We develop a comprehensive string diagrammatic treatment of electrical circuits.
Building on previous, limited case studies, we introduce controlled sources and meters as elements,
and the impedance calculus, a powerful toolbox for diagrammatic reasoning on circuit
diagrams. We demonstrate the power of our approach by giving comprehensive proofs of several
textbook results, including the superposition theorem and Thévenin's theorem.

§ INTRODUCTION

Classical electrical circuit theory [7], and the ubiquitous use of electrical circuits in the daily life of an engineer are testament to their power as a compositional, diagrammatic abstraction, denoting actual physical systems while at the same time intuitively reflecting network interconnections. Like other formalisms (e.g. signal flow graphs), theorems about circuits are proved by translating them—sometimes not entirely faithfully—to more traditional mathematics, with linear algebra playing a particularly important role, and compositionality taking a back seat. Nevertheless, diagrammatic reasoning at the level of circuits is the remit of several classical theorems, e.g. Thévenin's theorem or Norton's theorem.

We build on recent work [1, 6, 3]
that treats circuits compositionally. In particular, we use Coya's semantics [6]
that captures non-passive elements such as sources, which was subsequently axiomatised in the string-diagrammatic setting of Graphical Affine Algebra (GAA) [3], also building on the categorical approach to relational algebra developed by Carboni and Walters [5].

This is our starting point. Staying with affine relations as our ambient denotational universe, we enlarge the set of basic electrical elements with controlled sources and meters: voltmeters and ammeters. This lets us study both open and closed circuits; the latter have been ignored in existing compositional approaches since a closed circuit has only two possible denotations as an affine relation.

Our second innovation is what we call the impedance calculus. GAA and circuit syntax are intermingled via the use of impedance boxes: circuit elements that hold within them a GAA diagram. They reveal the symmetries of classical circuit elements more clearly than ordinary GAA diagrams, but they are more than just of aesthetic value. Indeed, they allow us to solve another common problem of string diagrammatic approaches: having to deal with large diagrams. Using the algebra of impedance boxes allows us to “compile” parts of the circuit to GAA, simplify, and translate back into circuit syntax, dramatically reducing the complexity of diagrammatic reasoning when solving typical textbook circuit problems.

Armed with our richer set of circuits and the impedance calculus, we are able to prove most of the well-known textbook (e.g. [7]) theorems of elementary circuit theory. For space reasons we have not been able to include many; nevertheless, we hope that the assortment on offer will give the reader a taste of the power of diagrammatic reasoning with the impedance calculus.
On the one hand, the diagrammatic syntax allows for rigorous yet elegant statements of results about circuits, often   revealing the underlying symmetries –  e.g. the independent measurement principle (Theorem <ref>) vs the Superposition Theorem (Theorem <ref>); the principle of relativity of potentials (Proposition <ref>) vs the principle of conservation of currents (Proposition <ref>). On the other, we show that the compositional framework—with its use of diagrammatic reasoning and the algebra of cartesian and abelian categories of relations—leads to elegant and  rigorous proofs. Unfortunately, most have been moved to the appendix because of space restrictions.

§ GRAPHICAL AFFINE ALGEBRA AND ELECTRICAL CIRCUITS

We begin with the basics of Graphical Linear Algebra (GLA) with its equational theory of Interacting Hopf Algebras ($\IH$) [4]. Fix a field $\mathsf{k}$.
GLA is a string diagrammatic syntax, which organises itself as the arrows of the free prop $\mathsf{GLA}_{\mathsf{k}}$ over the following monoidal signature:

\begin{align}%\label{eq:SFcalculusSyntax}
\{\;&\Bcounit,\ \Bcomult,\  \scalar,\ \Wmult,\ \Wunit,\  \label{eq:GLAsyntax1} \\
&\Bunit,\ \Bmult,\ \scalarop,\ \Wcomult,\ \Wcounit \;\}  \label{eq:GLAsyntax2}
\end{align}

where $k\in\mathsf{k}$ is a scalar.
Intuitively the black structure can be thought of as copying, the white as adding. This is borne out by the intended semantics, which we describe next.

The prop of linear relations $\LinRel{\mathsf{k}}$ has as arrows $m\to n$ relations $R\subseteq \mathsf{k}^m\times\mathsf{k}^n$, which are $\mathsf{k}$ vector spaces; i.e. closed under $\mathsf{k}$-linear combinations. In other words, arrows $m\to n$ are linear subspaces of $\mathsf{k}^m\times\mathsf{k}^n$, considered as a $\mathsf{k}$-vector space. Composition is standard relational composition, $R\mathrel{;}S = \{\, (u,w) \;|\; \exists v.\; (u,v)\in R \ \wedge\ (v,w)\in S\,\}$.
It is simple to show linear relations are closed under composition.

The semantics of GLA is a prop morphism$[-]_{\mathsf{k}}\mathrel{:}\mathsf{GLA}_\mathsf{k} \rightarrow \LinRel{\mathsf{k}}$.
Since $\mathsf{GLA}_\mathsf{k}$ is free, it is enough to describe its action on the generators.
We do so below for the generators in <ref>;
the corresponding generators in <ref> are sent to the opposite relations. In each case the variables range over $\mathsf{k}$.

