1} \colon [1]^n \to [1]^{n-1}$ for $i = 1, 2, \ldots, n-1$ given by

  \begin{align*} \gamma^n_{i,1} (x_1, x_2, \ldots, x_n) = (x_1, x_2, \ldots, x_{i-1}, \min\{ x_i , x_{i+1}\}, x_{i+2}, \ldots, x_n) . \end{align*}

These maps obey the following cubical identities:

\begin{gather*} \begin{array}{ll} \partial_{j, \varepsilon'} \partial_{i, \varepsilon} = \partial_{i+1, \varepsilon} \partial_{j, \varepsilon'} \quad \text{for } j \leq i; & \sigma_j \partial_{i, \varepsilon} = \begin{cases} \partial_{i-1, \varepsilon} \sigma_j & \text{for } j < i \\ \operatorname{id} & \text{for } j = i \\ \partial_{i, \varepsilon} \sigma_{j-1} & \text{for } j > i; \end{cases} \end{array}\\ \begin{array}{ll} \sigma_i \sigma_j = \sigma_j \sigma_{i+1} \quad \text{for } j \leq i; & \gamma_{j,\varepsilon'} \gamma_{i,\varepsilon} = \begin{cases} \gamma_{i,\varepsilon} \gamma_{j+1,\varepsilon'} & \text{for } j > i \\ \gamma_{i,\varepsilon}\gamma_{i+1,\varepsilon} & \text{for } j = i, \varepsilon' = \varepsilon; \end{cases} \end{array}\\ \begin{array}{ll} \gamma_{j,\varepsilon'} \partial_{i, \varepsilon} = \begin{cases} \partial_{i-1, \varepsilon} \gamma_{j,\varepsilon'} & \text{for } j < i-1 \\ \operatorname{id} & \text{for } j = i-1, \, i, \, \varepsilon = \varepsilon' \\ \partial_{j, \varepsilon} \sigma_j & \text{for } j = i-1, \, i, \, \varepsilon = 1-\varepsilon' \\ \partial_{i, \varepsilon} \gamma_{j-1,\varepsilon'} & \text{for } j > i; \end{cases} & \sigma_j \gamma_{i,\varepsilon} = \begin{cases} \gamma_{i-1,\varepsilon} \sigma_j & \text{for } j < i \\ \sigma_i \sigma_i & \text{for } j = i \\ \gamma_{i,\varepsilon} \sigma_{j+1} & \text{for } j > i . \end{cases} \end{array} \end{gather*}

This category enjoys many good properties, making it suitable for modeling homotopy theory. Formally speaking, these properties can be encapsulated in saying that $\mathord {\square }$ is an Eilenberg–Zilber category, which in particular implies that the object $[1]^n$ has no non-identity automorphisms. However, we will not explicitly rely on the notion of Eilenberg–Zilber categories.

A cubical set is a presheaf $X \colon \mathord {\square }^{\mathrm {op}} \to \mathsf {Set}$. A cubical map is a natural transformation of such presheaves. We write $\mathsf {cSet}$ for the category of cubical sets and cubical maps.

Given a cubical set $X$, we write $X_n$ for the value of $X$ at $[1]^n$ and refer to the elements of $X_n$ as $n$-cubes of $X$. We write cubical operators on the right, e.g. given an $n$-cube $x \in X_n$ of $X$, we write $x\partial ^{}_{1,0}$ for the $\partial ^{}_{1,0}$-face of $x$.

By a degenerate cube, we always mean a cube that is in the image of a degeneracy or a connection map. This nomenclature is borrowed from the theory of Reedy categories, where one can speak abstractly of degenerate elements in a presheaf ([Reference Riehl and VerityRV14], [Reference HoveyHov99, § 5.2], [Reference HirschhornHir03, Ch. 15]).

Observe $\mathord {\square ^{0}} \in \mathsf {cSet}$ is the terminal object in $\mathsf {cSet}$.

Example 2.2 Define a functor $\mathord {\square } \to \mathsf {Top}$ from the box category to the category of topological spaces which sends $[1]^n$ to $[0, 1]^n$, where $[0, 1]$ is the unit interval. The face and degeneracy maps are sent to the face inclusion and product projection maps, respectively. The negative connection $\gamma ^{}_{i,0}$ is sent to the map $[0, 1]^n \to [0, 1]^{n-1}$ defined by

\begin{align*} (x_1, \ldots, x_n) \mapsto (x_1, \ldots, x_{i-1}, \max(x_i, x_{i+1}), x_{i+2}, \ldots, x_n), \end{align*}

and the image of the positive connection $\gamma ^{}_{i,1}$ is defined analogously.

Left Kan extension along the Yoneda embedding gives the geometric realization functor $\lvert \mathord {- } \rvert \colon \mathsf {cSet} \to \mathsf {Top}$:

This functor is left adjoint to the singular cubical complex functor $\operatorname {Sing} \colon \mathsf {Top} \to \mathsf {cSet}$ defined by

\begin{align*} (\operatorname{Sing}{S})_n := \mathsf{Top} ([0, 1]^n, S). \end{align*}

Define a functor $\otimes \colon \mathord {\square } \times \mathord {\square } \to \mathord {\square }$ on the cube category which sends $([1]^m, [1]^n)$ to $[1]^{m+n}$. Post-composing with the Yoneda embedding and left Kan extending gives a monoidal product on cubical sets:

This is the geometric product of cubical sets. Although, for $[1]^m, [1]^n \in \mathord {\square }$, there is an isomorphism $[1]^m \otimes [1]^n \cong [1]^n \otimes [1]^m$, this isomorphism is not natural. As a result, the geometric product of cubical sets is not symmetric, i.e. $X \otimes Y$ is not in general isomorphic to $Y \otimes X$.

This product is however biclosed. For a cubical set $X$, we write $\operatorname {hom}_{L}(X, \mathord {- }) \colon \mathsf {cSet} \to \mathsf {cSet}$ and $\operatorname {hom}_{R} (X, \mathord {- }) \colon \mathsf {cSet} \to \mathsf {cSet}$ for the right adjoints to the functors $\mathord {- } \otimes X$ and $X \otimes \mathord {- }$, respectively. As the geometric product is not symmetric, the functors $\operatorname {hom}_{L}(X, -)$ and $\operatorname {hom}_{R}(X, -)$ are not naturally isomorphic.

Kan complexes

We write $\mathsf {Kan}$ for the full subcategory of $\mathsf {cSet}$ consisting of Kan complexes.

