# Refinement of Higher-Rank Graph Reduction

S Joseph Lippert

###### Abstract

Given a row-finite, source-free, graph of rank $k$, we extend the definition
of reduction from [EFG+22]. This constitutes a large step forward in the
extension of the geometric classification of finite directed graph
$C^{*}$-algebras presented in [ERRS21] to higher-rank graph $C^{*}$-algebras.
In particular, this new move acts as an inverse to delay, directly extends the
move called reduction in [EFG+22], and provides previously undocumented Morita
classes of $k$-graphs. Additionally, this move is more robust in its
application to product graphs than its predecessor. The methodology used to
refine reduction is of particular note. That is, we define a neighborhood of a
vertex which is vital in codifying the global influence that local changes
have on a $k$-graph.

##  1 Introduction

Recently, there have been an exciting number of advancements in the field of
$C^{*}$-algebra classification by $K$-theoretic invariants. Specifically, the
Elliott invariant has proven to be necessary and sufficient for classification
of simple unital $C^{*}$-algebras which have finite nuclear dimension and
satisfy the Universal Coefficient Theorem from [RS87]. That is, given two such
algebras $\mathbb{A}_{1}$ and $\mathbb{A}_{2}$, we know
$\mathbb{A}_{1}\cong\mathbb{A}_{2}$ if and only if their Elliott invariants
are isomorphic [TWW17, GLN20a, GLN20b]. An impressive amount of effort has
already been put towards extending this result to the non-unital case
[EGLN20].

Much effort has also been put into classifying $C^{*}$-algebras that are not
simple. The most straightforward of these examples would be the Cuntz-Krieger
algebras $\mathcal{O}_{A}$. If $A$ is not irreducible, then $\mathcal{O}_{A}$
is not a simple algebra [Cun81]. However, $\mathcal{O}_{A}$ can be classified
by ordered filtered $K$-theory [ERRS21]. The Cuntz-Krieger algebras are
closely related to graph $C^{*}$-algebras. Because of this relationship,
researchers have been able to interpret these classification results in a
purely geometric context.

In particular, the work done by Eilers, Restorff, Ruiz, and Sørensen [ERRS21]
determined a complete list of $6$ Morita equivalence preserving graph moves.
We call a move M, taking $E$ to $E_{M}$, Morita equivalence preserving if
$C^{*}(E_{M})\sim_{ME}C^{*}(E)$. Further, given two graphs $E$ and $F$ such
that $C^{*}(E)\sim_{ME}C^{*}(F)$ there exists a finite sequence of these
Morita equivalence preserving moves and their inverses converting $E$ into
$F$. In effect, [ERRS21] showed that the list of $6$ moves completely
classified graph $C^{*}$-algebras up to Morita equivalence.

That result was undoubtedly a huge step forward for the field, but there are a
number of $C^{*}$-algebras that cannot be realized as $C^{*}(E)$ for some $E$.
In particular, $C^{*}$-algebras with torsion in their $K_{1}$ group cannot be
expressed in this way (cf-[RS04, Equation 3.3]). This was one of the
motivations for the introduction of higher-rank graphs (sometimes called
$k$-graphs) and their $C^{*}$-algebras by Kumjian and Pask [KP00]. These are
natural analogues of graph $C^{*}$-algebras, but they have been shown to
include some $C^{*}$-algebras with torsion in their $K_{1}$ group [Eva08].
Additionally, higher-rank graph $C^{*}$-algebras were crucial to the proof
that every UCT Kirchberg algebra has nuclear dimension 1 [RSS15]. They relate
to solutions of the Yang-Baxter equations [Yan16, Vdo21] and have provided key
examples for noncommutative geometry [PRS08, HNP+18].

Because of the importance of these higher-rank graph $C^{*}$-algebras, it is
sensible to try and generalize the result from [ERRS21]. Eckhardt, Fieldhouse,
Gent, Gillaspy, Gonzales, and Pask began the project of generalizing the moves
of Eilers et al. to the realm of $k$-graphs [EFG+22]. This paper introduced
four $k$-graph moves that preserve Morita equivalence (insplitting, delay,
sink deletion, and reduction). The move delay was first introduced by Bates
and Pask [BP04] for directed graphs. However, in [ERRS21] it was observed that
up to Morita equivalence delay acts as an inverse of reduction and thus only
one of those moves was necessary for complete classification of graph
$C^{*}$-algebras. Ben Listhartke, in his PhD thesis work, continued with the
project and extended the move outsplitting to $k$-graphs [Lisar].

Notably, the move reduction in the $k$-graph setting no longer acts as an
inverse to delay. Additionally, the reduction presented in by Eckhardt et al.
is not robust under products, meaning it will imply certain Morita
equivalences that are not accounted for by this move. Lastly, this reduction
has a very strong technical hypothesis that makes its application difficult.

The aim of this paper is to further generalize the concept of reduction, so
that it depends on weaker hypotheses, acts as an inverse of delay, and is
robust under products, Definition 2.16. Primarily this will rely on distilling
the hypotheses of [EFG+22] down to the essential components. That is, the
definition of a complete edge from [EFG+22, Notation 6.1] can be relaxed to
Definition 4.3. Combining this with a natural extension of reducibility from
[Sør13] allows us to define reduction, Definition 4.1. Furthermore, this
generalization will not only act as an inverse for delay, but it will also
capture previously undocumented Morita equivalences (like those displayed in
Figure 4.6 and discussed in Proposition 5.9). To avoid confusion, the move
from [EFG+22] will herein be referred to as Complete Edge Reduction (CR).

Reduction in [ERRS21] is defined as occurring from $w$ to $v$ and involving
the removal of the vertex $w$. We begin by exploring the implications of this
removal and define the neighborhood of a vertex. This definition will utilize
the concept of a pullback developed in [KP00]. Then we work to adjust the
reducibility condition from [ERRS21] to fit with the $k$-graph setting and
take into account the neighborhood of $w$. We define the move reduction (R)
and show, in Theorem 4.13, that it preserves Morita equivalence of the
$k$-graph $C^{*}$-algebras. The final section demonstrates that this move is
robust under products and captures all relevant information provided by the
moves CR and D. This brings the list of known Morita equivalence preserving
moves for $k$-graphs down to $4$, namely, I, O, S, and R. This has the benefit
of mirroring the first collection of graph moves preserving Morita equivalence
found by Sørensen [Sør13].

It is the hope of the author that these techniques can be applied to improve
the moves I and O, which are not robust under products. These adjustments will
hopefully pave the way to extending the move C to higher-rank graphs.

Acknowledgements: The author would like to thank his PhD advisor, Elizabeth
Gillaspy, for many helpful discussions and invaluable guidance. He would also
like to thank Luke Davis for bringing this problem to their attention. This
research was partially supported by NSF grant DMS-1800749 to Elizabeth
Gillaspy.

##  2 Preliminaries

Throughout this paper we will fix a $k\in\mathbb{N}$ and suppose that
$0\in\mathbb{N}$. Thus, we can view $\mathbb{N}^{k}$ as a category with
composition of morphisms given by addition. Note that the category is Abelian
with a single object which we denote $0$ or $(0,\dots,0)$. Additionally, we
will refer to the standard basis of $\mathbb{N}^{k}$ as
$E:=\\{e_{1},\dots,e_{k}\\}$. We will begin with the definition of a higher-
rank graph.

######  Definition 2.1.

[KP00, Definitions 1.1] Let $\Lambda$ be a countable category and
$d:\Lambda\to\mathbb{N}^{k}$ a functor. If $(\Lambda,d)$ satisfies the
factorization property\- that is, for every morphism $\lambda\in\Lambda$ and
$n,m\in\mathbb{N}^{k}$ such that $d(\lambda)=n+m$, there exist unique
$\mu,\nu\in\Lambda$ such that $d(\mu)=m$, $d(\nu)=n$, and $\lambda=\mu\nu$\-
then $(\Lambda,d)$ is a $k$-graph (or graph of rank $k$).

A helpful source of examples is product graphs.

######  Proposition 2.2.

[KP00, Proposition 1.8] Let $\Lambda_{1}$ and $\Lambda_{2}$ be rank $k$ and
$\ell$ graphs respectively. Then the product category
$\Lambda_{1}\times\Lambda_{2}$ with the product functor $d_{1}\times
d_{2}(\lambda_{1},\lambda_{2})=(d_{1}(\lambda_{1}),d_{2}(\lambda_{2}))$ is a
$k+\ell$-graph.

######  Definition 2.3.

A functor $\phi:(\Lambda_{1},d_{1})\to(\Lambda_{2},d_{2})$ is a higher-rank
graph morphism if given composable pair
$\lambda_{1},\lambda_{2}\in\Lambda_{1}$ we have
$\phi(\lambda_{2}\lambda_{1})=\phi(\lambda_{2})\phi(\lambda_{1})$ and
$d_{2}(\phi(\lambda_{1}))=d_{1}(\lambda_{1})$. If $\phi$ is bijective then
$\phi$ is called an isomorphism and we write $\Lambda_{1}\cong\Lambda_{2}$.

######  Remark 2.4.

It is relatively straightforward to check that the natural projection
$\pi_{i}:\Lambda_{1}\times\Lambda_{2}\to\Lambda_{i}$ is a higher-rank graph
morphism.

For a set $A\subseteq\mathbb{N}^{k}$ we will define
$\Lambda^{A}=\\{\lambda\in\Lambda:d(\lambda)\in A\\}$. When using singleton
sets like $\\{e_{1}\\}$ we will suppress the $\\{\cdot\\}$. That is,
$\Lambda^{e_{1}}:=\Lambda^{\\{e_{1}\\}}$. We will regard $\Lambda^{0}$ as the
set of vertices of $\Lambda$. In particular, the factorization property
guarantees that for any $\lambda\in\Lambda$ there exist unique
$v,w\in\Lambda^{0}$ such that $w\lambda v=\lambda$. We use the convention
$s(\lambda)=v$ and $r(\lambda)=w$. This will allow us to define two more sets:

$w\Lambda=\\{\lambda\in\Lambda:r(\lambda)=w\\}$ | and | $\Lambda v=\\{\lambda\in\Lambda:s(\lambda)=v\\}$
---|---|---

.

We define $w\Lambda^{A}$ and $\Lambda^{A}v$ analogously.

######  Definition 2.5.

We say that a $k$-graph, $\Lambda$, is row-finite if for all $w\in\Lambda^{0}$
we have $|w\Lambda^{E}|<\infty$. Additionally we say that $\Lambda$ is source
free if $|w\Lambda^{E}|\neq 0$ for all $w\in\Lambda$.

######  Definition 2.6.

For a $k$-graph, $\Lambda$, we define the $1$-skeleton as the tuple
$G(\Lambda)=(\Lambda^{0},\Lambda^{E},s,r)$. This can be thought of as a
colored digraph with each edge in $f\in\Lambda^{E}$ having the associated
color $d(f)\in E$ (the standard basis of $\mathbb{N}^{k}$).

There is more information in $\Lambda$ than just this $1$-skeleton. For
instance, the functor $d$ and its factorization property are not evident from
the $1$-skeleton alone.

To remedy this, we consider the path category $G(\Lambda)^{*}$, the set of all
paths with composition of morphisms given by concatenation of paths. For any
$\mu\in G(\Lambda)^{*}$, we define the color order
$c(\mu)=(d(\mu_{|\mu|}),\dots,d(\mu_{1}))$, where
$\mu=\mu_{|\mu|}\cdots\mu_{1}\mu_{1}$ and each $\mu_{i}$ is an edge. We also
define the functor $d(\mu)=d(\mu_{|\mu|})+\cdots+d(\mu_{1})$. This functor
only has the factorization property when $k=1$. To account for this, we
introduce an equivalence relation on $G(\Lambda)^{*}$. This equivalence
relation is defined by extending the natural inclusion
$\iota:\Lambda^{E}\to\Lambda$ multiplicatively. That is,
$\iota(\mu_{|\mu|}\dots\mu_{1})=\iota(\mu_{|\mu|})\dots\iota(\mu_{1})$. We say
that, for $\gamma,\mu\in G(\Lambda)^{*}$, $\gamma\sim\mu$ if and only if
$\iota(\gamma)=\iota(\mu)$ in $\Lambda$. Because of the factorization property
in $\Lambda$, for any permutation of the color order $c(\mu)$ there exists a
unique path $\gamma\in G(\Lambda)^{*}$ determined by that permutation which
satisfies $\mu\sim\gamma$ in $\Lambda$.

This schematic can be taken further using the following theorem from [EFG+22].
In particular, we may start with a colored digraph and an equivalence
relation, and obtain a $k$-graph.

######  Theorem 2.7.

[EFG+22, Theorem 2.3] Let $G$ be an edge-colored digraph and suppose $\sim$ is
an $(r,s,d)$-preserving equivalence relation on $G^{*}$ satisfying:

  1. (KG0):

If $\lambda\in G^{*}$ is a path such that $\lambda=\lambda_{2}\lambda_{1}$,
then $[\lambda]=[p_{2}p_{1}]$ whenever $p_{1}\in[\lambda_{1}]$ and
$p_{2}\in[\lambda_{2}]$.

  1. (KG4):

For each $\lambda\in G^{*}$ and each permutation of the color order of
$\lambda$, there is a unique path $\mu\in[\lambda]$ with that permuted color
order.

Then $G^{*}/\sim$ is a well defined $k$-graph.

The upshot is that given a colored digraph $G$ with $k$ colors of edges, and
an equivalence relation on $G^{*}$ that statisfies $(KG0)$ and $(KG4)$, then
we may construct a graph of rank $k$ out of those objects. Further, each graph
of rank $k$ will produce a unique pair $G(\Lambda)^{*}$ and $\sim_{\Lambda}$
such that $G(\Lambda)^{*}/\sim_{\Lambda}\cong\Lambda$ [HRSW13, Theorem 4.4].
So, we may move between these two schematics of a $k$-graph with little to no
effort.

######  Remark 2.8.

Careful analysis of the product graph definition will reveal that
$G(\Lambda_{1}\times\Lambda_{2})=G(\Lambda_{1})\square G(\Lambda_{2})$, where
$\square$ represents the Cartesian product of edge colored directed graphs.

######  Remark 2.9.

Note that a functor between path categories $\phi:G(\Lambda_{1})^{*}\to
G(\Lambda_{2})^{*}$ such that $\mu\sim_{1}\nu$ implies
$\phi(\mu)\sim_{2}\phi(\nu)$ will induce a higher-rank graph morphism
$\hat{\phi}:\Lambda_{1}\to\Lambda_{2}$ such that
$\hat{\phi}[\mu]_{1}=[\phi(\mu)]_{2}$.