\[
\begin{array}{lllll}
[\Bcomult]_\mathsf{k} = \left\{\,\left(x,\, \scriptstyle {\scriptstyle x \choose \scriptstyle x}\right)%\,|\,x\in\mathsf{k}
\,\right\},
[\Bcounit]_\mathsf{k} = \{\,(x,\, \bullet)%\,|\,x\in \mathsf{k}
\,\},
[\scalar]_\mathsf{k} = \{\,(x,\, kx)%\,|\,x\in \mathsf{k}
\,\},
\\[5pt]
{[}\Wmult]_\mathsf{k} = \left\{\,\left(\scriptstyle {\scriptstyle x \choose \scriptstyle y},\, x+y\right)%\,|\,x,y\in\mathsf{k}
\,\right\},
[\Wunit]_\mathsf{k} = \{\,(\bullet,\, 0)\}
\end{array}
\]

The associated theory $\IH$ characterises linear relations. We give a brief overview below:

* both monoids $(\Bmult,\,\Bunit)$ $(\Wmult,\,\Wunit)$ and comonoids $(\Bcomult,\,\Bcounit)$, $(\Wcomult,\,\Wunit)$ are commutative

* monoids and comonoids of the opposite colour satisfy the equations of commutative bialgebras

* monoids and comonoids of the same colour satisfy the extra special Frobenius equations

* to pass between black and white cups and caps is to compose with $-1$. We shall often draw the $-1$ scalar as $\antipode$.

* all non-zero scalars are invertible, with the inverse
of $\scalar$ given by  $\scalarop$ for $k\neq 0$.

Graphical Affine Algebra was introduced in [3],

extending the expressivity of GLA to affine relations.

Let us recall the main concepts.
A translation $v+V$ of a linear subspace $V$ by a vector $v$ is the set $v+V=\{\, v+w \;|\; w\in V\,\}$.
An affine subspace $W$

is either empty, or it is the translation $v+V$ for some vector $v$ and subspace $V$. Note that the empty set is an affine subspace, but it is not a linear subspace.

The prop of affine relations $\AffRel{\mathsf{k}}$ has as
affine subspaces of $\mathsf{k}^m\times\mathsf{k}^n$ as arrows $m\to n$. It is the case that the composition of two affine relations yields an affine relation, and can be shown using the notion of homogenisation <cit.>, but we will not delve into the details here.

On the syntactic side, we extend the signature (<ref>), (<ref>) with a single generator

\begin{equation}\label{eq:GAAsyntax}
\One
\end{equation}

obtaining a richer string diagrammatic syntax as the arrows of the free prop $\mathsf{GAA}$. Abusing notation, the semantics
$[-]_\mathsf{k} \mathrel{:} \mathsf{GAA} \rightarrow \AffRel{\mathsf{k}}$ is defined the same way as for $\mathsf{GLA}$ on the shared generators, and $[\One]_\mathsf{k} = \{\,(\bullet,\, 1)\}$. To characterise $\AffRel{\mathsf{k}}$ we add the following equations:

\[
\tikzfig{one-copy} \quad\myeq{dup}\quad \tikzfig{one2} \qquad
\tikzfig{one-delete} \quad\myeq{del}\quad \tikzfig{empty-diag} \qquad
\tikzfig{one-false} \quad\myeq{empty}\quad \tikzfig{one-false-disconnect}
\]

Instead of using relying purely on equational reasoning, it is often convenient to work with inequations, which can lead to shorter calculations, and allow us to identify some interesting, higher-level categorical structures. Instead of working with ordinary props, one works with ordered props, that is props enriched over the category of posets. Similarly any equational theory can be presented as an inequational theory in the obvious way, by replacing an equation by two inequations.

Indeed, $\LinRel{\mathsf{k}}$
and $\AffRel{\mathsf{k}}$ can be considered as ordered props by using set-theoretical inclusion of relations as the homset order. On the syntactic side, it suffices [2] to add a single inequation:

\[
\Wunit \leq \Bunit
\]

The ordered setting also lends itself to higher level reasoning schema, in particular, $\LinRel{\mathsf{k}}$ is an abelian bicategory of relations and $\AffRel{\mathsf{k}}$ is a cartesian bicategory of relations, concepts developed in [5].

§.§ The prop of electrical circuits and its semantics

We recall the string diagrammatic development of electrical circuits
from [3],
with minor modifications to suit our development in subsequent sections.
The prop $\circprop$ is free on the following signature:

\begin{equation}\label{eq:circpropsig}
\left\{\resistor ,\, \vsource ,\, \csource,\, \inductor,\, \capacitor \right\}_{R,L,C\in\R_+,\,V,I\in\R} \cup \quad \left\{\junction,\, \circcounit,\, \cojunction,\, \circunit \right\}
\end{equation}

where the parameters range over the reals. Arrows $m\to n$ of $\circprop$ represent open linear electrical circuits with $m$ open terminals on the left
and $n$ open terminals on the right. Generator $\resistor$ represents a resistor, $\vsource$ a voltage source, $\csource$ a current source,
$\inductor$ an inductor and
$\capacitor$ a capacitor.

Circuits in $\circprop$ are translated to $\mathsf{GAA}$ over $\R(x)$: the field of fractions of polynomials with real coefficients (see [6, 3]).
The semantics is a strict monoidal functor
$\mathcal{I}:\circprop \to \mathsf{GAA}$ where, on objects $\mathcal{I}(1)=2$: the idea is that every electrical wire is represented by two $\mathsf{GAA}$ wires, from top to bottom, the voltage wire and the current wire.
We give the semantics in Figure <ref>, by showing the action on generators. Given circuits $c,d$ of $\circprop$ we write $c \semEq d$ when $\semElec{c}=\semElec{d}$ (in the equational theory of $\mathsf{GAA}$, or equivalently as affine relations in $\AffRel{\R(x)}$). Similarly, we write $c \semLeq d$ when $\semElec{c}\leq\semElec{d}$.