Example 2.4 For any $S \in \mathsf {Top}$, the cubical set $\operatorname {Sing}{S}$ is a Kan complex. A map $\mathord {\sqcap ^{n}_{i,\varepsilon }} \to \operatorname {Sing}{S}$ is, by adjointness, a map $\lvert \mathord {\sqcap ^{n}_{i,\varepsilon }} \rvert \to S$. The inclusion $\lvert \mathord {\sqcap ^{n}_{i,\varepsilon }} \rvert \hookrightarrow \lvert \mathord {\square ^{n}}\rvert$ has a retract in $\mathsf {Top}$. Pre-composing with this retract gives a map $\lvert \mathord {\square ^{n}}\rvert \to S$ which restricts to the open box map $\lvert \mathord {\sqcap ^{n}_{i,\varepsilon }} \rvert \to S$:

By adjointness, this gives a suitable map $\mathord {\square ^{n}} \to \operatorname {Sing}{S}$.

We move towards describing the fibration category of Kan complexes.

Given a fibration category $\mathscr {C}$ with finite coproducts and a terminal object, the category of pointed objects $1 \downarrow \mathscr {C}$ is a fibration category as well.

Anodyne maps

We move towards defining anodyne maps of cubical sets, i.e. maps which are both monomorphisms and weak equivalences.

For a saturated class of maps, closure under pushouts and transfinite composition gives the following proposition.

Proposition 2.13 [Reference HoveyHov99, Lemma 2.1.13]

Saturated classes of maps are closed under coproduct. That is, given a collection $\{s_i \colon A_i \to B_i \mid i \in I \}$ of morphisms in a saturated class ${S}$, the coproduct

\begin{align*} \coprod_{i \in I} s_i \colon \coprod_{i \in I} A_i \to \coprod_{i \in I} B_i \end{align*}

is in ${S}$.

Definition 2.14 A map of cubical sets is anodyne if it is in the saturation of open box inclusions

\begin{align*} \operatorname{Sat} \{ \mathord{\sqcap^{n}_{i,\varepsilon}} \hookrightarrow \mathord{\square^{n}} \mid n \geq 1, i = 0, \ldots, n, \varepsilon = 0, 1 \}. \end{align*}

We use the following property of anodyne maps, which follows since saturations are closed under the left lifting property.

Theorem 2.15 [Reference Doherty, Kapulkin, Lindsey and SattlerDKLS24, Theorem 1.34]

Let $g \colon A \to B$ be an anodyne map and $f \colon X \to Y$ be a Kan fibration. Given a commutative square,

there exists a map $B \to X$ so that the triangles

commute.

In particular, we use that anodyne maps are sent to weak homotopy equivalences under geometric realization.

Homotopies and homotopy groups

Using the geometric product, we may define a notion of homotopy between cubical maps.

Definition 2.18 Given cubical maps $f, g \colon X \to Y$, a homotopy from $f$ to $g$ is a map $H \colon X \otimes \mathord {\square ^{1}} \to G$ such that the diagram

commutes.

Let ${\mathsf {cSet}}_{2}$ denote the full subcategory of ${\mathsf {cSet}}^{{[1]}}$ spanned by monomorphisms. Explicitly, its objects are monic cubical maps $A \hookrightarrow X$. A morphism from $A \hookrightarrow X$ to $B \hookrightarrow Y$ is a pair of maps $(f, g)$ which form a commutative square of the following form:

We refer to the objects and morphisms of ${\mathsf {cSet}}_{2}$ as relative cubical sets and relative cubical maps, respectively. Following our convention for relative graph maps, we denote a relative cubical set $A \hookrightarrow X$ by $(X, A)$, suppressing the data of the map itself. If the domain of the map $A \hookrightarrow X$ is the 0-cube $A = \mathord {\square ^{0}}$, we denote it by $(X, x)$, where $x$ is the unique 0-cube in the image of the map $\mathord {\square ^{0}} \hookrightarrow X$.

For a relative cubical map $(f, g) \colon (X, A) \to (Y, B)$, the map $g$ is uniquely determined by $f$ since $B \hookrightarrow Y$ is monic. As a result, we denote a relative cubical map by $f \colon (X, A) \to (Y, B)$. Mirroring our convention for relative graph maps, we write $f$ for the map $X \to Y$ and ${f}|_{A}$ for the map $A \to B$.

We have a corresponding notion of homotopy between relative cubical maps.

It is essential in Proposition 2.20 that $B$ and $Y$ are Kan complexes. If not, the relation of relative homotopy is neither symmetric nor transitive (and in this case, one considers the symmetric transitive closure of relative homotopy).

We write $[(X, A), (Y, B)]$ for the set of relative homotopy classes of relative maps $(X, A) \to (Y, B)$. With this, we define the homotopy groups of a Kan complex.

We give an explicit description of multiplication in the first homotopy group $\pi _1(X,x)$.

Proposition 2.23 [Reference Carranza and KapulkinCK23, Theorem 3.11]

If $(X,x)$ is a pointed Kan complex then composition induces a well-defined binary operation

\begin{align*} [(\mathord{\square^{1}}, \partial \mathord{\square^{1}}), (X,x)] \times [(\mathord{\square^{1}}, \partial \mathord{\square^{1}}), (X,x)] \to [(\mathord{\square^{1}}, \partial \mathord{\square^{1}}), (X,x)] \end{align*}

on relative homotopy classes of relative maps which gives a group structure on $[(\mathord {\square ^{1}}, \partial \mathord {\square ^{1}}), (X,x)]$.

As with spaces, the homotopy groups of a Kan complex are the connected components of its loop space, which we define.

Proposition 2.25 [Reference Carranza and KapulkinCK23, Corollary 3.16]

For a pointed Kan complex $(X,x)$ and $0 \leq k \leq n$, we have an isomorphism

\begin{align*} \pi_{n}(X,x) \cong \pi_{n-k} (\Omega^{k}(X,x)) \end{align*}

natural in $X$.

Using Proposition 2.25, the group structure on higher homotopy groups is induced by the bijection $\pi _n(X,x) \cong \pi _1(\Omega ^{n-1}(X,x))$.

This definition of homotopy groups agrees with the homotopy groups of its geometric realization.

Theorem 2.26 [Reference Carranza and KapulkinCK23, Theorem 3.25]

There is an isomorphism

\begin{align*} \pi_n(X, x) \cong \pi_n(\lvert X \rvert , x) \end{align*}

natural in $X$.

We know that the loop space functor is exact.

Statement

Our main theorem is the following result.

Before proving Theorem 4.1, we explain the proof strategy and establish some auxiliary lemmas.