Since the move described in this paper is meant to be interpreted
geometrically, we will be using the $1$-skeleton picture almost exclusively.
For this reason, we will take a moment to reintroduce the definition of a
$k$-graph $C^{*}$-algebra in terms of the $1$-skeleton.

######  Definition 2.10.

[KP00, Definition 1.5] Let $\Lambda$ be a row-finite, source free $k$-graph. A
Cuntz-Krieger (CK) $\Lambda$-family is a set of partial isometries
$\\{T_{\lambda}:\lambda\in\Lambda^{E}\cup\Lambda^{0}\\}$ such that:

  1. (CK0):

$\\{T_{x}:x\in\Lambda^{0}\\}$ is a set of mutually orthogonal projections.

  2. (CK1):

If $\lambda\mu\sim\eta\gamma$, then $T_{\lambda}T_{\mu}=T_{\eta}T_{\gamma}$.

  3. (CK2):

For any $\lambda\in\Lambda^{E}$, we have that
$T^{*}_{\lambda}T_{\lambda}=T_{s(\lambda)}$.

  4. (CK3):

For any $x\in\Lambda^{0}$ and $e_{i}\in E$, we have that
$\displaystyle{\sum_{\lambda\in
x\Lambda^{e_{i}}}}T_{\lambda}T^{*}_{\lambda}=T_{x}$.

$C^{*}(\Lambda)$ is defined as the universal $C^{*}$-algebra generated by a
Cuntz-Krieger $\Lambda$-family, that is $C^{*}(\Lambda)$ is generated by a
Cuntz-Krieger $\Lambda$-family
$\\{s_{\lambda}:\lambda\in\Lambda^{E}\cup\Lambda^{0}\\}$ with the property
that for any CK $\Lambda$-family
$\\{T_{\lambda}:\lambda\in\Lambda^{E}\cup\Lambda^{0}\\}$ there exists an onto
$*$-homomorphism $\pi_{T}:C^{*}(\Lambda)\to C^{*}(T)$ such that
$\pi_{T}(s_{\lambda})=T_{\lambda}$.

The final piece of the puzzle is that our work is concerned with preserving
Morita equivalence between $C^{*}(\Lambda)$ and $C^{*}(\Lambda_{R})$. To that
end, we need to introduce some final results related to the general structure
of $C^{*}(\Lambda)$. Our first result relates to the canonical gauge action
$\alpha:\mathbb{T}^{k}\to\text{Aut}(C^{*}(\Lambda))$ defined by its action on
the generators, $\\{s_{\lambda}:\lambda\in\Lambda^{E}\cup\Lambda^{0}\\}$:

$\displaystyle\alpha_{z}(s_{\lambda})$
$\displaystyle=z^{d(\lambda)}s_{\lambda}$

where, for $z=(z_{1},\dots,z_{k})\in\mathbb{T}^{k}$, we define
$z^{(n_{1},n_{2},\dots,n_{k})}:=z_{1}^{n_{1}}z_{2}^{n_{2}}\cdots
z_{k}^{n_{k}}$.

######  Theorem 2.11.

[KP00, Theorem 3.4] Fix a row-finite, source free $k$-graph $\Lambda$ along
with a $*$-homomorphism $\pi:C^{*}(\Lambda)\to B$. If $\pi(p_{v})\neq 0$ for
all $v$, and there exists an action $\beta:\mathbb{T}^{k}\to\text{Aut}(B)$
such that

$\beta\pi=\pi\alpha,$

then $\pi$ is injective.

This result leads directly to many notable isomorphisms of $k$-graph
$C^{*}$-algebras. The one that is most useful for our purposes is the
following.

######  Corollary 2.12.

[KP00, Corollary 3.5] For row finite source free higher-rank graphs
$\Lambda_{1}$ and $\Lambda_{2}$

$C^{*}(\Lambda_{1}\times\Lambda_{2})\cong C^{*}(\Lambda_{1})\otimes
C^{*}(\Lambda_{2}).$

Using this theorem effectively relies on constructing the actions $\beta$. In
constructing such actions we will utilize the following technical lemma.

######  Lemma 2.13.

[EFG+22, Lemma 2.9] Let $(\Lambda,d)$ be a row-finite source-free $k$-graph.
Given a functor $R:\Lambda\to\mathbb{Z}^{k}$, the function
$\beta:\mathbb{T}^{k}\to\text{Aut}(C^{*}(\Lambda))$ which satisfies

$\beta_{z}(t_{\mu}t_{\nu}^{*})=z^{R(\mu)-R(\nu)}t_{\mu}t_{\nu}^{*}$

for all $\mu,\nu\in\Lambda$ and $z\in\mathbb{T}^{k}$, is an action of
$\mathbb{T}^{k}$ on $C^{*}(\Lambda)$.

Finally, we need a way of understanding Morita equivalence of $k$-graph
$C^{*}$-algebras. There is a well known result from [All08] that we will
restate here with added context to utilize later in our proofs. Specifically
these results depend on the idea of a saturation whose definition follows.

######  Definition 2.14.

Given a set $X\subseteq\Lambda^{0}$, we define its saturation, $\Sigma(X)$, as
the smallest set that contains $X$ and is:

  * •

Hereditary: If $v\in\Sigma(X)$ and $\lambda\in v\Lambda$, then
$s(\lambda)\in\Sigma(X)$.

  * •

Saturated: If for some $n\in\mathbb{N}^{k}$ we have
$s(v\Lambda^{n})\subseteq\Sigma(X)$, then $v\in\Sigma(X)$.

We now state a result that combines Remarks $3.2(2)$, Corollary $3.7$, and
Proposition $4.2$ from [All08].

######  Theorem 2.15.

[All08, 3.2(2), 3.7, 4.2] Let $\Lambda$ be a $k$-graph,
$X\subseteq\Lambda^{0}$, and define $P_{X}=\displaystyle{\sum_{v\in X}}p_{v}$.
If $\Sigma(X)=\Lambda^{0}$, then
$P_{X}C^{*}(\Lambda)P_{X}\sim_{ME}C^{*}(\Lambda)$.

It is a relatively well known result that for separable $C^{*}$-algebras,
$\mathbb{A}_{1}\sim_{ME}\mathbb{A}_{2}$ implies that
$\mathbb{A}_{1}\otimes\mathbb{A}_{3}\sim_{ME}\mathbb{A}_{2}\otimes\mathbb{A}_{3}$.
This is the motivation behind our final definition.

######  Definition 2.16.

A higher-rank graph move M is said to be Robust Under Products if for any
$k$-graph $\Lambda_{1}$ satisfying the hypotheses of M and any $\ell$-graph
$\Lambda_{2}$ the product graph satisfies the hypotheses of M and

$\left(\Lambda_{1}\times\Lambda_{2}\right)_{M}\cong(\Lambda_{1})_{M}\otimes\Lambda_{2}.$

It can be checked that of the higher-rank graph moves in [EFG+22] only the
moves D and S are robust under products. This is a desirable property for
higher-rank graph moves preserving Morita equivalence to have. Specifically,
it demonstrates that the move does not imply Morita equivalences that it fails
to account for.

We restate that the goal of this paper is to define a higher-rank graph move
Reduction (R) taking $\Lambda$ to $\Lambda_{R}$, Definition 4.1. We then
demonstrate Morita equivalence between $C^{*}(\Lambda)$ and
$C^{*}(\Lambda_{R})$, Theorem 4.13. Taking inspiration from [EFG+22], the
strategy to obtain this result will be to use the universal property of
$C^{*}(\Lambda_{R})$ to obtain a map from $C^{*}(\Lambda_{R})$ to
$C^{*}(\Lambda)$. Next we will use Theorem 2.11 to show that the map is indeed
injective. From there we will argue that the image is equal to
$P_{X}C^{*}(\Lambda)P_{X}$ for some choice of set $X$. Finally, showing that
$\Sigma(X)=\Lambda^{0}$ will give us the desired Morita equivalence between
$C^{*}(\Lambda)$ and $C^{*}(\Lambda_{R})$. As we will prove in section 5, this
refined reduction move will directly extend the work from [EFG+22], be robust
under products, and act as an inverse to delay.

##  3 Neighborhoods of a Vertex and Reducibility

In [KP00] the pullback of a $k$-graph is defined in the following way.

######  Definition 3.1.

[KP00, Definition 1.9] Let $f:\mathbb{N}^{\ell}\to\mathbb{N}^{k}$ be a monoid
morphism. If $(\Lambda,d)$ is a $k$-graph, the pullback is the $\ell$-graph
$f^{*}(\Lambda)$ defined by

$f^{*}(\Lambda)=\\{(\lambda,n):d(\lambda)=f(n)\\}$

with $d(\lambda,n)=n$, $s(\lambda,n)=s(\lambda)$ and
$r(\lambda,n)=r(\lambda)$.

Now consider some color set $\mathcal{E}\subseteq E$ and notice that we may
define a monoid morphism
$\Phi_{\mathcal{E}}:\mathbb{N}^{|\mathcal{E}|}\to\mathbb{N}^{k}$ which maps
the standard basis to the elements of the color set, $\mathcal{E}$. This means
that we may define the $|\mathcal{E}|$-graph $\Phi^{*}_{\mathcal{E}}(\Lambda)$
as above. This can be envisioned as the colored digraph
$\Lambda^{\mathcal{E}}$. Furthermore, there exists an injective higher-rank
graph morphism, $\Pi:\Phi_{\mathcal{E}}^{*}(\Lambda)\to\Lambda$, such that
$\Pi(\lambda,n)=\lambda$ [KP00, Corollary 3.5 (iii)]. Using this notation, we
define a neighborhood around a vertex $w$.

######  Definition 3.2.

Let $w\in\Lambda^{0}$ and $\mathcal{E}\subseteq E$. Consider the pullback
$\Phi^{*}_{\mathcal{E}}(\Lambda)$. Let
$\left(\Phi^{*}_{\mathcal{E}}(\Lambda)\right)_{w}$ denote the connected
component of $\Phi^{*}_{\mathcal{E}}(\Lambda)$ that contains $w$. Define the
neighborhood of $w$ with color set $\mathcal{E}$, as
$\Pi\left(\left(\Phi^{*}_{\mathcal{E}}(\Lambda)\right)_{w}\right)$.

There will be some situations where we wish to view a neighborhood as a
$|\mathcal{E}|$-subgraph and others where we wish to think of the neighborhood
as a subgraph of the $1$-skeleton of $\Lambda$. In order to make the
distinction between these two, $\Upsilon^{\mathcal{E}}_{w}$ will denote the
$|\mathcal{E}|$-graph and $U_{w}^{\mathcal{E}}$ will denote the $1$-skeleton.
That is

$\Upsilon_{w}^{\mathcal{E}}\cong\left(U_{w}^{\mathcal{E}}\right)^{*}/\sim.$

yvxwyvxw$\Lambda$$\Phi_{\mathcal{E}}^{*}\left(\Lambda\right)$$U_{w}^{\mathcal{E}}$$\left(\Phi_{\mathcal{E}}^{*}\left(\Lambda\right)\right)_{w}$

Figure 3.3: In the above figure $\Lambda$ is a $3$-graph with its $1$-skeleton
pictured, and we take $\mathcal{E}$ to be the color red (dashed). Notice that
$\Phi_{\mathcal{E}}^{*}(\Lambda)$ is indeed a $1$-graph and we may easily
identify the connected component containing $w$ (circled in green). This
component then corresponds to our neighborhood in $\Lambda$ (likewise circled
in green).

Our eventual goal is to define reduction at a vertex $w$. Just as in [Sør13],
we will need to establish criteria that make a vertex $w$ reducible.

######  Definition 3.4.

Let $s^{-1}_{B}(w):=s^{-1}(w)\cap\Lambda^{B}$ and define $r^{-1}_{B}$ and
$r_{b}^{-1}(w)$ analogously. We say that $w$ is reducible with color set
$\emptyset\neq B\subseteq E$ if there exists $v\in\Lambda^{0}$ such that the
following hold:

$\displaystyle\mu\in w\Lambda^{E}v$ $\displaystyle\implies d(\mu)\in B$
$\displaystyle r(s_{B}^{-1}(w))$ $\displaystyle=\\{x\\}\neq w$
$\displaystyle\forall b\in B,\hskip 5.69054ptr^{-1}_{b}(w)$
$\displaystyle=\\{f_{b}\\}$ $\displaystyle s(r^{-1}_{B}(w))$
$\displaystyle=\\{v\\}.$

We have an understanding of reducibility at a vertex $w$, but notably removing
a single vertex $w$ could disrupt conditions like $(KG4)$ in the resulting
graph. To account for this, we will remove the neighborhood around $w$ of a
certain color set $\mathcal{E}$. To build intuition for what the right choice
of $\mathcal{E}$ is, we will prove a small result about reducible vertices
that will give us a window into what makes them so special.

######  Proposition 3.5.

If $w$ is reducible with color set $B$ then for all edges $\mu\in
v\Lambda^{B}$, and for all $b\in B$, there exists some $\mu^{\prime}$ with
$d(\mu^{\prime})=b$ such that $f_{b}\mu\sim f_{d(\mu)}\mu^{\prime}$.

###### Proof.

Suppose that $\mu\in v\Lambda^{B}$. For any $b\neq d(\mu)\in B$ there exists a
path $f^{\prime}\mu^{\prime}$ such that $f_{b}\mu\sim f^{\prime}\mu^{\prime}$.
Note that $r(f^{\prime})=r(f_{b})=w$. Since $f^{\prime}\in
r^{-1}_{d(\mu)}(w)=\\{f_{d(\mu)}\\}$ we may conclude that
$f^{\prime}=f_{d(\mu)}$. ∎

This is not a terribly deep result, but it gives some valuable insight into
the behavior of the edges $\\{f_{b}:b\in B\\}$. In particular, we see that
they remain stationary when paired with other edges in their color set. This
tells us that a likely candidate for $\mathcal{E}$ is the complement of $B$
(i.e. $\mathcal{E}=E\setminus B$) since we want the reducibility of $w$ to
persist after the removal of the neighborhood $U_{w}^{\mathcal{E}}$. This
choice will also help enforce robustness with products.

######  Definition 3.6.

We say that $w$ is fully reducible with color set $B$ if after defining
$\mathcal{E}:=E\setminus B$ each $x\in(U_{w}^{\mathcal{E}})^{0}$ is reducible
with color set $B$. For any $x\in(U_{w}^{\mathcal{E}})^{0}$, we utilize the
notation $f_{b,x}$ for the unique edge of color $b\in B$ such that
$r(f_{b,x})=x$.