\[
\elementSemantics
{\phi_2-\phi_1 = Ri}
\]

\[
\elementSemantics
\elementSemantics
\]

\[
\!\!\!\!\!\!\!
\elementSemantics
{\phi_2-\phi_1 = Lxi}
\elementSemantics
{i = Cx(\phi_2-\phi_1)}
\]

\[
\electricalSemantics
{{\phi \choose i_1}, \scriptscriptstyle \left(\begin{smallmatrix} \scriptscriptstyle \phi \\ \scriptscriptstyle i_2 \\ \scriptscriptstyle \phi \\ \scriptscriptstyle i_3 \end{smallmatrix}\right)}
{i_1 + i_2 + i_3 = 0}
\electricalSemantics
{{\scriptscriptstyle \left(\begin{smallmatrix} \scriptscriptstyle \phi \\ \scriptscriptstyle i_2 \\ \scriptscriptstyle \phi \\ \scriptscriptstyle i_3 \end{smallmatrix}\right)}, {\phi \choose i_1}}
{i_1 + i_2 + i_3 = 0}
\]

\[
\electricalSemantics
{{\phi \choose i}, \bullet}
{i = 0}
\electricalSemantics
{\bullet, {\phi \choose i}}
{i = 0}
\]

Compositional semantics of circuits.

§ THE IMPEDANCE CALCULUS

We now exploit a pattern of the semantics in Figure <ref>
to simplify the passage between circuits and $\mathsf{GAA}$. This results in the impedance calculus, which can be used to simplify diagrammatic reasoning on circuits.

We extend the syntax (<ref>) of $\circprop$ with impedance boxes—illustrated below left—parametrised wrt arbitrary $\mathsf{GAA}$ circuits of type $\sort{1}{1}$: that is, with one wire on the left and one on the right. We then extend the semantic mapping $\semElec{\cdot}$ to cover impedance boxes, as below right.

\begin{equation}\label{eq:impedanceBox}
\impedanceBox,
\qquad \qquad
\semElec{\impedanceBox} \quad = \quad \lower10pt\hbox{$\includegraphics[height=1cm]{graffles/impedanceGLA.pdf}$}
\end{equation}

The following are now easy derivations in the equational theory of GAA:

\[
\begin{array}{rccrccrcc}
\wire       &\semEq& \lower13pt\hbox{$\includegraphics[height=1cm]{graffles/identityImpedance.pdf}$},
&\vsource   &\semEq& \lower13pt\hbox{$\includegraphics[height=1cm]{graffles/voltageImpedance.pdf}$},
&\csource   &\semEq& \lower13pt\hbox{$\includegraphics[height=1cm]{graffles/currentImpedance.pdf}$},
\\
\resistor   &\semEq& \lower13pt\hbox{$\includegraphics[height=1cm]{graffles/resistorImpedance.pdf}$},
&\inductor  &\semEq& \lower13pt\hbox{$\includegraphics[height=1cm]{graffles/inductorImpedance.pdf}$},
&\capacitor &\semEq& \lower13pt\hbox{$\includegraphics[height=1cm]{graffles/capacitorImpedance.pdf}$}.
\end{array}
\]

We now prove results that give the impedance calculus its power and allow us to manipulate impedance
boxes within circuits.
Henceforward we will use the syntactic sugar
$\lower6pt\hbox{$\includegraphics[height=.8cm]{graffles/cup.pdf}$} \Defeq \lower6pt\hbox{$\includegraphics[height=.8cm]{graffles/cupdef.pdf}$}$
$\lower6pt\hbox{$\includegraphics[height=.8cm]{graffles/cap.pdf}$} \Defeq \lower6pt\hbox{$\includegraphics[height=.8cm]{graffles/capdef.pdf}$}$.

\[\begin{array}{rcccrccc}
\text{(i)}&
\lower10pt\hbox{$\includegraphics[height=.7cm]{graffles/impedancecompLHS.pdf}$}
\lower17pt\hbox{$\includegraphics[height=1.2cm]{graffles/impedancecompRHS.pdf}$}
\text{(ii)}&
\lower18pt\hbox{$\includegraphics[height=1.5cm]{graffles/impedanceparLHS.pdf}$}
&\semEq &
\lower17pt\hbox{$\includegraphics[height=1.2cm]{graffles/impedanceparRHS.pdf}$}
\\
\text{(iii)}&
\lower17pt\hbox{$\includegraphics[height=1.3cm]{graffles/SLHS.pdf}$}
\lower8pt\hbox{$\includegraphics[height=.7cm]{graffles/SRHS.pdf}$}
\text{(iv)}&
\semElec{\lower8pt\hbox{$\includegraphics[height=.7cm]{graffles/closedLHS.pdf}$}}
\lower4pt\hbox{$\includegraphics[height=.4cm]{graffles/closedRHS.pdf}$}
% \\
% \text{(v)}&
% \semElec{\lower16pt\hbox{$\includegraphics[height=1.4cm]{graffles/loopLHS.pdf}$}}
% &=&
% \lower4pt\hbox{$\includegraphics[height=.4cm]{graffles/loopRHS.pdf}$}
\end{array}\]

See <ref>.