To show that the nerve of any graph is a Kan complex, we construct a map $\Phi \colon I_{3m}^{\otimes {n}} \to \lvert \mathord {\sqcap ^{n}_{i,\varepsilon }} \rvert _{m}$ so that the triangle

commutes. We show that every map $\mathord {\sqcap ^{n}_{i,\varepsilon }} \to \mathrm {N} G$ must factor through some $m$-nerve $\mathord {\sqcap ^{n}_{i,\varepsilon }} \to \mathrm {N}_{m} G \to \mathrm {N} G$. The map $\Phi$ gives a filler for the composite $\mathord {\sqcap ^{n}_{i,\varepsilon }} \to \mathrm {N}_{m} G \to \mathrm {N}_{3m} G$, thus a filler in $\mathrm {N} G$.

To show that the map $\mathrm {N}_{1} G \hookrightarrow \mathrm {N} G$ is anodyne, we show the maps $l^*, r^* \colon \mathrm {N}_{m} G \hookrightarrow \mathrm {N}_{m+1} G$ are anodyne. This is done by an explicit construction establishing $l^*$ and $r^*$ as a transfinite composition of pushouts along coproducts of open box inclusions. This implies $\mathrm {N}_{1} G \hookrightarrow \mathrm {N} G$ is anodyne as the nerve of $G$ is a transfinite composition of $l^*$ and $r^*$.

Proof of part (i)

Construction 4.2 Fix $m, n \geq 1$, $i \in \{ 1, \ldots, n \}$, and $\varepsilon \in \{ 0, 1 \}$. We construct the map $\Phi \colon I_{3m}^{\otimes {n}} \to \lvert \mathord {\sqcap ^{n}_{i,\varepsilon }} \rvert _{m}$. For $n = 1$, we compute $\lvert \mathord {\sqcap ^{1}_{i,\varepsilon }} \rvert _{m} \cong I_0$. In this case, the map $\Phi$ is immediate. Thus, we assume $n \geq 2$.

Define a map $\varphi \colon I_{3m} \to I_{3m}$ by

\begin{align*} \varphi v := \begin{cases} m & v \leq m \\ i & m \leq v \leq 2m \\ 2m & v \geq 2m. \end{cases} \end{align*}

It is straightforward to verify this definition gives a graph map. From this, we have a map $\varphi ^{\otimes {n-1}} \colon I_{3m}^{\otimes {n-1}} \to I_{3m}^{\otimes {n-1}}$ which sends $(v_1, \ldots, v_{n-1})$ to $(\varphi v_1, \ldots, \varphi v_{n-1})$.

For $v = (v_1, \ldots, v_{n-1}) \in I_{3m}^{\otimes {n-1}}$, let $d(v)$ denote the node distance between $v$ and $\varphi ^{\otimes {n-1}}v$. That is,

\begin{align*} d(v) := \sum_{i=1}^{n-1} |v_i - \varphi v_i|. \end{align*}

Observe this gives a graph map $d \colon I_{3m}^{\otimes {n-1}} \to I_\infty$ (Figure 7).

Figure 7. Vertices of $I_6^{\otimes {2}}$ labeled by their image under $d \colon I_6^{\otimes {2}} \to I_\infty$.

For $t \geq 0$, we have a graph map $\beta [t] \colon I_\infty \to I_{t}$ defined by

\begin{align*} \beta[t](v) := \begin{cases} 0 & v \leq 0 \\ v & 0 \leq v \leq t \\ t & v \geq t. \end{cases} \end{align*}

One thinks of $\beta [t]$ as bounding the graph $I_\infty$ between 0 and $t$. Recall that the map $c^m \colon I_{3m} \to I_{m}$ is given by

\begin{align*} c^m v := \begin{cases} 0 & v \leq m \\ v - m & m \leq v \leq 2m \\ m & v \geq 2m. \end{cases} \end{align*}

For a vertex $(v_1, \ldots, v_n) \in I_{3m}^{\otimes {n}}$, we write $\sigma ^{}_{i}v$ for the vertex $(v_1, \ldots, v_{i-1}, v_{i+1}, \ldots, v_n) \in I_{3m}^{\otimes {n-1}}$. As well, let $\alpha ^{v_i}_\varepsilon$ denote the value

\begin{align*} \alpha^{v_i}_\varepsilon := \varepsilon m + (1 - 2\varepsilon)(c^m v_i). \end{align*}

We may also write this as

\begin{align*} \alpha^{v_i}_\varepsilon = \begin{cases} c^m v_i & \varepsilon = 0 \\ m - c^m v_i & \varepsilon = 1. \end{cases} \end{align*}

We define $\Phi \colon I_{3m}^{\otimes {n}} \to \lvert \mathord {\sqcap ^{n}_{i,\varepsilon }} \rvert _{m}$ by

\begin{align*} &\Phi (v_1, \ldots, v_n)\\ &\quad := (c^m v_1, \ldots, c^m v_{i-1}, (1 - \varepsilon)m + (2\varepsilon - 1)(\beta[\alpha^{v_i}_{(1 - \varepsilon)}](d(\sigma^{}_{i}v) - \alpha^{v_i}_\varepsilon)), c^m v_{i+1}, \ldots, c^m v_{n}). \end{align*}

That is,

\begin{align*} &\Phi (v_1, \ldots, v_n)\\ &\quad = \begin{cases} (c^m v_1, \ldots, c^m v_{i-1}, m - \beta[m - c^m v_i](d(\sigma^{}_{i}v) - c^m v_i), c^m v_{i+1}, \ldots, c^m v_{n}) & \text{if } \varepsilon = 0\\ (c^m v_1, \ldots, c^m v_{i-1}, \beta[c^m v_i](d(\sigma^{}_{i}v) + c^m v_i - m), c^m v_{i+1}, \ldots, c^m v_{n}) & \text{if } \varepsilon = 1. \end{cases} \end{align*}

We first show this formula is well defined, i.e. that this tuple lies in the subgraph $\lvert \mathord {\sqcap ^{n}_{i,\varepsilon }} \rvert _{m}$ of $I_m^{\otimes {n}}$. Observe that if $c^m v_k = 0$ or $m$ for some $k \neq i$ in $\{ 1, \ldots, n \}$ then this tuple indeed lies in $\lvert \mathord {\sqcap ^{n}_{i,\varepsilon }} \rvert _{m}$. For $k < i$, this tuple lies on the $(k, 0)$- or $(k, 1)$-face. For $k > i$, this tuple lies on the $(k+1,0)$- or $(k+1,1)$-face. Otherwise, if $0 < c^m v_k < m$ for all $k \neq i$ then $d(\sigma ^{}_{i}v) = 0$ since $v_k = \varphi v_k$ for all $k \neq i$. From this, it follows that

\begin{align*} \beta[\alpha^{v_i}_{1 - \varepsilon}](d(\sigma^{}_{i}v) - \alpha^{v_i}_{\varepsilon}) = \beta[\alpha^{v_i}_{1 - \varepsilon}](-\alpha^{v_i}_{\varepsilon}) = 0, \end{align*}

thus $\Phi (v_1, \ldots, v_n)$ lies on the $(i,1-\varepsilon )$-face, i.e. the face opposite the missing face.