This definition is particularly important because, as we establish in the
following proposition, when $w$ is fully reducible with color set $B$ there
exists an $|\mathcal{E}|$-graph homomorphism
$\hat{\beta}_{b}:\Upsilon_{w}^{\mathcal{E}}\to\Upsilon_{v}^{\mathcal{E}}$ for
each $b\in B$.

######  Proposition 3.7.

If $w$ is fully reducible with color set $B$, then for each $b\in B$ there
exists a map $\beta_{b}:(U_{w}^{\mathcal{E}})^{*}\to(U_{v}^{\mathcal{E}})^{*}$
defined as

$\beta_{b}(\gamma)=\begin{cases}s(f_{b,\gamma}),&d(\gamma)=0\\\
\gamma^{\prime},&d(\gamma)\neq 0\end{cases}$

where $\gamma^{\prime}$ is the unique path such that $\gamma
f_{b,s(\gamma)}\sim f_{b,r(\gamma)}\gamma^{\prime}$ and
$c(\gamma)=c(\gamma^{\prime})$. This map intertwines with $s$ and $r$ (i.e.
preserves path composition) and preserves $\sim$. By Remark 2.9, $\phi$
induces a $|\mathcal{E}|$-graph homomorphism
$\hat{\beta}_{b}:\Upsilon_{w}^{\mathcal{E}}\to\Upsilon_{v}^{\mathcal{E}}$.
Further, we may explicitly state the image:

$\operatorname{im}(\beta_{b})=\\{\eta\in(U_{v}^{\mathcal{E}})^{*}:\text{
$f_{b,x}\eta$ is a valid path for some $x\in(U_{w}^{\mathcal{E}})^{0}$ }\\}.$

###### Proof.

This proof is organized into two parts. First, we will demonstrate that
$\operatorname{im}(\beta_{b})\subseteq(U_{v}^{\mathcal{E}})^{*}$. We then
demonstrate that $\beta_{b}$ will indeed extend to a $|\mathcal{E}|$-graph
homomorphism.

Since $\beta_{b}$ is defined piece-wise we will investigate each case
individually. We claim that $s(f_{b,x})\in U_{v}^{\mathcal{E}}$ for all $x\in
U_{w}^{\mathcal{E}}$. It is enough to show that for the
$\mathcal{E}$-neighbors of a reducible vertex $w$ this holds. Suppose that $x$
is an $\mathcal{E}$-neighbor of $w$, that is, there exists an edge $\mu\in
U_{w}^{\mathcal{E}}$, such that $s(\mu)=w$ and $r(\mu)=x$ or vice versa.
Suppose the former, and let $b\in B$. We know that $r(f_{b,w})=w$ and thus
$\mu f_{b,w}$ is a valid path. By $(KG4)$ there exists a unique path
$f^{\prime}\eta\sim\mu f_{b,w}$ such that $d(f^{\prime})=b$, $d(\eta)=d(\mu)$.
Since $r(\mu)=x\in\left(U_{w}^{\mathcal{E}}\right)^{0}$, we know that $x$ is a
reducible vertex with color set $B$ and thus $f^{\prime}=f_{b,x}$. Further, by
construction $r(\eta)\in U_{v}^{\mathcal{E}}$. We conclude that $s(f_{b,x})\in
U_{v}^{\mathcal{E}}$ as desired.

Suppose that $s(\mu)=x$ and $r(\mu)=w$. In this case we may form the path $\mu
f_{b,x}$, which is equivalent to some path $f^{\prime}\zeta$ with
$d(f^{\prime})=b$ and $d(\zeta)=d(\mu)$. Observing that $w$ is reducible we
know that $f^{\prime}=f_{b,w}$ so in particular this means that $r(\zeta)=v$.
Then, since $d(\zeta)=d(\mu)\in\mathcal{E}$, we may conclude that
$s(f_{b,x})=s(\zeta)\in U_{v}^{\mathcal{E}}$.

Now we suppose that $\gamma\in(U_{w}^{\mathcal{E}})^{*}$. We claim that there
exists a unique path $\gamma^{\prime}$ such that $\gamma f_{b,s(\gamma)}\sim
f_{b,r(\gamma)}\gamma^{\prime}$ and further
$\gamma^{\prime}\in(U_{v}^{\mathcal{E}})^{*}$. Define the color order
$c(\gamma)$, as the tuple $(d(\gamma_{|\gamma|}),\dots,d(\gamma_{1}))$. By
hypothesis, $s(\gamma)$ is reducible with color set $B$. In particular, this
means that $\gamma f_{b,s(\gamma)}$ is a well defined path in
$G(\Lambda)^{*}$. Thus $(KG4)$ guarantees a unique path
$f^{\prime}\gamma^{\prime}$ with color order $(b,c(\gamma))$ such that $\gamma
f_{b,s(\gamma)}\sim f^{\prime}\gamma^{\prime}$. We observe that $r(\gamma)$ is
reducible with color set $B$. Thus, $d(f^{\prime})=b$ implies that
$f^{\prime}=f_{b,r(\gamma)}$. Finally, since $c(\gamma^{\prime})=c(\gamma)$,
each edge of $\gamma$ was in the color set $\mathcal{E}$. From this together
with the fact that $s(f_{b,s(\gamma)})\in U_{v}^{\mathcal{E}}$, we may
conclude that $\gamma^{\prime}\in(U_{v}^{\mathcal{E}})^{*}$.

The uniqueness of $\gamma^{\prime}$ ensures that this function is well
defined. Let $\beta_{b}$ be defined as above. Notice that
$s(\beta_{b}(\gamma))=s(\gamma^{\prime})=s(f_{b,s(\gamma)})=\beta_{b}(s(\gamma))$.
Likewise we see that
$r(\gamma^{\prime})=s(f_{b,r(\gamma)})=\beta_{b}(r(\gamma))$. This establishes
the intertwining condition. Next recall that by construction
$c(\gamma)=c(\gamma^{\prime})$. This guarantees that
$d(\gamma)=d(\beta_{b}(\gamma))$. Let
$\gamma,\eta\in(U_{w}^{\mathcal{E}})^{*}$ and suppose that $\gamma\sim\eta$.
Observe that $s(\gamma)=s(\eta)\in(U_{w}^{\mathcal{E}})$. In particular,
$f_{b,s(\eta)}$ exists. So, $\gamma f_{b,s(\eta)}$ and $\eta f_{b,s(\eta)}$
are well defined paths. Further by $(KG0)$ we know that $\gamma
f_{b,s(\eta)}\sim\eta f_{b,s(\eta)}$. Further, we have $\gamma
f_{b,s(\eta)}\sim f_{b,r(\eta)}\beta_{b}(\gamma)$ and $\eta f_{b,s(\eta)}\sim
f_{b,r(\eta)}\beta_{b}(\eta)$. $(KG0)$ and $(KG4)$ together ensure that since
$f_{b,r(\eta)}\sim f_{b,r(\eta)}$ we have
$\beta_{b}(\gamma)\sim\beta_{b}(\eta)$. By Remark 2.9 it follows that
$\hat{\beta}_{b}$ is a $|\mathcal{E}|$-graph homomorphism

Finally, we explicitly state the image of $\beta_{b}$. For notation’s sake we
will define $X=\\{\eta\in(U_{v}^{\mathcal{E}})^{*}:\text{ $f_{b,x}\eta$ is a
valid path for some $x\in(U_{w}^{\mathcal{E}})^{0}$ }\\}$. The containment
$\operatorname{im}(\beta_{b})\subseteq X$ follows immediately since
$f_{b,r(\gamma)}\beta_{b}(\gamma)$ is a well defined path for each
$\gamma\in(U_{w}^{\mathcal{E}})^{*}$. Now suppose that $\eta\in X$. Next we
observe that $(KG4)$ will again guarantee a path, $\eta^{\prime}f^{\prime}$
such that $\eta^{\prime}f^{\prime}\sim f_{b,x}\eta$ with $d(f^{\prime})=b$ and
$c(\eta^{\prime})=c(\eta)$. Since $r(\eta^{\prime})=x\in U_{w}^{\mathcal{E}}$
and each edge of $\eta^{\prime}$ has degree in $\mathcal{E}$, we may conclude
that $s(\eta^{\prime})\in U_{w}^{\mathcal{E}}$. We conclude that
$f^{\prime}=f_{b,s(\eta^{\prime})}$ since $s(\eta^{\prime})$ is reducible with
color set $B$. We note that $\eta=\beta_{b}(\eta^{\prime})$ allowing us to
conclude that the image is as claimed. ∎

######  Corollary 3.8.

Let $w$ be fully reducible to $v$ with color set $B$ and suppose
$U_{v}^{\mathcal{E}}\cap U_{w}^{\mathcal{E}}=\emptyset$. If
$\mu\in\Lambda^{B}$ and $s(\mu)\in U_{w}^{\mathcal{E}}$, then $r(\mu)\notin
U_{w}^{\mathcal{E}}$.

###### Proof.

Let $\mu$ be as above. Suppose that $r(\mu)\in U_{w}^{\mathcal{E}}$. Then
$r(\mu)$ is reducible with color set $B$. We conclude that
$\mu=f_{d(\mu),r(\mu)}$. In particular this means that $s(\mu)\in
U_{v}^{\mathcal{E}}$, which is a contradiction since $U_{v}^{\mathcal{E}}\cap
U_{w}^{\mathcal{E}}=\emptyset$. ∎

Returning to Figure 3.3 we notice that $w$ is a reducible vertex with color
set black (solid) and blue (dotted). Further, with $\mathcal{E}$ taken as red
(dashed), $(U_{w}^{\mathcal{E}})^{0}=\\{w\\}$ and thus $w$ is fully reducible.
This example is of particular note because it is the $1$-skeleton of a product
graph. We are beginning to see why this construction is indeed robust under
products.

Before moving on, let’s look at one last example.

$0,0$$1,0$$2,0$$\vdots$$0,1$$1,1$$2,1$$\vdots$$0,2$$1,2$$2,2$$\vdots$$0,3$$1,3$$2,3$$\vdots$$\cdots$$\cdots$$\cdots$$\iddots$$\cdots$$\cdots$$\cdots$$\ddots$$\iddots$$\vdots$$\vdots$$\vdots$$\vdots$$\ddots$

Figure 3.9: The $2$-graph above is commonly referred to as $\Delta_{2}$.
Notice that every vertex $v_{i,j}$ is reducible in both the horizontal and
vertical directions. In particular, any $v_{i,j}$ is reducible to $v_{i-1,j}$
with color set blue (dotted). Furthermore, the red (dashed) neighborhood of
$v_{i,j}$ is the horizontal line intersecting $v_{i,j}$ and therefore the
vertices of this neighborhood are also reducible with color set blue.
Therefor, $v_{i,j}$ is fully reducible to $v_{i-1,j}$ with color set blue.
Similarly, it can be argued that $v_{i,j}$ is fully reducible to $v_{i,j-1}$
with color set red.

##  4 The Move Reduction (R)

We open this section with the definition of reduction at $w$ with color set
$B$. We offer definitions and motivations for the technical conditions
presented. The references to these definitions and motivating examples are
displayed next to each condition. We then demonstrate that $\Lambda_{R}$ is a
well defined $k$-graph, and we show that
$C^{*}(\Lambda)\sim_{ME}C^{*}(\Lambda_{R})$.

######  Definition 4.1.

Reduction (R): Let $\Lambda$ be a $k$-graph. Suppose that $w$ is fully
reducible with color set $B$, and suppose that for any
$x\in(U_{w}^{\mathcal{E}})^{0}$ the set $\mathcal{F}_{x}=\\{f_{b,x}:b\in
B\\}$, hereinafter referred to as the bridge edges to $x$, is a stationary set
with color set $B$ [Definition 4.3, Figure 4.2], and $U_{w}^{\mathcal{E}}\cap
U_{v}^{\mathcal{E}}=\emptyset$ [Figure 4.4]. We may define $\Lambda_{R}$ in
the following way:

$\displaystyle\Lambda_{R}^{0}$
$\displaystyle=\Lambda^{0}\setminus(U_{w}^{\mathcal{E}})^{0}$
$\displaystyle\Lambda_{R}^{E}$
$\displaystyle=\Lambda^{E}\setminus((U_{w}^{\mathcal{E}})^{E}\cup\\{r_{B}^{-1}(x):x\in(U_{w}^{\mathcal{E}})^{0}\\})$
$\displaystyle r_{R}(\mu)$ $\displaystyle=r(\mu)$ $\displaystyle s_{R}(\mu)$
$\displaystyle=\begin{cases}s(f_{b,s(\mu)}),&\text{ if }s(\mu)\in
U_{w}^{\mathcal{E}}\\\ s(\mu),&\text{ otherwise. }\end{cases}$

Note that in this construction each edge in $\Lambda_{R}^{E}$ shares a name
with an edge in $\Lambda$. Thus we may define the natural inclusion
$\iota:\Lambda_{R}^{E}\to\Lambda^{E}$, and when necessary this may be extended
multiplicatively. For a fixed $b\in B$, define
$\operatorname{par}:\Lambda_{R}^{E}\to\left(\Lambda^{E}\right)^{*}$:

$\operatorname{par}(\mu)=\begin{cases}\iota(\mu)f_{b,s(\iota(\mu))},&\text{ if
}s(\iota(\mu))\in(U_{w}^{\mathcal{E}})^{0}\\\ \iota(\mu),&\text{ otherwise.
}\end{cases}$

Extend $\operatorname{par}$ to paths multiplicatively. Define
$\gamma\sim_{R}\gamma^{\prime}$ if and only if
$\operatorname{par}(\gamma)\sim\operatorname{par}(\gamma^{\prime})$.

The first technical condition to address is stationary sets. This definition
is a generalization of complete edges [EFG+22, Notation 6.1]. Primarily the
definition is meant to guard against a situation like Figure 4.2.

wxv$\boldsymbol{\\{{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}a_{1}},{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}a_{2}}\\}}$$\boldsymbol{\\{{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}b_{1}},{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}b_{2}}\\}}$$\boldsymbol{\\{{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}c_{1}},{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}c_{2}}\\}}$$\boldsymbol{\\{{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}d_{1}},{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}d_{2}}\\}}$$\boldsymbol{\\{{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}e_{1}},{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}e_{2}}\\}}$

Figure 4.2: Equip the $1$-skeleton above with the factorization rule
$c_{2}b_{1}\sim e_{1}a_{2}$ and $e_{2}a_{1}\sim c_{1}b_{2}$ (the remaining
relations are fixed). Notice that $w$ is fully reducible with color set red
(dashed) and blue (dotted). If a smaller color set was chosen, notice that the
first condition of Definition 3.4 would not be satisfied. Observe that
reduction cannot be performed in any obvious way. That is, removal of $b_{1}$
will remove only a single edge of the commuting square $c_{2}b_{1}\sim
e_{1}a_{2}$. This means the resulting $1$-skeleton will not correspond to a
$2$-graph.