Using <Ref>, (iii) we can immediately derive several useful properties of circuits:

* Registers, inductors and capacitors are “directionless”:

\[
\lower15pt\hbox{$\includegraphics[height=1.3cm]{graffles/Sresistor.pdf}$}
\semEq
\resistor,
\qquad
\lower15pt\hbox{$\includegraphics[height=1.3cm]{graffles/Sinductor.pdf}$}
\semEq
\inductor,
\qquad
\lower15pt\hbox{$\includegraphics[height=1.3cm]{graffles/Scapacitor.pdf}$}
\semEq
\capacitor.
\]

* Reversing the direction of voltage and current sources flips polarities:

\[
\lower15pt\hbox{$\includegraphics[height=1.3cm]{graffles/Svsource.pdf}$}
\semEq\
\lower5pt\hbox{$\includegraphics[height=.8cm]{graffles/negvsource.pdf}$}
\ \Defeq\
\lower5pt\hbox{$\includegraphics[height=.8cm]{graffles/flippedvsource.pdf}$},
\qquad\qquad
\lower15pt\hbox{$\includegraphics[height=1.3cm]{graffles/Scsource.pdf}$}
\semEq\
\lower5pt\hbox{$\includegraphics[height=.8cm]{graffles/negcsource.pdf}$}
\ \Defeq\
\lower5pt\hbox{$\includegraphics[height=.8cm]{graffles/flippedcsource.pdf}$}.
\]

The impedance calculus is useful for proving
circuit equivalences. The following proposition are just a few examples of classic equivalences
one would find in any textbook.

* $\lower5pt\hbox{$\includegraphics[height=.8cm]{graffles/r1r2series.pdf}$} \semEq
\lower5pt\hbox{$\includegraphics[height=.8cm]{graffles/r3.pdf}$}$ where
$\lower3pt\hbox{$\includegraphics[height=.4cm]{graffles/r3gla.pdf}$} =
\lower9pt\hbox{$\includegraphics[height=.8cm]{graffles/r1plusr2gla.pdf}$} =
\lower3pt\hbox{$\includegraphics[height=.4cm]{graffles/r1plusr2.pdf}$}$

* $\lower14pt\hbox{$\includegraphics[height=1.4cm]{graffles/r1r2par.pdf}$} \semEq
\lower5pt\hbox{$\includegraphics[height=.8cm]{graffles/r3.pdf}$}$ where
$\lower3pt\hbox{$\includegraphics[height=.4cm]{graffles/r3gla.pdf}$} =
\lower9pt\hbox{$\includegraphics[height=.8cm]{graffles/r1r2pargla.pdf}$} =
\lower2.5pt\hbox{$\includegraphics[height=.3cm]{graffles/thisismyfinalform.pdf}$}$

* $\lower14pt\hbox{$\includegraphics[height=1.4cm]{graffles/csourceparresistor.pdf}$} \semEq
\lower5pt\hbox{$\includegraphics[height=.8cm]{graffles/vsourceseriesresistor.pdf}$}$

* $\lower14pt\hbox{$\includegraphics[height=1.4cm]{graffles/vsourceparvsource.pdf}$} \semEq
\lower5pt\hbox{$\includegraphics[height=.8cm]{graffles/v1source.pdf}$}$  if $V1=V2$,
otherwise its semantics is $\emptyset$ (the empty relation)

It is useful to contrast our treatment with the classical approach.
(i) and (iii) are standard and often-used equivalences. Part
(ii) is known classically,
but $R3$ is typically given a formula like $R1 R2/(R1+R2)$.
Classical formulas, however, don't work for all values of $R1$ and $R2$,
whereas the graphical one does.
Given that it mirrors the case
for resistors in series,
we argue that the graphical formula is the more natural one.
Finally, in (iv), the empty case is usually excluded by classical treatments.
A textbook deems such a circuit degenerate
and ignores that case when proving theorems.
In $\mathsf{GAA}$ however, the empty relation is first-class
and our theorems uniformly also include the empty case.

Part (i) is a simple exercise in the use of the impedance calculus:

\[
\lower5pt\hbox{$\includegraphics[height=.8cm]{graffles/r1r2series.pdf}$} \semEq
\lower12pt\hbox{$\includegraphics[height=1cm]{graffles/r1r2impedance.pdf}$} \semEq
\lower12pt\hbox{$\includegraphics[height=1cm]{graffles/r1plusr2impedance.pdf}$} \semEq
\lower5pt\hbox{$\includegraphics[height=.8cm]{graffles/r3.pdf}$}.
\]

For part (ii), we have $\lower14pt\hbox{$\includegraphics[height=1.4cm]{graffles/r1r2par.pdf}$} \semEq
\lower20pt\hbox{$\includegraphics[height=1.7cm]{graffles/r1r2parimpedance.pdf}$} \semEq
\lower12pt\hbox{$\includegraphics[height=1cm]{graffles/r1r2parimpedance2.pdf}$}.$

\begin{multline*}
\lower9pt\hbox{$\includegraphics[height=.8cm]{graffles/r1r2pargla.pdf}$}
\lower9pt\hbox{$\includegraphics[height=.8cm]{graffles/r1goesleft.pdf}$}
\lower9pt\hbox{$\includegraphics[height=.8cm]{graffles/r2goesright.pdf}$}
\lower2.5pt\hbox{$\includegraphics[height=.3cm]{graffles/thisismyfinalform.pdf}$}
\end{multline*}

extracts the classical formula: because $R1$ and $R2$ are nonnegative,
either $R1+R2\neq 0$, and
is a scalar, or $R1=R2=0$, and the formula is equal to $0$.
In both cases the result is a scalar.