To see this formula gives a graph map, suppose $(v_1, \ldots, v_n)$ and $(w_1, \ldots, w_n)$ are connected vertices in $I_{3m}^{\otimes {n}}$. By definition, there exists $k = 1, \ldots, n$ so that $v_k$ and $w_k$ are connected in $I_{3m}$ and $v_j = w_j$ for all $j \neq k$. We first consider the case where $k \neq i$. Observe that if $c^m v_k = c^m w_k$ then this is immediate. If $c^m v_k \neq c^m w_k$ then $\sigma ^{}_{i}v = \varphi (\sigma ^{}_{i}v)$. This gives that $d(\sigma ^{}_{i}v) = d(\sigma ^{}_{i}w)$, hence $\Phi (v_1, \ldots, v_n)$ and $\Phi (w_1, \ldots, w_n)$ are equal on all components except the $k$th component. That is, they are connected. In the case where $k = i$, we have that $d v$ and $d w$ differ by at most 1. This implies $(1 - e)m + (2\varepsilon - 1)(\beta [\alpha ^{v_i}_{1-\varepsilon }](dv - \alpha ^{v_i}_\varepsilon ))$ and $(1 - e)m + (2\varepsilon - 1)(\beta [\alpha ^{v_i}_{1-\varepsilon }](dw - \alpha ^{v_i}_\varepsilon ))$ differ by at most 1, thus $\Phi (v_1, \ldots, v_n)$ and $\Phi (w_1, \ldots, w_n)$ are connected.

Example 4.3 We look at the map $\Phi \colon I_{3m}^{\otimes {n}} \to \lvert \mathord {\sqcap ^{n}_{i,\varepsilon }} \rvert _{m}$ in the case of $n = 3$, $m = 2$, and $(i,\varepsilon ) = (3, 1)$ (Figure 8).

We may write the map $\Phi \colon I_6^{\otimes {3}} \to \lvert \mathord {\sqcap ^{3}_{3,1}} \rvert _{2}$ as

\begin{align*} \Phi (v_1, v_2, v_3) = \begin{cases} (c^2 v_1, c^2 v_2, 0) & \text{if }v_3 \leq 2 \\ (c^2 v_1, c^2 v_2, \beta[1](d(v_1, v_2) - 1)) & \text{if } v_3 = 3 \\ (c^2 v_1, c^2 v_2, \beta[2](d(v_1, v_2))) & \text{if } v_3 \geq 4. \end{cases} \end{align*}

If $v_3 \leq 2$ then $(v_1, v_2, v_3)$ is contained in the $\partial ^{}_{3,0}$-face of $\lvert \mathord {\sqcap ^{3}_{3,1}} \rvert _{2}$, which is the face opposite the missing face. For the cross-section where $v_3 = 3$, if $d(v_1, v_2) \geq 2$ then $(v_1, v_2, v_3)$ is sent to a vertex in the $v_3 = 1$ cross-section of $\lvert \mathord {\sqcap ^{3}_{3,1}} \rvert _{2}$. For $v_3 \geq 3$, if $d(v_1, v_2) = 1$ then $(v_1, v_2, v_3)$ is sent to a vertex in the $v_3 = 1$ cross-section. If $d(v_1, v_2) \geq 2$ then $(v_1, v_2, v_3)$ is sent to a vertex in the $v_3 = 2$ cross-section. In all three cross-sections, if $d(v_1, v_2) = 0$ then $(v_1, v_2, v_3)$ is contained in the face opposite the missing face of $\lvert \mathord {\sqcap ^{3}_{3,1}} \rvert _{2}$ (Figure 9).

Lemma 4.4$ The diagram

commutes.

Proof. For $n = 1$, we have $\lvert \mathord {\sqcap ^{1}_{i,\varepsilon }} \rvert _{m} \cong I_0$. The diagram then commutes as $I_0$ is terminal in $\mathsf {Graph}$.

Figure 8. The graph $\lvert \mathord {\sqcap ^{3}_{3,1}} \rvert _{2}$ as a net. Vertices connected by a dotted line are identical.

Figure 9. The map $\Phi \colon I_6^{\otimes {3}} \to \lvert \mathord {\sqcap ^{3}_{3,1}} \rvert _{2}$ split into cross-sections.

For $n \geq 2$, fix $(v_1, \ldots, v_n) \in \lvert \mathord {\sqcap ^{n}_{i,\varepsilon }} \rvert _{3m}$. It suffices to show

\begin{align*} (1 - \varepsilon)m + (2\varepsilon - 1)(\beta[\alpha^{v_i}_{1 - \varepsilon}](d(\sigma^{}_{i}v) - \alpha^{v_i}_\varepsilon)) = c^m v_i. \end{align*}

If $v_i = (1 - \varepsilon )(3m)$ then $d(\sigma ^{}_{i}v) = 0$. Thus,

\begin{align*} (1 - \varepsilon)m + (2\varepsilon - 1)(\beta[\alpha^{v_i}_{1 - \varepsilon}](d(\sigma^{}_{i}v) - \alpha^{v_i}_\varepsilon)) &= (1 - \varepsilon)m + (2\varepsilon - 1)(\beta[\alpha^{v_i}_{1 - \varepsilon}](-\alpha^{v_i}_\varepsilon))\\ &= (1 - \varepsilon)m + (2\varepsilon - 1)(0) \\ &= (1 - \varepsilon)m\\ &= c^m ((1 - \varepsilon)(3m)) \\ &= c^m v_i. \end{align*}

Otherwise, if $v_i \neq (1 - \varepsilon )(3m)$ then $d(\sigma ^{}_{i}v) \geq m$ (since $v_k = 0, 3m$ for some $k \neq i$). For $\varepsilon = 0$, we compute

\begin{align*} (1 - \varepsilon)m + (2\varepsilon - 1)(\beta[\alpha^{v_i}_{1 - \varepsilon}](d(\sigma^{}_{i}v) - \alpha^{v_i}_\varepsilon)) &= m - \beta[\alpha^{v_i}_1](d(\sigma^{}_{i}v) - \alpha^{v_i}_0)\\ &= m - \beta[m - c^m v_i](d(\sigma^{}_{i}v) - c^m v_i) \\ &= m - (m - c^m v_i) \\ &= c^m v_i. \end{align*}