######  Definition 4.3.

The set $\\{f_{b}\\}_{b\in B}$ of edges is stationary in $B$ if, for all
$\gamma\in\Lambda^{B}$, $\gamma f_{b}\sim\gamma^{\prime}g$ implies
$g=f_{d(\gamma)}$ and $f_{b}\gamma\sim g\gamma^{\prime}$ implies
$g=f_{d(\gamma)}.$

Notice that Proposition 3.5 shows that the set of bridge edges
$\mathcal{F}_{x}$ will satisfy the second statement for any $x\in
U_{w}^{\mathcal{E}}$. Unfortunately the former condition must still be checked
to guarantee that $\mathcal{F}_{x}$ is a stationary in $B$.

Recall from Corollary 3.8 that the condition $U_{w}^{\mathcal{E}}\cap
U_{v}^{\mathcal{E}}=\emptyset$ is not wholly unmotivated. When understanding
why this is a necessary hypothesis, a helpful example to keep in mind is the
following $3$-graph.

$w$$x$$v$

Figure 4.4: In the $3$-graph above the black (solid) and blue (dotted) edges
are proceeding clockwise while the red edges (dashed) circulate counter-
clockwise. This means that $w$ is fully reducible to $v$ with red as the color
set, yet $(U^{\mathcal{E}}_{w})^{0}=\Lambda^{0}$ and thus deleting $w$ and its
neighborhood will delete the entire graph.

Now that we have built up intuition about these technical conditions, it is
also important to understand $\operatorname{im}(\operatorname{par})$ and thus
understand $\operatorname{par}^{-1}$. Lifting through $\operatorname{par}$ is
very important when understanding $\sim_{R}$.

######  Remark 4.5.

Note that

$\operatorname{im}(\iota)=\\{\mu\in\Lambda^{E}:r(\mu)\notin
U_{w}^{\mathcal{E}}\\}.$

Further we may partition $\operatorname{im}(\operatorname{par})$ in the
following way:

$\operatorname{im}(\operatorname{par})=\\{\gamma\in\operatorname{im}(\iota):s(\iota)\notin
U_{w}^{\mathcal{E}}\\}\sqcup\\{\theta
f_{b,s(\theta)}:\theta\in\operatorname{im}(\iota)\cap
s^{-1}((U_{w}^{\mathcal{E}})^{0})\\}=:\Gamma\sqcup\Theta,$

and explicitly state $\operatorname{par}^{-1}$:

$\operatorname{par}^{-1}(\mu)=\begin{cases}\iota^{-1}(\gamma),&\mu=\gamma\in\Gamma\\\
\iota^{-1}(\theta),&\mu=\theta f_{b,s(\theta)}\in\Theta\end{cases}$

Further, we will often consider the path category $\Gamma^{*}$. These are the
paths in $G(\Lambda)^{*}$ such that every edge in the path is ranged and
sourced outside of $U_{w}^{\mathcal{E}}$. The path category $\Theta^{*}$ is
less interesting since Corollary 3.8 ensures no pair of elements in $\Theta$
are composable.

Before we get into the proofs that reduction produces a well defined $k$-graph
and preserves Morita equivalence of $C^{*}$-algebras, we should take a moment
to understand geometrically what is occurring.

yvxwyvx$\Lambda$$\Lambda_{R}$

Figure 4.6: Recall from earlier that $w$ was fully reducible with color set
blue (dotted) and black (solid). Further, you may notice that the
factorization rule, $\sim$, for $\Lambda$ is indeed fixed, and that
factorization rule makes $\mathcal{F}_{w}$ stationary. Lastly we notice that
the red (dashed) neighborhoods of $v$ and $w$ are disjoint, so we may perform
reduction and obtain the colored digraph pictured to the right.

It remains to show that the resulting $\Lambda_{R}$ is a well defined
$k$-graph. To this end, we will use Theorem 2.7. To see that our definition of
$\Lambda_{R}$ yields a $k$-graph, we will check that $\sim_{R}$ satisfies
$(KG0)$ and $(KG4)$. Since $(KG0)$ follows immediately from the construction
of $\operatorname{par}$, we will prove this first.

######  Proposition 4.7.

$\Lambda_{R}$ satisfies (KG0).

###### Proof.

Suppose that $\lambda=\lambda_{2}\lambda_{1}\in\Lambda_{R}$ and that
$\mu_{2}\in[\lambda_{2}]$ and $\mu_{1}\in[\lambda_{1}]$. This means that
$\operatorname{par}(\lambda_{i})\sim\operatorname{par}(\mu_{i})$ in $\Lambda$.
By the definition of $\operatorname{par}$ we know that
$r_{R}(\gamma)=r(\operatorname{par}(\gamma))$ and
$s_{R}(\gamma)=s(\operatorname{par}(\gamma))$ for all $\gamma\in\Lambda$. This
means that $\mu_{2}\mu_{1}$ is a valid path in $\Lambda_{R}$. Since
$\operatorname{par}$ is multiplicative and $\Lambda$ satisfies $(KG0)$ we have
that

$\operatorname{par}(\mu_{2}\mu_{1})=\operatorname{par}(\mu_{2})\operatorname{par}(\mu_{1})\sim\operatorname{par}(\lambda_{2})\operatorname{par}(\lambda_{1})=\operatorname{par}(\lambda).$

Thus $\lambda_{2}\lambda_{1}\sim_{R}\mu_{2}\mu_{1}$, and so $\Lambda_{R}$
satisfies $(KG0)$. ∎

To demonstrate $(KG4)$, we will need to establish a few results. In
particular, these minor results will allow us to closely follow the argument
for [EFG+22, Theorem 6.5]. For ease of notation, in the remainder of this
section we will have the standing assumption that $w$ is a fully reducible
vertex, $U_{w}^{\mathcal{E}}\cap U_{v}^{\mathcal{E}}=\emptyset$, and for each
$x\in(U_{w}^{\mathcal{E}})^{0}$ the set $\mathcal{F}_{x}$ is stationary.

######  Lemma 4.8.

Consider a path $\mu_{n}\dots\mu_{1}$ in $\Lambda$. If $s(\mu_{n})\in
U_{w}^{\mathcal{E}}$ then $\mu_{j}=f_{d(\mu_{j}),s(\mu_{j+1})}$ for some
$1\leq j<n$ or $s(\mu_{1})\in U_{w}^{\mathcal{E}}$.

###### Proof.

We shall proceed by induction on $n$. First suppose that $n=1$. In this case,
the implication is trivial. Suppose that the implication holds for all paths
of length $n\geq 1$. Consider some path, $\mu$, of length $n+1$ and
$s(\mu_{n+1})\in U_{w}^{\mathcal{E}}$. Since $s(\mu_{n+1})=r(\mu_{n})$ and
$(U_{w}^{\mathcal{E}})^{0}$ is completely reducible, if $d(\mu_{n})\in B$ then
$\mu_{n}=f_{d(\mu_{n}),s(\mu_{n+1})}$. If $d(\mu_{n})\notin B$ then
$s(\mu_{n})\in U_{w}^{\mathcal{E}}$. In particular, the inductive hypothesis
applies to the path $\mu_{n}\dots\mu_{1}$. Thus we obtain the desired result.
∎

######  Corollary 4.9.

If $\mu=\mu_{|\mu|}\dots\mu_{1}$ has $\ell$ edges with $s(\mu_{i})\in
U_{w}^{\mathcal{E}}$ and $d(\mu_{i})\in B$ then $\mu\sim\nu$ implies that
$\nu$ has $\ell$ edges with $s(\nu_{j})\in U_{w}^{\mathcal{E}}$ and
$d(\nu_{j})\in B$.

###### Proof.

We may simply demonstrate that the number of edges with source in
$U_{w}^{\mathcal{E}}$ and degree in $B$ is preserved under transpositions two
color paths (i.e. applying the factorization rule to re-write paths of length
2), since any color permutation is generated by transpositions. Suppose that
$\mu_{i}$ is an edge such that $s(\mu_{i})\in U_{w}^{\mathcal{E}}$ and
$d(\mu_{i})\in B$. By applying Lemma 4.8 to the path $\mu_{i}\mu_{i-1}$, we
know that either $\mu_{i-1}=f_{d(\mu_{i-1}),s(\mu_{i})}$ or $s(\mu_{i-1})\in
U_{w}^{\mathcal{E}}$. If $d(\mu_{i-1})\in B$ then reducibility of $s(\mu_{i})$
implies $\mu_{i}\mu_{i-1}=\mu_{i}f_{d(\mu_{i-1}),s(\mu_{i})}$. Since
$\mathcal{F}_{s(\mu_{i})}$ is stationary and by hypothesis $d(\mu_{i})\in B$,
$\mu_{i}f_{d(\mu_{i-1}),s(\mu_{i})}\sim\mu_{i}^{\prime}f_{d(\mu_{i}),s(\mu_{i})}$
with $d(\mu_{i}^{\prime})=b$. Since $s(\mathcal{F}_{s(\mu_{i})})\notin
U_{w}^{\mathcal{E}}$, we conclude that the number of edges with source in
$U_{w}^{\mathcal{E}}$ and degree in $B$ is preserved. If $d(\mu_{i-1})\notin
B$, then $s(\mu_{i-1})\in U_{w}^{\mathcal{E}}$ and thus
$\mu_{i}\mu_{i-1}\sim\mu^{\prime}_{i}\mu_{i-1}^{\prime}$ with
$d(\mu_{i-1}^{\prime})\in B$ and $d(\mu_{i}^{\prime})\notin B$. Therefore,
$\mu_{i-1}^{\prime}$ has source in $U_{w}^{\mathcal{E}}$ and degree in $B$.

Finally, consider $\mu_{i+1}\mu_{i}$. By Corollary 3.8 we know that
$r(\mu_{i})\notin U_{w}^{\mathcal{E}}$. Further, Definition 3.4 implies that
there are no edges with degree in $B$ sourced at $s(\mu_{i})$ that range
anywhere other than $r(\mu_{i})$. This means that if $d(\mu_{i+1})\in B$, then
the factorization rule preserves the fact that exactly one of the edges
involved in any path equivalent to $\mu_{i+1}\mu_{i}$ has source in
$U_{w}^{\mathcal{E}}$ and degree in $B$. If instead $d(\mu_{i+1})\notin B$,
then there exists a path $\mu_{i+1}^{\prime}\mu_{i}^{\prime}$ such that
$\mu_{i+1}^{\prime}\mu_{i}^{\prime}\sim\mu_{i+1}\mu_{i}$ with
$d(\mu_{i}^{\prime})=d(\mu_{i+1})\notin B$ and
$d(\mu_{i+1}^{\prime})=d(\mu_{i})\in B$. We conclude that $\mu_{i+1}^{\prime}$
is an edge with source in $U_{w}^{\mathcal{E}}$ and degree in $B$.

This case analysis allows us to conclude that in an arbitrary color
permutation of the path $\mu$ the number of edges with source in
$U_{w}^{\mathcal{E}}$ and degree in $B$ is preserved. ∎

######  Corollary 4.10.

If $\gamma\in G(\Lambda)^{*}$ and $\gamma$ contains no bridge edges then
$s(\gamma),r(\gamma)\notin U_{w}^{\mathcal{E}}$ implies $\gamma\in\Gamma^{*}$.

###### Proof.

Since $s(\gamma)\notin U_{w}^{\mathcal{E}}$, the contrapositive of Lemma 4.8
will guarantee that no edge in $\gamma$ is sourced in $U_{w}^{\mathcal{E}}$
since $\gamma$ contains no bridge edges. In particular, this will guarantee
that each edge, except the final edge, will be ranged outside of
$U_{w}^{\mathcal{E}}$, and the final edge has the same range as $\gamma$ which
is outside of $U_{w}^{\mathcal{E}}$. We conclude that
$\gamma\in(\operatorname{im}(\iota))^{*}$ and further the source condition
yields $\gamma\in\Gamma^{*}$. ∎

######  Lemma 4.11.

Consider a path $\mu$ with precisely one bridge edge such that
$\mu=\mu_{n}\dots\mu_{i}f_{b,s(\mu_{i})}\mu_{i-1}\dots\mu_{1}\sim\gamma_{n+1}\dots\gamma_{1}=\gamma$.
There is a unique $1\leq j\leq n+1$ such that
$\gamma_{j}=f_{d(\gamma_{j}),r(\gamma_{j})}$. In effect, the number of bridge
edges is conserved by the factorization rule.

###### Proof.

Let $c(\mu)$ be the color order of $\mu$, that is the ordered tuple
$(d(\mu_{|\mu|}),d(\mu_{|\mu|-1}),\dots,d(\mu_{1}))$, and likewise
$c(\gamma)$. By $(KG4)$, there exists some $\pi\in S_{|\mu|}$ such that
$\pi(c(\mu))=c(\gamma)$. Since $S_{|\mu|}$ is generated by transpositions, all
we need to do is check that transpositions of the color order conserve the
number of bridge edges.

To show this, we will first demonstrate that bridge edges cannot spontaneously
appear. Suppose $\mu\gamma$ is a two color path containing no bridge edges.
Further let $\zeta\eta$ be the path such that $\zeta\eta\sim\mu\gamma$. If
$d(\mu),d(\gamma)\notin B$, then certainly neither $\zeta$ nor $\eta$ could be
bridge edges. If instead $d(\mu)\in B$ then we are simply checking that
$\mu\gamma\nsim\zeta f_{b,s(\zeta)}$ or likewise if $d(\gamma)\in B$ then we
are checking that $\mu\gamma\nsim\ f_{b,r(\mu)}\eta$. These claims follow
directly from the next arguments.