Part (iii) is another simple calculation:

\begin{multline*}
\lower12pt\hbox{$\includegraphics[height=1.4cm]{graffles/csourceparresistor.pdf}$}
\semEq \lower17pt\hbox{$\includegraphics[height=1.4cm]{graffles/ctov1.pdf}$}
\semEq \lower14pt\hbox{$\includegraphics[height=1.3cm]{graffles/ctov2.pdf}$}
\semEq \lower14pt\hbox{$\includegraphics[height=1.3cm]{graffles/ctov3.pdf}$} \\
\semEq \lower14pt\hbox{$\includegraphics[height=1.3cm]{graffles/ctov4.pdf}$}
\semEq \lower12pt\hbox{$\includegraphics[height=.9cm]{graffles/ctov5.pdf}$}
\semEq \lower5pt\hbox{$\includegraphics[height=.8cm]{graffles/vsourceseriesresistor.pdf}$}
\end{multline*}

For part (iv), we can simplify using the impedance calculus as follows:

\[
\lower14pt\hbox{$\includegraphics[height=1.4cm]{graffles/vsourceparvsource.pdf}$}
\semEq \lower18pt\hbox{$\includegraphics[height=1.4cm]{graffles/vsources2.pdf}$}
\semEq \lower18pt\hbox{$\includegraphics[height=1.4cm]{graffles/vsources3.pdf}$}
\]

Now $\lower9pt\hbox{$\includegraphics[height=.9cm]{graffles/v1v2convolution.pdf}$}$
is just $V1$ if $V1=V2$, and $\lower1pt\hbox{$\includegraphics[height=.2cm]{graffles/zerozero.pdf}$}$
otherwise. In that case, the circuit evaluates to
$\lower14pt\hbox{$\includegraphics[height=1cm]{graffles/vsources4.pdf}$}$ which is denoted by the empty relation.

We shall see that, as a consequence of our Representation Theorem (Theorem <ref>), any $\sort{\bluebullet}{\bluebullet}$ circuit can be represented by an impedance box.

§ MEASURING CLOSED CIRCUITS

Thus far, we have kept the language of circuits $\circprop$ and the language of
$\mathsf{GAA}$ neatly separated by impedance boxes (<ref>) and the prop morphism
$\semElec{\cdot}\mathrel{:}\circprop \to \mathsf{GAA}$. It's time to tear down the wall.

While impedance boxes are useful for open circuit calculations, engineers
often study closed circuits.
The problem is that these are mapped to $0 \to 0$ affine relations,
and there are only two: the singleton and the empty set.
In other words, the mapping only tells us whether the circuit has a solution,
and nothing else. Indeed, no interesting behavior can be observed in a closed circuit in our existing framework.

Classically, a closed circuit is annotated with names for the currents and voltage differences of interest.
In some presentations voltmeters and ammeters are even added as explicit elements.
This is exactly what we do: add meters to our syntax.
Instead of assigning names, however, our meters have an outgoing “information” wire, thought of as an ordinary $\mathsf{GAA}$ wire. Information wires
allow us to probe closed circuits, i.e. those with no open electric wires,
and to state and prove several theorems of circuit theory. Our diagrams will now have two types of wires: electric wires and information wires. Instead of keeping the props $\circprop$ and $\mathsf{GAA}$ separate, we unite the two and work with a coloured (multisorted) prop $\ecircprop$.

The coloured prop $\ecircprop$ has as objects words over the alphabet $\{\bluebullet,\,\bullet\}$. It is freely generated
by the union of all the generators ((<ref>), (<ref>), (<ref>))
of $\mathsf{GAA}$, the generators (<ref>) of $\circprop$, and the following:

\[
\{\,\voltmeter,\,\ampmeter,\,\cvsource,\,\ccsource,\,\lower12pt\hbox{$\includegraphics[height=1.4cm]{graffles/mnimpedanceBox.pdf}$}\,\}_{m,n\in \mathbb{N},\, c\from m+1\rightarrow n+1 \text{ in }\mathsf{GAA}}.
\]

These are a voltmeter $\voltmeter\typ\sort{\bluebullet}{\bullet\bluebullet}$, ammeter $\ampmeter\typ\sort{\bluebullet}{\bullet\bluebullet}$, controlled voltage source
$\cvsource\typ\sort{\bullet\bluebullet}{\bluebullet}$, controlled current source $\ccsource\typ\sort{\bullet\bluebullet}{\bluebullet}$ and a generalised impedance box
$\sort{\bullet^m\bluebullet}{\bullet^n\bluebullet}$, parametrised over arbitrary $\mathsf{GAA}$ diagrams $c\mathrel{:}m+1\to n+1$.

We abuse notation and denote the translation of $\ecircprop$ to $\mathsf{GAA}$ by $\mathcal{I}\from\ecircprop\rightarrow\mathsf{GAA}$.
On objects,
$\semElec{\bluebullet} = 2$, since—as before—en electrical wire is represented by two wires in $\mathsf{GAA}$,
but the information wire “is” a GAA wire, i.e. $\semElec{\bullet} = 1$. Next, $\mathcal{I}$ acts on the generators of $\mathsf{GAA}$ as identity and on the basic electrical components (<ref>) as described in Figure <ref>. We delay the translations of $\{\,\voltmeter,\,\ampmeter,\,\cvsource,\,\ccsource\,\}$ to later in this section and first focus on the generalised impedance boxes. Their translation is:

\[
\semElec{\lower12pt\hbox{$\includegraphics[height=1.4cm]{graffles/mnimpedanceBox.pdf}$}}
\quad = \quad
\lower20pt\hbox{$\includegraphics[height=1.8cm]{graffles/mnimpedanceboxGLA.pdf}$}.
\]