For $\varepsilon = 1$, we compute

\begin{align*} (1 - \varepsilon)m + (2\varepsilon - 1)(\beta[\alpha^{v_i}_{1 - \varepsilon}](d(\sigma^{}_{i}v) - \alpha^{v_i}_\varepsilon)) &= \beta[\alpha^{v_i}_0](d(\sigma^{}_{i}v) - \alpha^{v_i}_1)\\ &= \beta[c^m v_i](d(\sigma^{}_{i}v) + c^m v_i - m) \\ &= c^m v_i. \end{align*}

Proof. Fix a map $f \colon \mathord {\sqcap ^{n}_{i,\varepsilon }} \to \mathrm {N}{G}$. We know that $\mathrm {N}{G} \cong \operatorname {colim}(\mathrm {N}_{1}{G} \hookrightarrow \mathrm {N}_{3}{G} \hookrightarrow \mathrm {N}_{5}{G} \hookrightarrow \cdots )$ by Proposition 3.5. Recall that in a presheaf category, any map from a representable presheaf to a colimit must factor through some component of the colimit cone by the Yoneda lemma. Thus, for any $k \geq 0$ and $x \colon \mathord {\square ^{k}} \to \mathord {\sqcap ^{n}_{i,\varepsilon }}$, the map $fx \colon \mathord {\square ^{k}} \to \mathrm {N}{G}$ factors through an inclusion $\mathrm {N}_{m} G \hookrightarrow \mathrm {N} G$ for some $m \geq 1$. As $\mathord {\sqcap ^{n}_{i,\varepsilon }}$ has only finitely many non-degenerate cubes, $f$ factors through the natural inclusion $\mathrm {N}_{m}{G} \hookrightarrow \mathrm {N} G$ as a map $g \colon \mathord {\sqcap ^{n}_{i,\varepsilon }} \to \mathrm {N}_{m}{G}$ for some $m \geq 0$.

By adjointness, $g$ corresponds to a map $\overline {g} \colon \lvert \mathord {\sqcap ^{n}_{i,\varepsilon }} \rvert _{m} \to G$. Lemma 4.4 shows that $\overline {g}\Phi \colon I_{3m}^{\otimes {n}} \to G$ is a lift of the composite map $\overline {g}(c^*)^m \colon \lvert \mathord {\sqcap ^{n}_{i,\varepsilon }} \rvert _{3m} \to G$:

By adjointness, this gives a filler $\mathord {\square ^{n}} \to \mathrm {N}_{3m}{G}$ for the map $(c^*)^m g \colon \mathord {\sqcap ^{n}_{i,\varepsilon }} \to \mathrm {N}_{3m}{G}$. Post-composing with the natural inclusion $\mathrm {N}_{3m}{G} \hookrightarrow \mathrm {N} G$, this gives a filler $\mathord {\square ^{n}} \to \mathrm {N}{G}$ of $f$.

Via Theorem 4.5, we may speak of the homotopy groups of $\mathrm {N} G$. We prove that the A-homotopy groups of $G$ are exactly the cubical homotopy groups of $\mathrm {N} G$.

Remark 4.7 Theorem 4.6 shows why the functor $\mathrm {N} \colon \mathsf {Graph} \to \mathsf {cSet}$ fails to preserve arbitrary limits, even though, by Proposition 3.8, it preserves finite ones. To see that, we first note that

\begin{align*} A_1\bigg(\prod_{k \geq 5} (C_k, 0)\bigg) \cong \bigoplus_{k \geq 5} \mathbb{Z}, \end{align*}

since a map $I_\infty \to \prod _{k \geq 5} C_k$ that stabilizes outside of a finite interval must necessarily be null-homotopic in all but finitely many $C_k$. That is, $A_1 \colon \mathsf {Graph}_* \to \mathsf {Grp}$ does not preserve infinite products. Theorem 4.6 shows that $A_1 \cong \pi _1 \circ \mathrm {N}$, and $\pi _1 \colon \mathsf {cSet}_* \to \mathsf {Grp}$ preserves infinite products by [Reference Carranza and KapulkinCK23, Proposition 4.1]. Thus, we conclude that

\begin{align*} \mathrm{N}\bigg(\prod_{k \geq 5} (C_k, 0)\bigg) \not \cong \prod_{k \geq 5} (\mathrm{N} C_k, 0). \end{align*}

Proof of part (ii)

Now we show that the inclusions $l^*, r^* \colon \mathrm {N}_{m} G \hookrightarrow \mathrm {N}_{m+1} G$ are anodyne. We first explain the intuition for why this statement holds.

The 1-nerve $\mathrm {N}_{1} G$ of a graph $G$ contains, as 1-cubes, all paths of length 1 in $G$ (i.e. paths with two vertices and one edge). Consider the image of the embedding $l^* \colon \mathrm {N}_{1} G \to \mathrm {N}_{2} G$ as a cubical subset $X \subseteq \mathrm {N}_{2} G$ of the 2-nerve of $G$. A 1-cube of $\mathrm {N}_{2} G$ is exactly a path of length 2 in $G$; a 1-cube of $X$ is a path of length 1 regarded as a path of length 2 whose first two vertices are the same. Given a path $f \colon I_2 \to G$ of length 2 in $G$, we may define a $3 \times 3$ square $g \colon I_2^{\otimes {2}} \to G$ in $G$ by

\begin{align*} g(v_1, v_2) := \begin{cases} f(l(v_1)) & v_2 < 2 \\ f(v_1) & v_2 = 2. \end{cases} \end{align*}

Observe that the $\partial ^{}_{1,0}$-, $\partial ^{}_{1,1}$-, and $\partial ^{}_{2,0}$-faces of $g$ are 1-cubes of $X$, i.e. they are paths of length 2 whose first two vertices are the same, whereas the $\partial ^{}_{2,1}$-face of $g$ is $f$. That is, we have shown the restriction ${g}|_{\lvert \mathord {\sqcap ^{2}_{2,1}} \rvert _{2}} \colon \lvert \mathord {\sqcap ^{2}_{2,1}} \rvert _{2} \to G$ of $g$ to the open box corresponds to a map $\mathord {\sqcap ^{2}_{2,1}} \to \mathrm {N}_{2} G$ whose image is contained in $X$ (Figure 10).

Figure 10. The square $g \colon I_2^{\otimes {2}} \to G$ constructed from the path $f \colon I_2 \to G$.

Let $Y \subseteq \mathrm {N}_{2} G$ denote the cubical subset generated by $X$ and $g$, i.e. the cubical subset containing $X$, the 2-cube $g \in (\mathrm {N}_{2} G)_2$, and all faces, degeneracies, and connections of $g$. The square

is a pushout by definition of $Y$. This gives exactly that the inclusion $X \hookrightarrow Y$ is anodyne. Following this approach, one may construct an anodyne inclusion $X \hookrightarrow X_n$ such that $X_n$ contains all $n$-cubes of $\mathrm {N}_{2} G$. With this, the colimit of the sequence $X \hookrightarrow X_1 \hookrightarrow X_2 \hookrightarrow \cdots$ is exactly $\mathrm {N}_{2} G$ and the inclusion $X \hookrightarrow \mathrm {N}_{2} G$ is anodyne by closure under transfinite composition.