Let $\zeta,\eta\in\Lambda^{E}$ such that $s(\zeta)\in U_{w}^{\mathcal{E}}$
(i.e $\zeta f_{b,s(\zeta)}$ is a valid path). Suppose further, there exists
some $x\in(U_{w}^{\mathcal{E}})^{0}$ such that $s(f_{b,x})=r(\eta)$. If
$d(\zeta)\notin B$, by Proposition 3.7, $\zeta f_{b,s(\zeta)}\sim
f_{b,r(\zeta)}\beta_{b}(\zeta)$. When $d(\eta)\notin B$, then
$\eta\in\operatorname{im}(\beta_{b})$ so this also covers the $f_{b,x}\eta$
case.

The final case to consider is if $d(\zeta),d(\eta)\in B$. Then we observe that
$\zeta f_{b,s(\zeta)}\sim\zeta^{\prime}f_{d(\zeta),s(\zeta)}$ and
$f_{b,x}\eta\sim f_{d(\eta),x}\eta^{\prime}$ since $\mathcal{F}_{s(\zeta)}$ is
a stationary set in $B$. In either case, notice that both results still
contain exactly one $f_{b^{\prime},v^{\prime}}$ with $b^{\prime}\in B$ and
$v^{\prime}\in U_{w}^{\mathcal{E}}$. We conclude that $\sim$ conserves the
number of bridge edges since they may not spontaneously appear or disappear. ∎

These tools will provide us with everything necessary to extend the argument
for [EFG+22, Theorem 6.5] to this more general setting. In essence, these
combined results allow us to conclude that any path in $G(\Lambda)^{*}$ that
passes through the neighborhood $U_{w}^{\mathcal{E}}$ is equivalent to a path
in $G(\Lambda)^{*}$ that only ever passes through a single vertex of the
neighborhood. Further, that path will have a lift through $\operatorname{par}$
into $(\Lambda_{R}^{E})^{*}$.

######  Theorem 4.12.

If $\Lambda$ is a $k$-graph satisfying the hypotheses of R, then
$(\Lambda_{R}^{E})^{*}/\sim_{R}:=\Lambda_{R}$ is a well defined $k$-graph.

###### Proof.

Recall from Proposition 4.7 that $\sim_{R}$ satisfies $(KG0)$. It remains to
show that $\sim_{R}$ satisfies $(KG4)$.

The general strategy for this proof is to use $(KG4)$ inside of
$G(\Lambda)^{*}$, where it is guaranteed since $\Lambda$ is a $k$-graph, to
obtain a path equivalent to $\operatorname{par}(\mu)$, called $\gamma$. Then
we check that this path has a lift through $\operatorname{par}$ with the
permuted color order. In this set up, the injectivity of $\operatorname{par}$
and the uniqueness of $\gamma$ will guarantee that this lift is unique in
$(\Lambda_{R}^{E})^{*}$. It becomes very important to know when and how
$\gamma$ may lift through $\operatorname{par}$. In particular, we will be
utilizing the picture of $\operatorname{im}(\iota)$ and
$\operatorname{im}(\operatorname{par})$ presented in Remark 4.5.

Let $\mu_{n}\dots\mu_{1}$ be a path in $(\Lambda_{R}^{E})^{*}$ of color order
$(a_{n},\dots,a_{1})$ and let $(b_{n},\dots,b_{1})$ be a permutation of this
color order.

We suppose first that for all $1\leq i\leq n$ it is the case that
$s(\iota(\mu_{i}))\notin U_{w}^{\mathcal{E}}$. That is,
$\operatorname{par}(\mu_{n}\dots\mu_{1})=\iota(\mu_{n})\dots\iota(\mu_{1})=\iota(\mu)$.
Since $(G(\Lambda)^{*},\sim)$ satisfies $(KG4)$ there exists a unique path
$\gamma$ in $G(\Lambda)^{*}$, with color order $(b_{n},\dots,b_{1})$, such
that $\iota(\mu)\sim\gamma$. Observe that since $\iota(\mu)$ contains no
bridge edges, by Lemma 4.11 $\gamma$ also contains no bridge edges. Thus by
Corollary 4.10 we conclude that $\gamma\in\Gamma^{*}$. By Remark 4.5 $\gamma$
lifts through $\operatorname{par}$ to the path $\iota^{-1}(\gamma)$. Thus
$\mu\sim_{R}\iota^{-1}(\gamma)$. Recall that this lift, $\iota^{-1}(\gamma)$
is unique since $\gamma$ was unique and $\operatorname{par}$ was injective.

Now suppose that there exists an $i$ such that $s(\iota(\mu_{i}))\in
U_{w}^{\mathcal{E}}$. For notation’s sake we will assume that there is only
one, however, this style of argument will work for multiple as well. In this
case

$\operatorname{par}(\mu_{n}\dots\mu_{1})=\iota(\mu_{n})\dots\iota(\mu_{i})f_{b,s(\mu_{i})}\iota(\mu_{i-1})\dots\iota(\mu_{1}).$

This path has color order $(a_{n},\dots,a_{i},b,a_{i-1},\dots,a_{1})$. Since
$G(\Lambda)^{*}$ satisfies $(KG4)$, there exists a unique path
$\gamma_{n+1}\dots\gamma_{1}$ with color order
$(b_{n},\dots,b_{i},b,b_{i-1},\dots,b_{1})$ such that
$\iota(\mu_{n})\dots\iota(\mu_{i})f_{b,s(\mu_{i})}\iota(\mu_{i-1})\dots\iota(\mu_{1})\sim\gamma_{n+1}\dots\gamma_{1}$.
By Lemma 4.11 we know that for some $j$,
$\gamma_{j}=f_{d(\gamma_{j}),r(\gamma_{j})}$. We will proceed in cases
determined by $j$.

If $j=i$, then
$\gamma_{n+1}\dots\gamma_{1}=\gamma_{n+1}\dots\gamma_{i+1}f_{b,s(\gamma_{i+1})}\gamma_{i-1}\dots\gamma_{1}$.
We claim that the color $b_{k}$ is in $B$ for some $i\leq k\leq n$. For sake
of contradiction, suppose not. In this case, we get the relation
$\gamma_{n+1}\dots\gamma_{i+1}f_{b,s(\gamma_{i+1})}\sim
f_{b,r(\gamma_{n+1})}\beta_{b}(\gamma_{n+1}\dots\gamma_{i+1})$ by Proposition
3.7. This is a contradiction since
$r(f_{b,r(\gamma_{n+1})})=r(\gamma_{n+1})=r(\iota(\mu_{n}))\notin
U_{w}^{\mathcal{E}}$.

Let $k$ be the smallest $i\leq k\leq n$ such that $b_{k}\in B$. Again,
Proposition 3.7 tells us that

$\displaystyle\gamma_{n+1}\dots\gamma_{1}$
$\displaystyle=\gamma_{n+1}\dots\gamma_{i+1}f_{b,s(\gamma_{i+1})}\gamma_{i-1}\dots\gamma_{1}$
$\displaystyle\sim\gamma_{n+1}\dots\gamma_{k+1}f_{b,s(\gamma_{k+1})}\beta_{b}(\gamma_{k-1}\dots\gamma_{i+1})\gamma_{i}\dots\gamma_{1}.$

Define the path
$\xi:=\beta_{b}(\gamma_{k-1}\dots\gamma_{i+1})\gamma_{i}\dots\gamma_{1}$ and
note that by assumption $\xi$ contains no bridge edges. Thus, by Corollary
4.10, $\xi\in\Gamma^{*}$. Further, note that since $d(\gamma_{k+1})=b_{k}\in
B$, Corollary 3.8 yields $r(\gamma_{k+1})\notin U_{w}^{\mathcal{E}}$. Thus we
define $\eta:=\gamma_{n+1}\dots\gamma_{k+2}$ and note that
$s(\eta)=r(\gamma_{k+1})\notin U_{w}^{\mathcal{E}}$. We conclude that
$\eta\in\Gamma^{*}$ by Corollary 4.10. Observing that
$\gamma_{k+1}f_{b,s(\gamma_{k+1})}\in\Theta$ demonstrates that the path
$\eta\gamma_{k+1}f_{b,s(\gamma_{k+1})}\xi$ lifts through $\operatorname{par}$
to a path of color order $(b_{n},\dots,b_{1})$.

Now suppose that $j<i$. This means that the path to the left of $\gamma_{j}$
has color order $(b_{n},\dots,b_{i},b,b_{i-1},\dots,b_{j+1})$. Since $\Lambda$
satisfies $(KG4)$, we know that there exists some unique path
$\eta_{n+1}\dots\eta_{j+2}\zeta$ with color order $(b_{n},\dots,b_{j+1},b)$
equivalent to $\gamma_{n+1}\dots\gamma_{j+1}$. We will refer to the edges
after $\zeta$ as the path $\eta$. In particular, we have that
$\operatorname{par}(\mu)\sim\eta\zeta
f_{b_{j},r(\gamma_{j})}\gamma_{j-1}\dots\gamma_{1}$. The stationary condition,
Definition 4.3, allows us to exchange the color of $\zeta$, which is $b\in B$,
and $f_{b_{j},s(\eta_{j+1})}$ resulting in the path
$\eta\zeta^{\prime}f_{b,s(\zeta)}\gamma_{j-1}\dots\gamma_{1}$. Similar
reasoning as above will show that this path lifts via $\operatorname{par}$ to
a path in $(\Lambda_{R}^{E})^{*}$ with color order $(b_{n},\dots,b_{1})$.

Consider the case where $j>i$. This means that the color order to the right of
$\gamma_{j}$ is $(b_{j-1},\dots,b_{i+1},b,b_{i-1},\dots,b_{1})$. Again, we use
the fact that $\Lambda$ satisfies $(KG4)$ and so there is a unique path
$\zeta\eta_{j-2}\dots\eta_{1}$ with color order $(b,b_{j-1},\dots,b_{1})$
which is equivalent to $\gamma_{j-1}\dots\gamma_{1}$. In particular, we get
that
$\operatorname{par}(\mu)\sim\gamma_{n+1}\dots\gamma_{j+1}f_{b_{j},r(\gamma_{j})}\zeta\eta$.
We again exchange the color of $f_{b_{j},r(\gamma_{j})}$ with $\zeta$ giving a
path which will lift through $\operatorname{par}$ to a path of the appropriate
color order. In particular, this lift is again unique since
$\operatorname{par}$ was injective and the path in $\Lambda$ that we are
lifting is also unique.

We may conclude that since the pair $(\Lambda_{R}^{E},\sim_{R})$ satisfy
$(KG4)$ and $(KG0)$, $(\Lambda^{E}_{R})^{*}/\sim_{R}=:\Lambda_{R}$ is a well
defined $k$-graph. ∎

Now we have all of the tools necessary to demonstrate our main result. That
is, we have established a well-defined $k$-graph move, and we will demonstrate
that this transformation yields a $C^{*}$-algebra Morita equivalent to the
$C^{*}$-algebra of the original $k$-graph, $\Lambda$. As alluded to earlier,
this argument is similar to that for [EFG+22, Theorem 6.6] in construction and
it will rely heavily on the lemmas and corollaries we have obtained up to this
point.

######  Theorem 4.13.

Let $\Lambda$ be a source free, row-finite $k$-graph, and $w\in\Lambda^{0}$ be
fully reducible with color set $B$. Further suppose $U_{w}^{\mathcal{E}}\cap
U_{v}^{\mathcal{E}}=\emptyset$ and $\mathcal{F}_{x}$ is stationary with color
set $B$ for all $x\in(U_{w}^{\mathcal{E}})^{0}$. Then
$C^{*}(\Lambda)\sim_{ME}C^{*}(\Lambda_{R})$.

###### Proof.

Let $\\{s_{\lambda}:\lambda\in\Lambda^{0}\cup\Lambda^{E}\\}$ be the canonical
Cuntz-Krieger $\Lambda$-family generating $C^{*}(\Lambda)$. Define

$T_{\lambda}=s_{\operatorname{par}(\lambda)}\text{, for
}\lambda\in\Lambda_{R}^{0}\cup\Lambda_{R}^{E}.$

We now proceed to show that this is a Cuntz-Krieger $\Lambda_{R}$-family.
$(CK0)$ follows directly from $\operatorname{par}(x)=x$ for vertices $x$.
$(CK1)$ follows from the fact that $\mu\sim_{R}\lambda$ if and only if
$\operatorname{par}(\mu)\sim\operatorname{par}(\lambda)$.

Fix some $\mu\in\Lambda_{R}^{E}$, and observe that it can be easily checked
that $s(\operatorname{par}(\mu))=s_{R}(\mu)$. So,

$T_{\mu}^{*}T_{\mu}=s_{\operatorname{par}(\mu)}^{*}s_{\operatorname{par}(\mu)}=s_{s(\operatorname{par}(\mu))}=s_{s_{R}(\mu)}=T_{s_{R}(\mu)}.$

This shows that $\\{T_{\lambda}\\}$ satisfies $(CK2)$.

All that remains is $(CK3)$. To demonstrate this, we will consider some
$x\in\Lambda_{R}^{0}$ and $e_{j}\in\mathbb{N}^{k}$. We claim that for any
$\mu\in\Lambda_{R}^{e_{j}}$ we have
$T_{\mu}T_{\mu}^{*}=s_{\iota(\mu)}s_{\iota(\mu)}^{*}$. This amounts to
recognizing first that
$s_{\iota(\mu)f_{b,s(\mu)}}s^{*}_{\iota(\mu)f_{b,s(\mu)}}=s_{\iota(\mu)}s_{f_{b,s(\iota(\mu))}}s_{f_{b,s(\iota(\mu))}}^{*}s_{\iota(\mu)}$.
Observe that since $s(\iota(\mu))$ is reducible with color set $B$ and
$\\{s_{\lambda}:\lambda\in\Lambda^{0}\cup\Lambda^{E}\\}$ satisfies $(CK3)$, we
have that
$s_{f_{b,s(\iota(\mu))}}s_{f_{b,s(\iota(\mu))}}^{*}=s_{s(\iota(\mu))}$.

Consider the sum

$\sum_{\mu\in x\Lambda_{R}^{e_{j}}}T_{\mu}T_{\mu}^{*}=\sum_{\mu\in
x\Lambda_{R}^{e_{j}}}s_{\iota(\mu)}s_{\iota(\mu)}^{*}.$

Note that $x\notin U_{w}^{\mathcal{E}}$ and according to Remark 4.5 we have
that $\iota(x\Lambda_{R}^{e_{j}})=x\Lambda^{e_{j}}$. We re-index this sum to
obtain

$\sum_{\mu\in x\Lambda^{e_{j}}}s_{\mu}s_{\mu}^{*}=s_{x}=T_{x},$

and conclude that $\\{T_{\lambda}\\}$ satisfies $(CK3)$.