We again use $\semEq$ and $\semLeq$ as circuit relations to mean $=$ and $\leq$ on their translations to $\mathsf{GAA}$, using its (in)equational theory.
The properties of Lemma <ref> easily generalise to these extended impedance boxes. We omit the details, and only mention one simple, but useful fact: $\mathsf{GAA}$ diagrams can breach
impedance boxes. Suppose that $d\from m\rightarrow m'$, $e\from n\rightarrow n'$ in $\ecircprop$ are built up using only the $\mathsf{GAA}$ generators, then:

\[
\lower20pt\hbox{$\includegraphics[height=1.5cm]{graffles/leftbreachLHS.pdf}$}
\semEq
\lower20pt\hbox{$\includegraphics[height=1.7cm]{graffles/leftbreachRHS.pdf}$}
\ \text{and}\
\lower20pt\hbox{$\includegraphics[height=1.7cm]{graffles/rightbreachLHS.pdf}$}
\semEq
\lower20pt\hbox{$\includegraphics[height=1.5cm]{graffles/rightbreachRHS.pdf}$}.
\]

§.§ Meters and controlled sources

We now turn to the translations of the voltmeter and the ammeter.

\[
\semElec{\voltmeter} \quad = \quad \lower15pt\hbox{\includegraphics[height=1.5cm]{graffles/voltmeterGLA.pdf}}
\qquad\qquad
\semElec{\ampmeter} \quad = \quad \lower15pt\hbox{\includegraphics[height=1.5cm]{graffles/ampmeterGLA.pdf}}
\]

Note that
follows the conventions
used so far:
current and voltage difference are right to left.

We can finally observe the behavior of closed circuits!
Consider a simple circuit, with just a battery and a resistor. Having no
open electrical wires, it is closed, but has one outgoing information wire.
Mapping to $\mathsf{GAA}$, we get a diagram with one output, giving us the computed value for the current.

\begin{multline*}
\semElec{\lower15pt\hbox{\includegraphics[height=1.5cm]{graffles/measuring1.pdf}}}
\lower25pt\hbox{\includegraphics[height=2cm]{graffles/measuring2.pdf}}
\!\!=
\!\lower22pt\hbox{\includegraphics[height=1.8cm]{graffles/measuring3.pdf}}
\!\!\!\!\!=
\lower12pt\hbox{\includegraphics[height=1.2cm]{graffles/measuring4.pdf}}
\!\!\!\!=\
\lower2pt\hbox{\includegraphics[height=.25cm]{graffles/measuring5.pdf}}
\end{multline*}

Although elementary enough above, these kinds of calculations can be drastically simplified using the impedance calculus by keeping the GAA diagrams small, as we shall demonstrate below.

First let us turn to controlled sources.
Those are like independent sources, but their value is controlled by an information wire.
Their translations to $\mathsf{GAA}$ are given below.

\[
\semElec{\cvsource} \quad = \quad \lower12pt\hbox{\includegraphics[height=1.2cm]{graffles/cvsourceGLA.pdf}}\ ,\,\
\semElec{\ccsource} \quad = \quad \lower12pt\hbox{\includegraphics[height=1.2cm]{graffles/ccsourceGLA.pdf}}.
\]

As in the previous section, it's usually simpler to combine impedances directly
rather than map fully to GAA. This can be extended to work with meters too, giving
a convenient way to solve many circuits.
Using the generalised impedance boxes, we have:

\[
\voltmeter \semEq \lower10pt\hbox{\includegraphics[height=1.1cm]{graffles/voltmeterimpedance.pdf}}
\ \quad\
\ampmeter \semEq \lower10pt\hbox{\includegraphics[height=1.1cm]{graffles/ampmeterimpedance.pdf}}
\ \quad\
\ccsource \semEq \lower10pt\hbox{\includegraphics[height=1.1cm]{graffles/ccsourceimpedance.pdf}}
\ \quad\
\cvsource \semEq \lower10pt\hbox{\includegraphics[height=1.1cm]{graffles/cvsourceimpedance.pdf}}
\]

We can redo our simple example using this new technology:

\[
\lower10pt\hbox{\includegraphics[height=1.3cm]{graffles/simplified1.pdf}}
\semEq
\lower16pt\hbox{\includegraphics[height=1.6cm]{graffles/simplified2.pdf}}
\semEq
\lower18pt\hbox{\includegraphics[height=1.8cm]{graffles/simplified3.pdf}}
\semEq
\lower17pt\hbox{\includegraphics[height=1.7cm]{graffles/simplified4.pdf}}
\semEq
\lower12pt\hbox{\includegraphics[height=1.2cm]{graffles/simplified5.pdf}}
\]

and so

\[
\semElec{\lower14pt\hbox{\includegraphics[height=1.3cm]{graffles/simplified1.pdf}}}
\lower9pt\hbox{\includegraphics[height=.9cm]{graffles/simplified6.pdf}}
\lower3pt\hbox{\includegraphics[height=.3cm]{graffles/simplified7.pdf}}
\]

An important property of meters and sources is that they can be considered one at a time.
This is often used classically to keep calculations manageable.
The following two theorems make this

§.§ Independent Measurement Theorem

The first theorem is quite intuitive:
measuring somewhere in a circuit ought not to affect measurements elsewhere.
In other words, we can extract the full behavior of a circuit by considering measurements one at a time.
In fact, this is so natural that it is just assumed to be true classically,
and not mentioned in textbooks.
We are being more rigorous here:
the result is more subtle than one might think.