For $n > 0$ and $j \in \{ 0, \ldots, n \}$, the maps $l, r \colon I_{m+1} \to I_m$ yield maps

\begin{align*} (\operatorname{id}_{I_{m+1}}^{\otimes{j}} \otimes \,l^{\otimes{n-j}}), ( \operatorname{id}_{I_{m+1}}^{\otimes{j}} \otimes \,r^{\otimes{n-j}}) \colon I_{m+1}^{\otimes{n}} \to I_{m+1}^{\otimes{j}} \otimes I_m^{\otimes{n-j}}. \end{align*}

For $j \in \{ 0, \ldots, n-1 \}$, we define maps $\lambda ^{n}_{m,j}, \rho ^{n}_{m,j} \colon I_{m+1}^{\otimes {n}} \otimes I_1 \to I_{m+1}^{\otimes {j+1}} \otimes I_m^{\otimes {n-j-1}}$ by

\begin{align*} \lambda^{n}_{m,j}(v_1, \ldots, v_n, v_{n+1}) &= \begin{cases} (v_1, \ldots, v_j, lv_{j+1}, lv_{j+2}, \ldots, lv_n) & v_{n+1} = 0 \\ (v_1, \ldots, v_{j+1}, lv_{j+2}, \ldots, lv_n) & v_{n+1} = 1, \end{cases}\\ \rho^{n}_{m,j}(v_1, \ldots, v_n, v_{n+1}) &= \begin{cases} (v_1, \ldots, v_j, rv_{j+1}, rv_{j+2}, \ldots, rv_n) & v_{n+1} = 0 \\ (v_1, \ldots, v_{j+1}, rv_{j+2}, \ldots, rv_n) & v_{n+1} = 1. \end{cases} \end{align*}

That is, the restriction of $\lambda ^{n}_{m,j}$ to $I_{m+1}^{\otimes {n}} \otimes \{ 0 \}$ is the map $\operatorname {id}_{I_{m+1}}^{\otimes {j}} \otimes \,l^{\otimes {n-j}}$ and its restriction to $I_{m+1}^{\otimes {n}} \otimes \{ 1 \}$ is $\operatorname {id}_{I_{m+1}}^{\otimes {j+1}} \otimes \,l^{\otimes {n-j-1}}$ (likewise for $\rho ^{n}_{m,j}$).

Figure 11. The graph $I_2 \otimes I_1$ with vertices labeled by their image under $\lambda ^{n}_{1,0} \colon I_2 \otimes I_1 \to I_2$.

Figure 12. Cross-sections of the graph $I_2^{\otimes {2}} \otimes I_1$ with vertices labeled by their image under the maps $\lambda ^{2}_{1,0} \colon I_2^{\otimes {2}} \otimes I_1 \to I_2 \otimes I_1$ and $\lambda ^{2}_{1,1} \colon I_2^{\otimes {2}} \otimes I_1 \to I_2^{\otimes {2}}$.

The maps $l^m, r^m \colon I_{m+1} \to I_1$ denote application of the maps $l, r$ a total of $m$ times. That is,

\begin{align*} l^m(v) &= \begin{cases} 0 & v < m+1 \\ 1 & v = m+1, \end{cases} \\ r^m(v)& = \begin{cases} 0 & v = 0 \\ 1 & v > 0. \end{cases} \end{align*}

We write $\bar {\lambda }^{n}_{m,j}\colon I_{m+1}^{\otimes {n+1}} \to I_{m+1}^{\otimes {j+1}} \otimes I_m^{\otimes {n-j-1}}$ for the composition of $\operatorname {id}_{I_{m+1}}^{\otimes {n}}\otimes \, l^m \colon I_{m+1}^{\otimes {n+1}} \to I_{m+1}^{\otimes {n}} \otimes I_1$ followed by $\lambda ^{n}_{m,j} \colon I_{m+1}^{\otimes {n}} \otimes I_1 \to I_{m+1}^{\otimes {j+1}} \otimes I_m^{\otimes {n-j-1}}$ (Figure 11), and we write ${\bar {\rho }^{\, n}_{m,j}}\colon I_{m+1}^{\otimes {n+1}} \to I_{m+1}^{\otimes {j+1}} \otimes I_m^{\otimes {n-j-1}}$ for the composition of $\operatorname {id}_{I_{m+1}}^{\otimes {n}} \otimes \, r^n \colon I_{m+1}^{\otimes {n+1}} \to I_{m+1}^{\otimes {n}} \otimes I_1$ followed $\rho ^{n}_{m,j} \colon I_{m+1}^{\otimes {n}} \otimes I_1 \to I_{m+1}^{\otimes {j+1}} \otimes I_m^{\otimes {n-j-1}}$, respectively (Figure 12).

Proof. We consider the result for $\bar {\lambda }^{n}_{m,j}$, as the result for ${\bar {\rho }^{n}_{m,j}}$ is analogous.

Fix $(v_1, \ldots, v_n) \in I_{m+1}^{\otimes {n}}$. If $i < j + 1$ then we have

\begin{align*} \bar{\lambda}^{n}_{m,j} \partial^{}_{i,\varepsilon}(v_1, \ldots, v_n) &= \bar{\lambda}^{n}_{m,j} (v_1, \ldots, v_{i-1}, \varepsilon (m+1), v_{i}, \ldots, v_n)\\ &= \lambda^{n}_{m,j} (v_1, \ldots, v_{i-1}, \varepsilon (m+1), v_i, \ldots, v_{n-1}, l^m v_n) \\ &= \begin{cases} (v_1, \ldots, v_{i-1}, \varepsilon (m+1), v_i, \ldots, v_{j}, l v_{j+1}, \ldots, l v_{n-1}) & \text{if } l^m v_n = 0\\ (v_1, \ldots, v_{i-1}, \varepsilon (m+1), v_i, \ldots, v_{j+1}, l v_{j+2}, \ldots, l v_{n-1}) & \text{if } l^m v_n = 1 \end{cases} \\ &= \partial^{n}_{i,\varepsilon} \lambda^{n}_{m,j} (v_1, \ldots, v_{n-1}, l^m v_n) \\ &= \partial^{n}_{i,\varepsilon} \bar{\lambda}^{n}_{m,j} (v_1, \ldots, v_n). \end{align*}

Thus, $\bar {\lambda }^{n}_{m,j} \partial ^{}_{i,\varepsilon } = \partial ^{}_{i,\varepsilon }\bar {\lambda }^{n-1}_{m,j}$.