Consequently, there is an onto homomorphism $\pi:C^{*}(\Lambda_{R})\to
C^{*}(\Lambda)$ such that if
$C^{*}(\Lambda_{R})=\\{t_{\lambda}:\lambda\in\Lambda_{R}^{0}\cup\Lambda_{R}^{1}\\}$,
we have $\pi(t_{\lambda})=s_{\operatorname{par}(\lambda)}$ for all
$\lambda\in\Lambda_{R}^{0}\cup\Lambda_{R}^{1}$.

Next we must show that $\pi$ is injective. Define the map
$R:G(\Lambda)^{*}/\sim\to\mathbb{Z}^{k}$ by

$\displaystyle R(\mu)$ $\displaystyle=d(\mu)\text{, for
}\mu\in\Lambda^{E}\text{ s.t. }s(\mu)\notin U_{w}^{\mathcal{E}}\text{ or
}d(\mu)\notin B$ $\displaystyle R(\mu)$ $\displaystyle=d(\mu)-b\text{, for
}\mu\in\Lambda^{E}\text{ s.t. }s(\mu)\in U_{w}^{\mathcal{E}}\text{ and
}d(\mu)\in B$ $\displaystyle R(\lambda)$
$\displaystyle=\sum_{i=1}^{|\lambda|}R(\lambda_{i})\text{, for
}\lambda=\lambda_{|\lambda|}\dots\lambda_{1}\in G(\Lambda)^{*}$ $\displaystyle
R(x)$ $\displaystyle=0\text{, for }x\in\Lambda^{0}.$

To show $R$ is well defined, we suppose that $\mu\sim\nu$. By Corollary 4.9
there are $\ell$ edges in $\mu$ and $\nu$ with source in $U_{w}^{\mathcal{E}}$
and degree in $B$. Thus we have $R(\mu)=d(\mu)-\ell b$ and likewise
$R(\nu)=d(\nu)-\ell b$. Since $d(\mu)=d(\nu)$ we may conclude that
$R(\mu)=R(\nu)$.

By Lemma 2.13 we have an action
$\beta:\mathbb{T}^{k}\to\text{Aut}(C^{*}(\Lambda))$ defined by
$\beta_{z}(s_{\mu}s_{\nu}^{*})=z^{R(\mu)-R(\nu)}s_{\mu}s_{\nu}^{*}$. Let
$\alpha$ be the canonical gauge action on $C^{*}(\Lambda_{R})$.

Let $\mu\in\Lambda_{R}^{E}$ be an edge such that either $s(\iota(\mu))\notin
U_{w}^{\mathcal{E}}\text{ or }d(\iota(\mu))\notin B$. We claim that
$s(\iota(\mu))\notin U_{w}^{\mathcal{E}}$. For sake of contradiction suppose
that $s(\iota(\mu))\in U_{w}^{\mathcal{E}}$. This would imply that
$d(\iota(\mu))\notin B$ and so $r(\iota(\mu))\in U_{w}^{\mathcal{E}}$. This
contradicts the fact that $\mu\in\Lambda_{R}^{E}$. We conclude
$s(\iota(\mu))\notin U_{w}^{\mathcal{E}}$. In particular,
$R(\operatorname{par}(\mu))=d(\iota(\mu))=d_{R}(\mu)$ and hence

$\displaystyle\pi(\alpha_{z}(t_{\mu}))=\pi(z^{d_{R}(\mu)}t_{\mu})$
$\displaystyle=z^{d(\iota(\mu))}T_{\mu}=z^{d(\iota(\mu))}s_{\iota(\mu)}$
$\displaystyle=z^{R(\operatorname{par}(\mu))}s_{\operatorname{par}(\mu)}=\beta_{z}(s_{\operatorname{par}(\mu)})$
$\displaystyle=\beta_{z}(\pi(t_{\mu})).$

Alternatively, consider an edge $\mu\in\Lambda^{E}_{R}$ such that
$s(\iota(\mu))\in U_{w}^{\mathcal{E}}$ and $d(\iota(\mu))\in B$. In
particular, $R(\operatorname{par}(\mu))=d(\iota(\mu))=d_{R}(\mu)$ and thus a
similar calculation will show that $\beta_{z}$ intertwines with $\pi$. We
conclude that $\pi$ is injective according to Theorem 2.11.

At this point we have established that $\pi:C^{*}(\Lambda_{R})\to
C^{*}(\Lambda)$ is an isomorphism onto its image. To demonstrate Morita
equivalence we must find a set $X\subseteq\Lambda^{0}$ such that
$P_{X}C^{*}(\Lambda)P_{X}=\operatorname{im}(\pi)$ and demonstrate that
$\Sigma(X)=\Lambda^{0}$.

Define the set
$X=\iota(\Lambda_{R}^{0})=\Lambda^{0}\setminus(U_{w}^{\mathcal{E}})^{0}$. We
claim that that $P_{X}C^{*}(\Lambda)P_{X}=\operatorname{im}(\pi)$. Note the
set equality:

$P_{X}C^{*}(\Lambda)P_{X}=\overline{\operatorname{span}}\\{s_{\mu}s_{\lambda}^{*}:r(\mu),r(\lambda)\in
X\text{ and }s(\mu)=s(\lambda)\\}.$

For the inclusion $\operatorname{im}(\pi)\subseteq P_{X}C^{*}(\Lambda)P_{X}$
we observe that for all $\lambda\in\Lambda_{R}$, both
$s(\operatorname{par}(\lambda))$ and $r(\operatorname{par}(\lambda))$ lie in
$X$. Therefore
$s_{\operatorname{par}(\lambda)}=s_{\operatorname{par}(\lambda)}s_{s(\operatorname{par}(\lambda))}^{*}\in
P_{X}C^{*}(\Lambda)P_{X}$. We conclude that $\operatorname{im}(\pi)\subseteq
P_{X}C^{*}(\Lambda)P_{X}$.

Suppose that $\mu,\nu\in\Lambda^{E}$ such that $s(\mu)=s(\nu)\in
U_{w}^{\mathcal{E}}$. Further suppose that $r(\mu),r(\nu)\notin
U_{w}^{\mathcal{E}}$, that is $\mu f_{b,s(\mu)},\nu f_{b,s(\nu)}\in\Theta$. We
see that
$s_{\operatorname{par}(\iota^{-1}(\mu))}s_{\operatorname{par}(\iota^{-1}(\nu))}^{*}=s_{\mu}s_{f_{b,s(\mu)}}s^{*}_{f_{b,s(\mu)}}s_{\nu}^{*}=s_{\mu}s_{\nu}^{*}$.

Now fix some $d\in B$ and consider the path $\mu f_{d,s(\mu)}$. If $d=b$ then
$\mu f_{d,s(\mu)}\in\Theta$ and thus $s_{\mu
f_{d,s(\mu)}}\in\operatorname{im}(\pi)$. Instead suppose that $d\neq b$. From
[EFG+22, Theorem 5.7] we may assume without loss of generality that
$\Lambda^{b}s(\mu)\neq\emptyset$. If this was not the case, we apply sink
deletion to $\Lambda$ and $\Lambda_{R}$ while appealing to the transitivity of
$\sim_{ME}$. Choose $\xi\in\Lambda^{b}s(\mu)$. Since $\xi
f_{d,s(\mu)}\sim\xi^{\prime}f_{b,s(\mu)}$, we know that
$s_{\xi}^{*}s_{\xi^{\prime}}s_{f_{b,s(\mu)}}=s_{f_{d,s(\mu)}}$. We conclude
that $s_{\mu}s_{f_{d,s(\mu)}}=s_{\mu}s_{\xi}^{*}s_{\theta}$ for some
$\theta\in\Theta$. In particular, this and the previous paragraph imply that
$s_{\mu}s_{f_{d,s(\mu)}}\in\operatorname{im}(\pi)$

Suppose that $\mu\in G(\Lambda)^{*}$ with $s(\mu),r(\mu)\notin
U_{w}^{\mathcal{E}}$ and without loss of generality, let’s suppose that $\mu$
contains at most one bridge edge. If $\mu\in\Gamma^{*}$ then we conclude
$s_{\mu}\in\operatorname{im}(\pi)$. Instead suppose that $\mu$ contains one
bridge edge. As shown in the proof for Theorem 2.7, $\mu$ is equivalent to a
path $\eta\gamma f_{d,s(\gamma)}\xi$ for $\xi,\eta\in\Gamma^{*}$. We may
conclude, using the argument of the previous paragraph, that
$s_{\mu}\in\operatorname{im}(\pi)$.

To finish the inclusion we must consider some $\mu,\nu\in G(\Lambda)^{*}$ with
$r(\mu),r(\nu)\notin U_{w}^{\mathcal{E}}$ and $s(\mu)=s(\nu)\in
U_{w}^{\mathcal{E}}$. Again, these have equivalent paths $\mu\sim\eta\gamma$
and $\nu\sim\xi\zeta$ with $\eta,\xi\in\Gamma^{*}$. Note that
$s_{\mu}s_{\nu}^{*}=(s_{\eta})(s_{\gamma}s_{\zeta}^{*})(s_{\xi})$, each of
which we have already demonstrate are contained in $\operatorname{im}(\pi)$.
We conclude that $P_{X}C^{*}(\Lambda)P_{X}=\operatorname{im}(\pi)\cong
C^{*}(\Lambda_{R})$.

Finally we demonstrate that $\Sigma(X)=\Lambda^{0}$. This follows from the
fact that $\Sigma(X)$ is hereditary. Let $y\in(U_{w}^{\mathcal{E}})^{0}$.
Since $y$ is a reducible vertex we know that $r(s_{B}^{-1}(y))=x\notin
U_{w}^{\mathcal{E}}$, which allows us to conclude that $y\in\Sigma(X)$. By
Theorem 2.15, we conclude that
$\operatorname{im}(\pi)\sim_{ME}C^{*}(\Lambda)$.

Since $\pi:C^{*}(\Lambda_{R})\to C^{*}(\Lambda)$ was an isomorphism onto its
image we may conclude that $C^{*}(\Lambda)\sim_{ME}C^{*}(\Lambda_{R})$. We
conclude that the move R preserves Morita equivalence of higher-rank graph
$C^{*}$-algebras. ∎

##  5 Complete Edge Reduction, Delay, and Robustness

As put forth in the introduction, this section will be dedicated to showing
that two of the moves from [EFG+22] are indeed special cases of this new form
of reduction. Specifically, they lie on opposite sides of a spectrum
determined by the size of the color set $B$. Additionally, we will demonstrate
that R is robust under products, Definition 2.16.

We begin with the move CR which requires the definition of a complete edge. As
stated earlier, this was a significant source of motivation when defining
stationary sets.

######  Definition 5.1.

[EFG+22, Notation 6.1] We say a collection of edges, $G\subseteq\Lambda^{E}$,
is a complete edge if it has the following properties:

  1. (1)

$G$ contains precisely one edge of each color;

  2. (2)

$s(g_{1})=s(g_{2})$ and $r(g_{1})=r(g_{2})$ for every $g_{1},g_{2}\in G$;

  3. (3)

if $g_{1}\in G$ and $\mu,\eta,g\in\Lambda^{E}$ satisfy $\mu g_{1}\sim g\eta$
or $\mu g_{1}\sim\eta g$, then $g\in G$.

One can check that a complete edge is indeed a stationary set with color set
$E$.

######  Definition 5.2.

Complete Edge Reduction (CR)[EFG+22, Definition 6.3]:  Let $\Lambda$ be a
$k$-graph and fix $w\in\Lambda^{0}$ such that $\Lambda^{E}w$ and
$w\Lambda^{E}$ are complete edges and $w\neq r(\Lambda^{E}w)=:x$. Define

$\displaystyle\Lambda_{CR}^{0}$ $\displaystyle=\Lambda^{0}\setminus\\{w\\}$
$\displaystyle\Lambda_{CR}^{E}$
$\displaystyle=\Lambda^{E}\setminus\Lambda^{E}w$ $\displaystyle s_{CR}(e)$
$\displaystyle=s(e)$ $\displaystyle r_{CR}(e)$
$\displaystyle=\begin{cases}r(e),&r(e)\neq w\\\ x,&r(e)=w\end{cases}$

Then fix $g\in\Lambda^{E}w$ and define $\operatorname{par}$ on
$y\in\Lambda_{CR}^{0}$ and $\gamma\in\Lambda_{CR}^{E}$ in the following way:

$\displaystyle\operatorname{par}(y)$ $\displaystyle=\iota(y)$
$\displaystyle\operatorname{par}(\gamma)$
$\displaystyle=\begin{cases}\iota(\gamma),&r(\iota(\gamma))\neq w\\\
g\iota(\gamma),&r(\iota(\gamma))=w.\end{cases}$

Now extend multiplicatively and define $\mu\sim_{CR}\lambda$ if and only if
$\operatorname{par}(\mu)\sim\operatorname{par}(\lambda)$.

This move shares enough in common with R that we are simply able to
demonstrate directly that $\Lambda_{R}\cong\Lambda_{CR}$.

######  Proposition 5.3.

If a $k$-graph $\Lambda$ satisfies the hypotheses of (CR) then the vertex $w$
is fully reducible with color set $E$,
$U_{w}^{\mathcal{E}}=\\{w\\}\neq\\{v\\}=U_{v}^{\mathcal{E}}$, and
$\Lambda^{E}w=\\{f_{b,w}:b\in B\\}$ is a stationary set. Thus we may reduce
$w$ with color set $E$ and obtain $\Lambda_{R}$. Moreover,
$\Lambda_{CR}\cong\Lambda_{R}$.

###### Proof.

Since $w\Lambda^{E}$ is a complete edge, by definition for each $e_{i}$ we
have $r^{-1}_{e_{i}}(w)=\\{f_{e_{i},w}\\}$. Since $x=r(\Lambda^{E}w)\neq w$ we
know that $v=s(w\Lambda^{E})\neq w$ (if it was this would contradict the first
statement). Now we quickly conclude that $w$ is reducible with color set $E$.
Since $\mathcal{E}=\emptyset$, we know that $U_{w}^{\mathcal{E}}=\\{w\\}$ and
thus we have that $w$ is fully reducible with color set $E$. Observe that
$U_{v}^{\mathcal{E}}=\\{v\\}$ and thus we have the disjoint condition. We note
that since $w\Lambda^{E}$ is a complete edge, $\\{f_{b,w}:b\in B\\}$ is a
stationary set with color set $E$.