To get a single measurement we discard (plug $\Bcounit$ into) all meters but one.
It is easy to show that an ignored ammeter is equivalent to a wire, and an ignored voltmeter to an open circuit.

\[
\lower5pt\hbox{\includegraphics[height=.7cm]{graffles/ampmeterignored.pdf}}
\ \semEq\ \wire \qquad
\lower5pt\hbox{\includegraphics[height=.7cm]{graffles/voltmeterignored.pdf}}
\ \semEq\ \opencircuit
\]

Given a closed circuit $C$ with $n$ meter outputs,
the behavior of the $n$ measurements simultaneously is a subset of the behavior
of one measurement at a time, ignoring the others.
The proof is a simple derivation, using cartesian bicategory structure:

\[
\lower25pt\hbox{\includegraphics[height=2.1cm]{graffles/imLHS.pdf}}
\ = \
\lower13pt\hbox{\includegraphics[height=1cm]{graffles/im1.pdf}}
\semEq
\lower35pt\hbox{\includegraphics[height=2.8cm]{graffles/im2.pdf}}
\ \semLeq \
\lower33pt\hbox{\includegraphics[height=2.5cm]{graffles/im3.pdf}}
\ = \
\lower55pt\hbox{\includegraphics[height=4.5cm]{graffles/imRHS.pdf}}
\]

The diagram above illustrates this for ammeters,
but it applies to any mixture of meters,
remembering that a discarded voltmeter is an open circuit.
If moreover each individual measurement is a function of the inputs
(i.e. single-valued and total), then the inclusion is an equality.

The inclusion above is an equality for all “well-behaved” circuits.

Classically, equality is postulated.

It is, of course, reasonable to do so:
if the inclusion is strict,
it must be the case that
either (i) that some meter can return a nonzero value with all sources off,
i.e. it is not measuring anything physical,
or (ii) that some settings of the sources are disallowed and can cause the semantics to be empty,
which is degenerate.
For example, in the following example we have accidentally short-circuited the source,
meaning that we can't turn it on with a nonzero value.
This, indeed, is a degenerate case and we don't have equality.

\[
\lower25pt\hbox{\includegraphics[height=2cm]{graffles/imexample1.pdf}}
\semEq\
\lower10pt\hbox{\includegraphics[height=.8cm]{graffles/imexample2.pdf}}
\ \neq\
\lower10pt\hbox{\includegraphics[height=.8cm]{graffles/imexample3.pdf}}
\ \semEq
\lower50pt\hbox{\includegraphics[height=3.5cm]{graffles/imexample4.pdf}}
\]

It does not appear easy to characterise graphically
the circuits with such behavior.
Instead we give the result in its general form,
trusting that equality is easy to notice when calculating the right-hand side.

§.§ Superposition Theorem

The second theorem is amongst the most useful results of classical circuit theory.
The superposition theorem ensures that the behavior of a circuit with multiple source
can be calculated from its behaviors with one source turned on at a time.

To do this, we set all sources but one to zero by plugging $\Wunit$.
It is easy to show that a zero current source is equivalent to an open circuit,
and a zero voltage source is equivalent to a plain wire.

\[
\lower5pt\hbox{\includegraphics[height=.7cm]{graffles/zerocurrentsource.pdf}}
\ \semEq\ \opencircuit \qquad
\lower5pt\hbox{\includegraphics[height=.7cm]{graffles/zerovoltagesource.pdf}}
\ \semEq\ \wire
\]

In a precise sense it's the dual of the independent measurement theorem:
they are related by swapping colors and vertical reflection,
which is a powerful operation of $\mathsf{GLA}$
that often generates elegant dualities.

The behaviour of closed circuit $C$ with $m$ source inputs and no independent sources

is a superset of the sum of its behaviors
with one source turned on at a time:

\[
\lower55pt\hbox{\includegraphics[height=4.5cm]{graffles/superLHS.pdf}}
\ = \
\lower33pt\hbox{\includegraphics[height=2.5cm]{graffles/superposition1.pdf}}
\ \semLeq \
\lower35pt\hbox{\includegraphics[height=2.8cm]{graffles/superposition2.pdf}}
\semEq
\lower13pt\hbox{\includegraphics[height=1cm]{graffles/superposition3.pdf}}
\ = \
\lower25pt\hbox{\includegraphics[height=2.1cm]{graffles/superRHS.pdf}}
\]

The diagram illustrates this with voltage sources,
but it applies to any mixture of sources,
given that a turned off current source is an open circuit.
If moreover the outputs are a function (i.e. single-valued and total)
of each source when activated individually,
then the inclusion is an equality.

Since $C$ has no independent sources, its mapping to $\mathsf{GAA}$ lands in the $\mathsf{GLA}$ fragment.
Thus we can transpose (color-swap and flip) the result from the independent measurement theorem,
noting that being a function is a self-transpose property,
and color-swap reverses the direction of inclusion. Alternatively, one can
use the fact that $GLA$ modulo its inequational theory is an abelian category of relations [5].

The analysis of the equality case in the previous section also applies here via the duality.
Note that the theorem as stated forbids independent sources.
However it's easy to extend it to work with them too:

\[
\lower40pt\hbox{\includegraphics[height=3.2cm]{graffles/superex1.pdf}}
\ \semEq
\lower41pt\hbox{\includegraphics[height=3.3cm]{graffles/superex2.pdf}}
\semLeq\
\lower22pt\hbox{\includegraphics[height=1.7cm]{graffles/superex3.pdf}}
\semEq\
\lower22pt\hbox{\includegraphics[height=1.7cm]{graffles/superex4.pdf}}
\]

§ STRUCTURAL THEOREMS

We conclude with three results
that showcase the graphical language's power to state and prove non-trivial invariants. Indeed, given that our circuits are a bona-fide syntax, we can use structural induction.

§.§ Representation theorem

Kirchhoff's laws imply two global invariants are satisfied by all circuits.
They can be elegantly stated and proved graphically,
and imply a very useful result:
all one-port circuits (those with one electrical input and one electrical output)
are representable by an impedance as described in <ref>.
Thus the impedance calculus can be used with any one-port circuit that we encounter.