Otherwise, we have $i > j+1$. Consider the embedding $\iota \colon I_{m+1}^{\otimes {n-1}} \to I_{m+1}^{\otimes {n}}$ given by

\begin{align*} \iota(v_1, \ldots, v_{n-1}) = (v_1, \ldots, v_{i-1}, \varepsilon m, v_i, \ldots, v_{n-1}). \end{align*}

With this, we may write

\begin{align*} \bar{\lambda}^{n}_{m,j} \partial^{n+1}_{i,0}(v_1, \ldots, v_n) &= \bar{\lambda}^{n}_{m,j} (v_1, \ldots, v_{i-1}, \varepsilon (m+1), v_{i}, \ldots, v_n)\\ &= \lambda^{n}_{m,j} (v_1, \ldots, v_{i-1}, \varepsilon (m+1), v_i, \ldots, v_{n-1}, l^m v_n) \\ &= \begin{cases} (v_1, \ldots, v_j, l v_{j+1}, \ldots, l v_{i-1}, l (\varepsilon (m+1)), l v_i, \ldots, l v_{n-1}) & \text{if } l^m v_n = 0\\ (v_1, \ldots, v_{j+1}, l v_{j+2}, \ldots, l v_{i-1}, l (\varepsilon (m+1)), l v_i, \ldots, l v_{n-1}) & \text{if } l^m v_n = 1 \end{cases} \\ &= \begin{cases} (v_1, \ldots, v_j, l v_{j+1}, \ldots, l v_{i-1}, \varepsilon m, l v_i, \ldots, l v_{n-1}) & \text{if } l^m v_n = 0\\ (v_1, \ldots, v_{j+1}, l v_{j+2}, \ldots, l v_{i-1}, \varepsilon m, l v_i, \ldots, l v_{n-1}) & \text{if } l^m v_n = 1 \end{cases} \\ &= \iota \lambda^{n}_{m,j} (v_1, \ldots, v_{n-1}, l^m v_n) \\ &= \iota \bar{\lambda}^{n}_{m,j} (v_1, \ldots, v_n). \end{align*}

Thus, $\bar {\lambda }^{n}_{m,j} \partial ^{}_{i,\varepsilon } = \iota \bar {\lambda }^{n-1}_{m,j}$.

Proof. We show the result for $\bar {\lambda }^{n}_{m,j}$ as the result for ${\bar {\rho }^{n}_{m,j}}$ is analogous.

Fix $(v_1, \ldots, v_n) \in I_{m+1}^{\otimes {n}}$. We compute

\begin{align*} \bar{\lambda}^{n}_{m,j} \partial^{}_{j+1,0}(v_1, \ldots, v_n) &= \bar{\lambda}^{n}_{m,j} (v_1, \ldots, v_{j}, 0, v_{j+1}, \ldots, v_n)\\ &= \lambda^{n}_{m,j}(v_1, \ldots, v_{j}, 0, v_{j+1}, \ldots, l^m v_n) \\ &= \begin{cases} (v_1, \ldots, v_{j}, l 0, l v_{j+1}, \ldots, l v_{n-1}) & \text{if } l^m v_n = 0 \\ (v_1, \ldots, v_j, 0, l v_{j+1}, \ldots, l v_{n-1}) & \text{if } l^m v_n = 1 \end{cases} \\ &= (v_1, \ldots, v_j, 0, l v_{j+1}, \ldots, l v_{n-1}) \\ &= \partial^{}_{j+1,0}\sigma^{}_{n}(\operatorname{id}_{I_{m+1}}^{\otimes{j}} \otimes \,l^{\otimes{n-j}})(v_1, \ldots, v_n), \end{align*}

thus $\bar {\lambda }^{n}_{m,j} \partial ^{}_{j+1,0} = \partial ^{}_{j+1,0}\sigma ^{}_{n}(\operatorname {id}_{I_{m+1}}^{\otimes {j}} \otimes \,l^{\otimes {n-j}})$.

Consider the map $f \colon I_{m+1}^{\otimes {j}} \otimes I_m^{\otimes {n-j}} \to I_{m+1}^{\otimes {j+1}} \otimes I_m^{\otimes {n-j-1}}$ defined by

\begin{align*} f(a_1, \ldots, a_j, b_1, \ldots, b_{n-j}) = \begin{cases} (a_1, \ldots, a_j, m, b_1, \ldots, b_{n-j-1}) & \text{if } l^{m-1} b_{n-j} = 0 \\ (a_1, \ldots, a_j, m+1, b_1, \ldots, b_{n-j-1}) & \text{if } l^{m-1} b_{n-j} = 1. \end{cases} \end{align*}

It is straightforward to verify this is a graph map. With this, we may write

\begin{align*} \bar{\lambda}^{n}_{m,j} \partial^{}_{j+1,1} (v_1, \ldots, v_n) &= \bar{\lambda}^{n}_{m,j} (v_1, \ldots, v_j, m+1, v_{j+1}, \ldots, v_n)\\ &= \lambda^{n}_{m,j} (v_1, \ldots, v_j, m+1, v_{j+1}, \ldots, l^m v_n) \\ &= \begin{cases} (v_1, \ldots, v_j, l (m+1), l v_{j+1}, \ldots, l v_{n-1}) & \text{if } l^m v_n = 0 \\ (v_1, \ldots, v_j, m+1, l v_{j+1}, \ldots, l v_{n-1}) & \text{if } l^m v_n = 1 \end{cases} \\ &= \begin{cases} (v_1, \ldots, v_j, m, l v_{j+1}, \ldots, l v_{n-1}) & \text{if } l^m v_n = 0 \\ (v_1, \ldots, v_j, m+1, l v_{j+1}, \ldots, l v_{n-1}) & \text{if } l^m v_n = 1 \end{cases} \\ &= f(v_1, \ldots, v_j, l v_{j+1}, \ldots, l v_{n}) \\ &= f(\operatorname{id}_{I_{m+1}}^{\otimes{j}} \otimes \,l^{\otimes{n-j}})(v_1, \ldots, v_n). \end{align*}

Thus, $\bar {\lambda }^{n}_{m,j} \partial ^{}_{j+1,1} = f(\operatorname {id}_{I_{m+1}}^{\otimes {j}} \otimes \,l^{\otimes {n-j}})$.

We have an analogous result for ${\bar {\rho }^{n}_{m,j}}$ as well.