We may therefore construct $\Lambda_{R}$. For ease of notation we will label
$\Lambda^{E}w=\\{g_{b}:b\in B\\}$. Notice first that
$\Lambda_{CR}^{0}=\Lambda_{R}^{0}$ since $U_{w}^{\mathcal{E}}=\\{w\\}$.
Compare $\Lambda_{CR}^{E}=\Lambda^{E}\setminus\Lambda^{E}w$ and
$\Lambda_{R}^{E}=\Lambda^{E}\setminus r^{-1}(w)$, and note that
$r^{-1}(w)=\\{f_{b,w}:b\in B\\}$ and $\Lambda^{E}w=\\{g_{b}:b\in B\\}$. We
will associate $g_{b}\mapsto f_{b,w}$ and see that this yeilds a bijection
between $\Lambda_{CR}^{E}$ and $\Lambda_{R}^{E}$. We need to check that
$s_{R},s_{CR},r_{R},$ and $r_{CR}$ all agree under this bijection. Observe
that

$s_{R}(g_{b})=s(f_{b,w})=s_{CR}(f_{b,w})\qquad\text{ and }\qquad
r_{R}(g_{b})=r(g_{b})=x=r_{CR}(f_{b,w}).$

We now check that $\sim_{R}$ gives the same equivalence classes as
$\sim_{CR}$. This amounts to checking that after fixing $g\in\Lambda^{E}w$ for
$\operatorname{par}_{CR}$ and $d(g)\in E$ for $\operatorname{par}_{R}$, we
have $\operatorname{par}_{R}(g_{b})=g_{b}f_{d(g),w}\sim
gf_{b,w}=\operatorname{par}_{CR}(f_{b,w})$. We may now conclude that
$\Lambda_{CR}\cong\Lambda_{R}$. ∎

To discuss the move Delay (D) and its relationship to R, we first present
[EFG+22, Definition 4.1], with some notation changes. This definition is
particularly technical. In essence, the goal is to choose an edge
$f\in\Lambda^{e_{1}}$ and replace it with a path of length $2$, $f^{2}f^{1}$.
However, as we saw with reduction, local changes like this have a global
effect on the $k$-graph. To account for this, D adds edges $C^{e_{i}}_{D}$ for
each $k\geq i>1$ and also delays the edges in the set $C^{e_{1}}$. To prime
the reader for this definition, we will state the following proposition which
will be proved after the definition.

######  Proposition 5.4.

$\Lambda_{D}$ may be reduced from $v_{f}$ to $s(f)$. In particular,

  1. (I)

$r_{D}(f^{1})=:v_{f}$ is reducible with color set $\\{e_{1}\\}$.

  2. (II)

$U_{v_{f}}^{\mathcal{E}}=\displaystyle{\bigcup_{i=2}^{k}C_{D}^{e_{i}}}$.

  3. (III)

$(U_{v_{f}}^{\mathcal{E}})^{0}=\\{v_{g}:g\in C^{e_{1}}\\}$ and each of these
are reducible with color set $\\{e_{1}\\}$ to $s(g)$.

  4. (IV)

$\mathcal{F}_{v_{g}}=\\{g^{1}\\}$ and thus is stationary with color set
$\\{e_{1}\\}$.

  5. (V)

$U_{s(f)}^{\mathcal{E}}\subseteq G(\Lambda)$ and thus
$U_{s(f)}^{\mathcal{E}}\cap U_{v_{f}}^{\mathcal{E}}=\emptyset$.

######  Definition 5.5.

Delay (D)[EFG+22, Definition 4.1]: Let $(\Lambda,d)$ be a $k$-graph and
$G=(\Lambda^{0},\Lambda^{E},r,s)$ its underlying directed graph. Fix
$f\in\Lambda^{E}$; without loss of generality, assume $d(f)=e_{1}$. Define the
sets:

$\displaystyle A_{1}$ $\displaystyle=\\{f\\}\cup\\{g\in\Lambda^{e_{1}}:ag\sim
fb\text{ or }ga\sim bf\text{ for some }a,b\in\Lambda^{e_{i}}\text{ for }2\leq
i\leq k\\},$ $\displaystyle A_{m}$
$\displaystyle=\\{e\in\Lambda^{e_{1}}:ag\sim eb\text{ or }ga\sim be\text{
where }a,b\in\Lambda^{e_{i}}\text{ for }2\leq i\leq k,~{}g\in A_{m-1}\\},$
$\displaystyle C^{e_{1}}$
$\displaystyle=\bigcup_{j=1}^{\infty}A_{j}\subseteq\Lambda^{e_{1}}.$

Define $C^{e_{i}}:$

$C^{e_{i}}=\\{[ga]\in\Lambda:g\in C^{e_{1}},a\in\Lambda^{e_{i}}\\}.$

From these sets in $\Lambda$ we define the sets to be added to form
$\Lambda_{D}$.

$C_{D}^{e_{1}}=\\{g^{1},g^{2}:g\in C^{e_{1}}\\}$

$C_{D}^{e_{i}}=\\{e_{\alpha}:\alpha\in C^{e_{i}}\\}.$

Define the $k$-colored graph
$G_{D}=(\Lambda_{D}^{0},\Lambda_{D}^{1},r_{D},s_{D})$ by

$\begin{split}\Lambda_{D}^{0}&=\Lambda^{0}\cup\\{v_{g}\\}_{g\in
C^{e_{1}}},\quad\Lambda_{D}^{e_{1}}=(\Lambda^{e_{1}}\setminus C^{e_{1}})\cup
C_{D}^{e_{1}}\text{, with}\\\ &\hskip
28.45274pts_{D}(g^{1})=s(g),s_{D}(g^{2})=v_{g},\hskip
8.53581ptr_{D}(g^{1})=v_{g},r_{D}(g^{2})=r(g);\\\
\Lambda_{D}^{e_{i}}&=\Lambda^{e_{i}}\cup C_{D}^{e_{i}}\text{, with}\\\ &\hskip
28.45274pts_{D}(e_{\alpha})=v_{g}\text{ such that }bg\text{ represents
}\alpha\text{ and }d(g)=e_{1},\\\ &\hskip
28.45274ptr_{D}(e_{\alpha})=v_{h}\text{ such that }ha\text{ represents
}\alpha\text{ and }d(h)=e_{1}.\end{split}$

Let $\iota_{D}:G_{D}\to G$ be the partially defined inclusion map with domain
$(\Lambda_{D}^{0}\cup\Lambda_{D}^{1})\setminus\left(\\{\bigcup\limits_{i=1}^{k}C_{D}^{e_{i}}\\}\cup\\{v_{e}:e\in
C^{e_{1}}\\}\right)$. Then, for edges
$g\in\Lambda^{1}_{D}\backslash\displaystyle{\bigcup_{i=1}^{k}}C_{D}^{e_{i}}$,
we can define

$s_{D}(g)=s(\iota_{D}(g)),~{}r_{D}(g)=r(\iota(g)),~{}d_{D}(g)=d(\iota(g)).$

Let $G^{*}_{D}$ be the path category for $G_{D}$ and define the equivalence
relation $\sim_{D}$ on bi-color paths $\mu=\mu_{2}\mu_{1}\in G_{D}^{2}$
according to the following rules.

Case 1: Assume $\mu_{1},\mu_{2}\notin\bigcup_{i=1}^{k}C_{D}^{e_{i}}$. Then we
set $[\mu]_{D}=\iota^{-1}([\iota(\mu)])$.

Case 2: Suppose $\mu_{j}$ lies in $C^{e_{1}}_{D}$, so that
$\mu_{j}\in\\{g^{1},g^{2}\\}$ for some edge $g\in C^{e_{1}}$. If $j=1$ and
$\mu_{1}=g^{2}$, then
$r(\mu_{1})=s(\mu_{2})=\iota^{-1}(r(g))\in\iota^{-1}(\Lambda^{0})$, and the
edges in $G_{D}$ with source in $\iota^{-1}(\Lambda^{0})$ and degree $e_{i}$
for $i\not=1$ are in $\iota^{-1}(\Lambda^{1})$. Therefore
$\mu_{2}\in\iota^{-1}(\Lambda^{e_{i}})$, and $\iota(\mu_{2})g$ is a bi-color
path in $G$, so $\iota(\mu_{2})g\sim ha$ for edges $h\in
C^{e_{1}},a\in\Lambda^{e_{i}}$. There is then an edge
$e_{[\mu_{2}g]}\in\Lambda^{e_{i}}_{D}$ with source $s(\mu_{1})=v_{g}$ and
range $v_{h}=s(h^{2})$; we define
$\mu_{2}\mu_{1}=\mu_{2}g^{2}\sim_{D}h^{2}e_{[\mu_{2}g]}$.

${\bullet}$${\bullet}$${\bullet}$${\bullet}$$\scriptstyle{\definecolor[named]{.}{rgb}{0,0,1}\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}a}$$\scriptstyle{g}$$\scriptstyle{h}$$\scriptstyle{\definecolor[named]{.}{rgb}{0,0,1}\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}b}$

$\longrightarrow$
${\bullet}$${\bullet}$${\bullet}$${\bullet}$${\bullet}$${\bullet}$$\scriptstyle{\definecolor[named]{.}{rgb}{0,0,1}\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}a}$$\scriptstyle{g^{1}}$$\scriptstyle{h^{1}}$$\scriptstyle{\definecolor[named]{.}{rgb}{0,0,1}\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}e_{[bg]}}$$\scriptstyle{g^{2}}$$\scriptstyle{h^{2}}$$\scriptstyle{\definecolor[named]{.}{rgb}{0,0,1}\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}b}$

Figure 5.6: A commuting square in $G$ and its “children” in $G_{D}$, when
$h,g\in C^{e_{1}}$.

If $j=1$ and $\mu_{1}=g^{1}$, the only edges in $G_{D}$ with source
$r(g^{1})=v_{g}$ and degree $e_{i}$ for $i\not=1$ are of the form
$e_{[bg]}=e_{[ha]}$ for some commuting square $bg\sim ha$ in $\Lambda$. In
this case, we will have $h\in C^{e_{1}}$, and $r(h^{1})=v_{h}=r(e_{[bg]})$, so
we set $e_{[bg]}g^{1}\sim_{D}h^{1}a$.

A similar argument shows that if $j=2$, the path $\mu_{2}\mu_{1}$ will be of
the form $h^{1}a$ or $h^{2}e_{[ha]}$, whose factorizations we have already
described.

Case 3: Assume $\mu$ is of the form $e_{\beta}e_{\alpha}$ for $\alpha\in
C_{D}^{e_{i}}$, and $\beta\in C_{D}^{e_{j}}$ with $i\neq j$. Now
$s_{D}(e_{\beta})=r_{D}(e_{\alpha})=v_{g}$ for some $g\in C^{e_{1}}$, and
consequently $\alpha,\beta\in\Lambda$ are linked as shown on the left of
Figure 5.7. Since $\Lambda$ is a $k$-graph, the 3-color path outlining
$\beta\alpha$ generates a 3-cube in $\Lambda$, which is depicted on the right
of Figure 5.7.

${\bullet}$${\bullet}$${\bullet}$${\bullet}$${\bullet}$${\bullet}$$\scriptstyle{\definecolor[named]{.}{rgb}{0,0,0}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}g}$${\beta}$${\alpha}$

$~{}~{}~{}\sim~{}~{}~{}$
${\bullet}$${\bullet}$${\bullet}$${\bullet}$${\bullet}$${\bullet}$${\bullet}$${\bullet}$${\beta}$${\alpha}$${\gamma}$${\delta}$

Figure 5.7: The commuting squares of edges from
$\bigcup_{i=2}^{k}C_{D}^{e_{i}}$.

Let $\delta$ and $\gamma$ denote the faces of this cube which lie,
respectively, opposite $\beta$ and $\alpha$. Since $g\in C^{e_{1}}$, all of
the vertical edges of this cube are in $C^{e_{1}},$ and so $\delta\in
C^{e_{j}},\gamma\in C^{e_{i}}.$ Moreover, the path $e_{\gamma}e_{\delta}$ is
composable in $\Lambda_{D}$, and has the same source and range as
$e_{\beta}e_{\alpha}$. Set $e_{\beta}e_{\alpha}\sim_{D}e_{\gamma}e_{\delta}$.

Before proving Proposition 5.4 it is important to point out that [EFG+22,
Theorem 4.2, 4.3] showed that $\Lambda_{D}$ is a well defined $k$-graph and
$C^{*}(\Lambda_{D})\sim_{ME}C^{*}(\Lambda)$.

###### Proof of Proposition 5.4.

In this proof, all edges and vertices are in $\Lambda_{D}$. So, for notation’s
sake we will write $s$ and $r$ instead of $s_{D}$ and $r_{D}$.

  1. (I)

This will follow directly from the definitions of $s$ and $r$ put forth above.
Observe that $v_{f}\Lambda^{E}_{D}s(f)=\\{f_{1}\\}$ and this edge has color
$e_{1}$. Note that $r(s^{-1}_{e_{1}}(v_{f}))=r(f^{2})=r(f)\neq v_{f}$.
Further, $r^{-1}_{e_{1}}(v_{f})=\\{f^{1}\\}$ as required. Finally, we see that
$s(r^{-1}_{e_{1}}(v_{f}))=s(f^{1})=s(f)$. By Definition 3.4, we see that
$v_{f}$ is reducible with color set $\\{e_{1}\\}$.

  2. (II)

We first consider $\Phi^{*}_{\mathcal{E}}(\Lambda_{D})$. This is the pullback
of $\Lambda_{D}$ under the map
$\Phi_{\mathcal{E}}:\mathbb{N}^{k-1}\to\mathbb{N}^{k}$ given by
$\Phi_{\mathcal{E}}(e_{i})=e_{i+1}$. This, in effect, deletes all edges of
degree $e_{1}$. In particular, this removes the edges in $C^{e_{1}}_{D}$.
Notice from the given definitions of $s$ and $r$ these are the only edges with
source $v_{g}$, for some $g\in C^{e_{1}}$, and range in $\Lambda$. Further,
every edge of the connected component,
$(\Phi_{\mathcal{E}}^{*}(\Lambda_{R}))_{v_{f}}$, is contained in
$C^{e_{i}}_{D}$ for some $1<i\leq k$, since the inductive definition is
inherently undirected. This returns the desired equality.

  3. (III)

Suppose that $g\in C^{e_{1}}$. By definition, there exists some edge $a$ with
$d(a)\neq e_{1}$ such that $ga$ is a well defined path. In particular,
$[ga]\in C^{d(a)}$. So $e_{[ga]}\in C^{d(a)}_{D}$ and further
$r(e_{[ga]})=v_{g}$ and $s(e_{[ga]})=v_{h}$ for some $h\in C^{e_{1}}$. We
conclude that
$(U_{v_{f}}^{\mathcal{E}})^{0}=s\left((U_{v_{f}}^{\mathcal{E}})^{E}\right)\cup
r\left((U_{v_{f}}^{\mathcal{E}})^{E}\right)=\\{v_{g}:g\in C^{e_{1}}\\}$. From
here we need only notice that a given $v_{g}$ is reducible to $s(g)$ with
color set $\\{e_{1}\\}$ by way of $g^{1}$, following the argument of (I).

  4. (IV)

This follows from the argument above that for any $v_{g}$ we have
$r^{-1}_{e_{1}}(v_{g})=g^{1}$. Further note from the definition of stationary
that a singleton edge connecting $s(g)$ and $v_{g}$ is vacuously stationary.

  5. (V)

Recall that $\Phi_{\mathcal{E}}^{*}(\Lambda_{D})$ removed the edges
$\\{g^{1}:g\in C^{e_{1}}\\}$ and so the connected component
$(\Phi_{\mathcal{E}}^{*}(\Lambda_{D}))_{s(f)}$ cannot contain any of the
edges, or vertices added to $\Lambda$ to form $\Lambda_{D}$.

∎

With D it is clear that objects are added to the $k$-graph while R removes
edges and vertices. This means that its relationship with R is clearly not a
simple matter of $\Lambda_{D}\cong\Lambda_{R}$. Instead we recall that in
[ERRS21] the sequences of graph moves between $E$ and $F$ can utilize these
moves or their inverses (i.e. the path between $E$ and $F$ is possibly
undirected). This means that if we show $(\Lambda_{D})_{R}\cong\Lambda$, then
we may conclude that the Morita equivalence between
$C^{*}(\Lambda_{D})\sim_{ME}C^{*}(\Lambda)$ can be thought of as a direct
consequence of the fact that
$C^{*}((\Lambda_{D})_{R})\sim_{ME}C^{*}(\Lambda_{D})$. To this end we present
the following proposition.

######  Proposition 5.8.

For any $k$-graph, $\Lambda$, if $\Lambda_{D}$ is obtained by delaying
$f\in\Lambda^{e_{1}}$ and reduction is performed from $v_{f}$ to $s(f)$ to
obtain $(\Lambda_{D})_{R}$, we have that $(\Lambda_{D})_{R}\cong\Lambda$.

###### Proof.

From Proposition 5.4 we know that $(\Lambda_{D})_{R}$ is well defined. Further
we may note that

$\displaystyle(\Lambda_{D})_{R}^{0}$
$\displaystyle=\Lambda_{D}^{0}\setminus\\{v_{g}:g\in
C^{e_{1}}\\}=(\Lambda^{0}\cup\\{v_{g}:g\in C^{e_{1}}\\})\setminus\\{v_{g}:g\in
C^{e_{1}}\\}=\Lambda^{0}$ $\displaystyle(\Lambda_{D})_{R}^{E}$
$\displaystyle=\Lambda_{D}^{E}\setminus((U_{v_{f}}^{\mathcal{E}})^{E}\cup\\{g^{1}:g\in
C^{e_{1}}\\})=\Lambda_{D}^{E}\setminus\left(\left(\displaystyle{\bigcup_{i=2}^{k}C_{D}^{e_{i}}}\right)\cup\\{g^{1}:g\in
C^{e_{1}}\\}\right)$ $\displaystyle=(\Lambda^{E}\setminus\\{g\in
C^{e_{1}}\\})\cup\\{g^{2}:g\in C^{e_{1}}\\}$

At first it is worrying that we do not recover $\Lambda^{E}$ exactly, but this
can be easily remedied by the identification $g^{2}\mapsto g$. This is the
right thing to do, because
$r_{R}(g^{2})=r_{D}(\iota_{R}(g^{2}))=r_{D}(g^{2})=r(g)$. Recall also that
$g^{2}\in(\Lambda_{D})_{R}$ is such that $s_{D}(\iota_{R}(g^{2}))=v_{g}$ and
thus $s_{R}(g^{2})=s_{D}(g^{2}g^{1})=s(g)$.

Being more precise, there is a bijective functor between the path categories
$\phi:G^{*}((\Lambda_{D})_{R})\to G(\Lambda)^{*}$ such that for all $g^{2}\in
C^{e_{1}}_{D}$ we have $\phi(\iota_{R}^{-1}(g^{2}))=g$ and, for the remaining
edges, $\mu\mapsto\iota_{D}(\iota_{R}(\mu))$. It remains to show that this
identification respects the relations $\sim_{DR}$ and $\sim$. That is, given
$\mu\sim_{DR}\nu$ then $\phi(\mu)\sim\phi(\nu)$. This proof relies primarily
on unpacking the definition of $\sim_{DR}$ and observing that $\phi$ acts this
way by construction.

Using the notation from Remark 4.5, we may observe that
$\Theta=\\{g^{2}g^{1}:g\in C^{e_{1}}\\}$. In particular this means that

$\operatorname{par}(\mu)=\begin{cases}g^{2}g^{1},&\iota_{R}(\mu)=g^{2}\text{
for some $g\in C^{e_{1}}$}\\\ \iota_{R}(\mu),&\text{otherwise.}\end{cases}$

Let’s consider some path of length $2$, $\mu=\mu_{2}\mu_{1}$, in
$(\Lambda_{D})_{R}$ and observe that
$[\mu]_{DR}=[\operatorname{par}(\mu)]_{D}$. We will investigate cases based on
whether $\operatorname{par}(\mu)\in\Gamma^{*}$ or
$\operatorname{par}(\mu)\notin\Gamma^{*}$.

Suppose that $\operatorname{par}(\mu)\in\Gamma^{*}$. Then
$\operatorname{par}(\mu)=\iota_{R}(\mu)$. Further, since
$\operatorname{par}(\mu)\in\Gamma^{*}$, we conclude that
$s_{D}(\iota_{R}(\mu_{1})),s_{D}(\iota_{R}(\mu_{2}))\notin\\{v_{g}:g\in
C^{e_{1}}\\}$ and thus we fall under Case 1 for $\sim_{D}$. We conclude that:

$[\mu]_{DR}=[\operatorname{par}(\mu)]_{D}=[\iota_{R}(\mu)]_{D}=[\iota_{D}(\iota_{R}(\mu))]=[\phi(\mu)].$

Now suppose that either $\operatorname{par}(\mu_{1})\in\Theta$ or
$\operatorname{par}(\mu_{2})\in\Theta$. We will closely examine the former,
and note that a similar argument holds for the latter. If
$\operatorname{par}(\mu_{1})\in\Theta$ then $\mu_{1}=\iota^{-1}_{R}(g^{2})$
for some $g\in C^{e_{1}}$ and
$\operatorname{par}(\mu_{2})=\iota_{R}(\mu_{2})$. So,
$\operatorname{par}(\mu)=g^{2}g^{1}\iota_{R}(\mu_{2})$. By the construction of
Case 2 we know that

$[\mu_{2}\mu_{1}]_{DR}=[\iota_{R}(\mu)g^{2}g^{1}]_{D}=[\iota_{D}(\iota_{R}(\mu_{2}))g]=[\phi(\mu_{2})\phi(\mu_{1}))].$

We conclude that $\phi$ induces an isomorphism of $k$-graphs
$\Lambda\cong(\Lambda_{D})_{R}$.

∎

From the above results we see that the moves CR and D from [EFG+22] are
special cases of the move R developed in this paper. This allows us to update
the list of necessary Morita equivalence preserving $k$-graph moves.

  * •

Insplitting (I) [EFG+22, §3]

  * •

Outsplitting (O) [Lisar, §4]

  * •

Sink Deletion (S) [EFG+22, §6]

  * •

Reduction (R)

As previously mentioned, the moves S and D are robust under products,
Definition 2.16. If R is meant to replace D, it would be preferable that this
move too was robust under products. We end by showing that the move R
presented in this paper is robust under products.

######  Proposition 5.9.

If $\Lambda_{1}$ is a $k$ graph such that $w$ is fully reducible with color
set $B$, $U_{w}^{\mathcal{E}}\cap U_{v}^{\mathcal{E}}=\emptyset$, and
$\mathcal{F}_{x}$ is stationary for all $x\in(U_{w}^{\mathcal{E}})^{0}$, then
for any $\ell$-graph $\Lambda_{2}$ and any vertex $y\in\Lambda_{2}^{0}$ the
vertex $(w,y)\in\Lambda_{1}\times\Lambda_{2}$ satisfies the reduction
hypotheses. Further,

$(\Lambda_{1})_{R}\times\Lambda_{2}\cong(\Lambda_{1}\times\Lambda_{2})_{R}.$

###### Proof.

For sake of notation we will partition the standard basis of
$\mathbb{N}^{k+\ell}$ into the sets $E_{1}=\\{e_{1},\dots,e_{k}\\}$ and
$E_{2}=\\{e_{k+1},\dots,e_{k+\ell}\\}$. We will abuse notation for $r$ and $s$
slightly, but the context should be enough to guide the reader.

Let $y\in\Lambda_{2}^{0}$. We claim that the vertex
$(w,y)\in\Lambda_{1}\times\Lambda_{2}$ is reducible with color set $B$. Since
$w$ was fully reducible with color set $B$ in $\Lambda_{1}$, we conclude, in
particular, that there exists a vertex $v\in\Lambda_{1}$ satisfying the
conditions of Definition 3.4. Consider $(v,y)\in\Lambda_{1}\times\Lambda_{2}$.

Since $s$ and $r$ are performed coordinate-wise note that
$(w,y)(\Lambda_{1}\times\Lambda_{2})^{E}(v,y)=w\Lambda_{1}^{E}v\times\\{y\\}$
and so condition one of reducibility is inherited directly. Next observe that
since $B\subseteq E_{1}$, we have the set equality
$r^{-1}_{B}(w,y)=r_{B}^{-1}(w)\times\\{y\\}$, and likewise for $s^{-1}_{B}$.
We conclude that $(w,y)$ inherits reducibility with color set $B$ directly
from $w$.

Note that since $y$ was arbitrary the vertices $\\{w\\}\times\Lambda_{2}^{0}$
are also reducible with color set $B$. Next define the set
$\mathcal{E}=E_{1}\setminus B$ and
$\mathcal{E}^{\prime}=\\{e_{1},\dots,e_{k+\ell}\\}\setminus B=\mathcal{E}\cup
E_{2}$. We must consider the neighborhood $U_{(w,y)}^{\mathcal{E}^{\prime}}$.
Observe that since $\mathcal{E}^{\prime}=\mathcal{E}\cup E_{2}$ we may
conclude that
$\Phi_{\mathcal{E}^{\prime}}^{*}(\Lambda_{1}\times\Lambda_{2})=\Phi_{\mathcal{E}}^{*}(\Lambda_{1})\times\Lambda_{2}$.
In particular, this means that the connected component containing $(w,y)$ is
the connected component
$\Phi_{\mathcal{E}}^{*}(\Lambda_{1})_{w}\times\Lambda_{2}$. We conclude that
$\Upsilon_{(w,y)}^{\mathcal{E}^{\prime}}=\Upsilon_{w}^{\mathcal{E}}\times\Lambda_{2}$.
That is,
$(\Upsilon_{(w,y)}^{\mathcal{E}^{\prime}})^{0}=(\Upsilon_{w}^{\mathcal{E}})^{0}\times\Lambda_{2}^{0}$.
Since each $x\in(\Upsilon_{w}^{\mathcal{E}})^{0}$ is reducible with color set
$B$ we conclude that $(w,y)$ is fully reducible with color set $B$.

The disjointness follows immediately from the facts that
$\Upsilon_{(v,y)}^{\mathcal{E}^{\prime}}=\Upsilon_{v}^{\mathcal{E}}\times\Lambda_{2}$
and $\Upsilon_{v}^{\mathcal{E}}\cap\Upsilon_{w}^{\mathcal{E}}=\emptyset$ by
hypothesis. The stationary condition follows from
$\mathcal{F}_{(x,y)}=\mathcal{F}_{x}\times\\{y\\}$.

We conclude that $(\Lambda_{1}\times\Lambda_{2})_{R}$ is well defined. The
final part is to show the isomorphism

$(\Lambda_{1})_{R}\times\Lambda_{2}\cong(\Lambda_{1}\times\Lambda_{2})_{R}.$

We fix some $b\in B$, and define
$\operatorname{par}_{1}:(\Lambda_{1})_{R}\to\Lambda_{1}$ and
$\operatorname{par}_{2}:(\Lambda_{1}\times\Lambda_{2})_{R}\to\Lambda_{1}\times\Lambda_{2}$.
Consider the following diagram.

${(\Lambda_{1})_{R}\times\Lambda_{2}}$${\Lambda_{1}\times\Lambda_{2}}$${(\Lambda_{1}\times\Lambda_{2})_{R}}$$\scriptstyle{\operatorname{par}_{1}\times\iota}$$\scriptstyle{\operatorname{par}_{2}}$

We need only demonstrate that
$\operatorname{par}_{2}=\operatorname{par}_{1}\times\iota$ to obtain our
isomorphism. Recall that $\operatorname{par}_{2}$ is multiplicative so
$\operatorname{par}_{2}(\lambda,\mu)=\operatorname{par}_{2}(\lambda,r(\mu))\operatorname{par}_{2}(s(\lambda),\mu)$.
Consider $(s(\lambda),\mu)$. Since
$(s(\lambda),s(\mu))\in(\Lambda_{1}\times\Lambda_{2})^{0}_{R}$, we may
conclude that $s(\lambda)\notin U_{w}^{\mathcal{E}}$ and thus
$\operatorname{par}_{2}(s(\lambda),\mu)=(\iota(s(\lambda)),\iota(\mu))$. Next
consider $\operatorname{par}_{2}(\lambda,r(\mu))$ and notice that
$s(\iota(\lambda),\iota(r(\mu))=(s(\iota(\lambda)),\iota(r(\mu)))$. This means
that the output of $\operatorname{par}_{2}$ is solely dependent of
$\iota(\lambda)$. We conclude that
$\operatorname{par}_{2}(\lambda,r(\mu))=(\operatorname{par}_{1}(\lambda),\iota(r(\mu))$.
This gives us
$\operatorname{par}_{2}(\lambda,\mu)=(\operatorname{par}_{1}(\lambda),\iota(\mu))$
as desired. ∎

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