Proof. We show that $l^*$ is anodyne as the result for $r^*$ is analogous. Let $X_0$ denote the image of $\mathrm {N}_{m}{G}$ in $\mathrm {N}_{m+1}{G}$ under the embedding $l^* \colon \mathrm {N}_{m}{G} \hookrightarrow \mathrm {N}_{m+1}{G}$. We show $\mathrm {N}_{m+1}{G}$ can be obtained from $X_0$ by a transfinite composition of pushouts along coproducts of open box inclusion. For $n > 0$ and $j \in \{ 1, \ldots, n \}$, let $X_{n,j}$ be the subobject of $\mathrm {N}_{m+1}{G}$ generated by:

• – $X_0$;

• – (if $n > 1$) for any $x \colon I_{m+1}^{\otimes {l}} \otimes I_m^{\otimes {k-l}} \to G$ where $k < n$ and $l \leq k$, the $(k+1)$-cube $x \bar {\lambda }^{k}_{m,l-1} \colon I_{m+1}^{\otimes {n+1}} \to G$ of $\mathrm {N}_{m+1} G$;

• – for any $x \colon I_{m+1}^{\otimes {i}} \otimes I_m^{\otimes {n-i}} \to G$ where $i \leq j$, the $(n+1)$-cube $x \bar {\lambda }^{n}_{m,i-1} \colon I_{m+1}^{\otimes {n+1}} \to G$ of $\mathrm {N}_{m+1} G$.

By construction, there is a sequence of embeddings

\begin{align*} X_0 \hookrightarrow X_{1,1} \hookrightarrow X_{2,1} \hookrightarrow X_{2,2} \hookrightarrow X_{3,1} \hookrightarrow \cdots \end{align*}

Note that the subobject $X_{n,n}$ contains all $n$-cubes of $\mathrm {N}_{m+1} G$. This is because any $n$-cube $x \colon I_{m+1}^{\otimes {n}} \to G$ is the $\partial ^{}_{n+1,1}$-face of $x \bar {\lambda }^{n}_{m,n-1} \colon I_{m+1}^{\otimes {n+1}} \to G$. By construction, $X_{n,n}$ contains $x \bar {\lambda }^{n}_{m,n-1}$. Thus, it contains all faces of $x\bar {\lambda }^{n}_{m,n-1}$, including $x$. With this, we have

\begin{align*} \mathrm{N}_{m+1}{G} \cong \operatorname{colim}(X_0 \hookrightarrow X_{1,1} \hookrightarrow X_{2,1} \hookrightarrow X_{2,2} \hookrightarrow \cdots). \end{align*}

It remains to show $X_{n,j+1}$ is a pushout of $X_{n,j}$ along a coproduct of open box inclusions and $X_{n+1,1}$ is a pushout of $X_{n,n}$ along a coproduct of open box inclusions.

Fix $n > 0$ and $j \in \{ 1, \ldots, n-1 \}$. Let $S_{n,j+1}$ be the set of $n$-cubes $I_{m+1}^{\otimes {n}} \to G$ which factor through the map $\operatorname {id}_{I_{m+1}}^{\otimes {j+1}} \otimes \,l^{\otimes {n-j-1}} \colon I_{m+1}^{\otimes {n}} \to I_{m+1}^{\otimes {j+1}} \otimes I_m^{\otimes {n-j-1}}$ and are not contained in $X_{n,j}$. We write an element of $S_{n,j+1}$ as $x(\operatorname {id}_{I_{m+1}}^{\otimes {j+1}} \otimes \,l^{\otimes {n-j-1}})$ for some $x \colon I_{m+1}^{\otimes {j+1}} \otimes I_m^{\otimes {n-j-1}} \to G$. By construction, $X_{n,j+1}$ contains all $n$-cubes of $S_{n,j+1}$ as such an $n$-cube $x(\operatorname {id}_{I_{m+1}}^{\otimes {j+1}} \otimes \,l^{\otimes {n-j-1}})$ is the $\partial ^{}_{n+1,1}$-face of $x\bar {\lambda }^{n}_{m,j}$ (which is contained in $X_{n,j+1}$ by construction). This gives a map

\begin{align*} \coprod_{x(\operatorname{id}_{I_{m+1}}^{\otimes{j+1}} \otimes \,l^{\otimes{n-j-1}}) \in S_{n,j+1}} \mathord{\square^{n+1}} \xrightarrow{x\bar{\lambda}^{n}_{m,j} } X_{n,j+1}. \end{align*}

For each $x(\operatorname {id}_{I_{m+1}}^{\otimes {j+1}} \otimes \,l^{\otimes {n-j-1}}) \in S_{n,j+1}$, the restriction of the map $x\bar {\lambda }^{n}_{m,j}$ to the open box $\mathord {\sqcap ^{n+1}_{n+1,1}}$ factors through $X_{n,j}$ by Lemma 4.10:

\begin{align*} \coprod_{x(\operatorname{id}_{I_{m+1}}^{\otimes{j+1}} \otimes \,l^{\otimes{n-j-1}}) \in S_{n,j+1}} \mathord{\sqcap^{n+1}_{n+1,1}} \xrightarrow{{x\bar{\lambda}^{n}_{m,j} }|_{\mathord{\sqcap^{}_{}}}} X_{n,j}. \end{align*}

As $x(\operatorname {id}_{I_{m+1}}^{\otimes {j+1}} \otimes \,l^{\otimes {n-j-1}})$ is not in $X_{n,j}$, the ($n+1$)-cube $x\bar {\lambda }^{n}_{m,j}$ is also not in $X_{n,j}$ (as one of its faces is $x(\operatorname {id}_{I_{m+1}}^{\otimes {j+1}} \otimes \,l^{\otimes {n-j-1}})$). The generating cubes of $X_{n,j+1}$ which are not contained in $X_{n,j}$ are exactly those of the form $x\bar {\lambda }^{n}_{m,j}$ for some $x \in S_{n,j+1}$. Thus, we may write $X_{n,j+1}$ as the following pushout.

Thus, the map $X_{n,j} \to X_{n,j+1}$ is a pushout along a coproduct of open box inclusions.

We now show the map $X_{n,n} \to X_{n+1,1}$ is a pushout along a coproduct of open box inclusions. Let $S_{n+1,1}$ be the set of $(n+1)$-cubes which factor through the map $\operatorname {id}_{I_{m+1}} \otimes \,l^{\otimes {n}} \colon I_{m+1}^{\otimes {n+1}} \to I_{m+1} \otimes I_m^{\otimes {n}}$ and are not contained in $X_{n,n}$. Similar to before, the generating cubes of $X_{n+1,1}$ which are not contained in $X_{n,n}$ are exactly those of the form $x\bar {\lambda }^{n+1}_{m,0}$ for some $x \in S_{n+1,1}$. Thus, Lemma 4.10 similarly shows that $X_{n+1,1}$ may be written as the following pushout.

Thus, the map $X_{n,n} \to X_{n+1,1}$ is a pushout along a coproduct of open box inclusions.