A circuit constrains voltage differences, not absolute voltages.
That is, adding the same voltage difference to all open wires of a circuit does not change
its behavior.

\[
\lower30pt\hbox{\includegraphics[height=3cm]{graffles/potentialsLHS.pdf}}
\quad \semEq\quad
\lower28pt\hbox{\includegraphics[height=2.2cm]{graffles/potentialsRHS.pdf}}
\]

By induction on the structure of circuits.
See <ref> for more details.

The sum of the currents going into a circuit is equal to the sum of the outgoing currents.

\[
\lower30pt\hbox{\includegraphics[height=3cm]{graffles/currentsLHS.pdf}}
\quad \semEq\quad
\lower30pt\hbox{\includegraphics[height=2.3cm]{graffles/currentsRHS.pdf}}
\]

The proof proceeds exactly like the previous one,
the only difference being an inversion of the colours,
which does not affect the proof.

A one-port circuit $c$
is representable by an impedance:

\[
\lower5pt\hbox{\includegraphics[height=.5cm]{graffles/oneportcircuit.pdf}}
\semEq
\lower20pt\hbox{\includegraphics[height=1.6cm]{graffles/oneportimpedance.pdf}}
\]

See <ref>.

§.§ Expressivity

A nervous reader might ask:
given that we have liberally extended the basic calculus with
arbitrary affine relations, via the information wires,
do we risk modeling unphysical circuits?
The answer is a surprising one:
textbook elements are already expressive enough;
they can express most affine relations.

We start by observing a surprising fact:
adding only controlled sources and meters to the basic circuit elements (<ref>)
is already enough to build a circuit for most affine relations.
Indeed, already we can build any $\mathsf{GAA}$ diagram on the information wires, since
every generator can be constructed with a small circuit:

\[
\Wmult \semEq \lower13pt\hbox{\includegraphics[height=1.3cm]{graffles/wmultcircuit.pdf}}
\quad
\Wunit \semEq \lower2pt\hbox{\includegraphics[height=.5cm]{graffles/wunitcircuit.pdf}}
\quad
\Bcomult \semEq \lower13pt\hbox{\includegraphics[height=1.1cm]{graffles/bcomultcircuit.pdf}}
\quad
\Bcounit \semEq \lower2pt\hbox{\includegraphics[height=.5cm]{graffles/bcounitcircuit.pdf}}
\]

\[
\Wcomult \semEq \lower13pt\hbox{\includegraphics[height=1.1cm]{graffles/wcomultcircuit.pdf}}
\quad
\Wcounit \semEq \lower2pt\hbox{\includegraphics[height=.5cm]{graffles/wcounitcircuit.pdf}}
\quad
\Bmult \semEq \lower13pt\hbox{\includegraphics[height=1.1cm]{graffles/bmultcircuit.pdf}}
\quad
\Bunit \semEq \lower2pt\hbox{\includegraphics[height=.5cm]{graffles/bunitcircuit.pdf}}
\]

\[
\One \semEq \lower10pt\hbox{\includegraphics[height=1.1cm]{graffles/onecircuit.pdf}}
\quad
\antipode \semEq \lower10pt\hbox{\includegraphics[height=1.1cm]{graffles/antipodecircuit.pdf}}
\quad
\lower2pt\hbox{\includegraphics[height=.25cm]{graffles/coscalar.pdf}}
\semEq
\lower10pt\hbox{\includegraphics[height=1.1cm]{graffles/coscalarcircuit.pdf}}
\quad
\circuitXop
\semEq
\lower10pt\hbox{\includegraphics[height=1.1cm]{graffles/xopcircuit.pdf}}
\quad
\circuitX
\semEq
\lower10pt\hbox{\includegraphics[height=1.1cm]{graffles/xcircuit.pdf}}
%\scalar = \lower20pt\hbox{\includegraphics[height=1.3cm]{graffles/scalarcircuit.pdf}}
\]

So, restricting to just information wires, we have the full expressivity of $\mathsf{GAA}$.
The following result characterises expressivity, also taking electrical wires into account.

Using only basic circuit elements (<ref>),
meters and controlled sources,
we can express all affine relations that respect the invariants of
This includes arbitrary affine relations on the information wires.

We sketched the main idea above. See <Ref> for more details.

Now take a circuit with no open information wires
made up of only basic elements, meters and sources.
It is important to note that, although we can construct circuits equivalent to an arbitrary affine relations on the information wires, there is no direct way to copy information wires in a circuit.
Since they cannot be copied,
every meter is connected to exactly one source.
This is essentially equivalent to adding textbook elements called
measurement-controlled sources.
The voltage-controlled current source is constructed as follows;
other combinations (e.g. current-controlled voltage source) are similar.

\[
\lower12pt\hbox{\includegraphics[height=1.2cm]{graffles/vccsource.pdf}}
\quad \Defeq \quad
\lower13pt\hbox{\includegraphics[height=1.3cm]{graffles/vccsourcedef.pdf}}
\]

But this also means that,
using only basic elements and measurement-controlled sources,
we can build a circuit for any relation that respects the invariants
(with no open information wires).
There is no need to fear, therefore:
textbook elements already capture enough affine relations.

§.§ Thévenin's theorem

Last but not least,
we prove Thévenin's theorem,
a well-known real-life example of compositional, diagrammatic reasoning.
It allows one to replace a one-port circuit by a simpler, equivalent one.
It can be seen both as a consequence and a stronger version of the representation theorem

If $C$ is a one-port circuit made only of resistors and independent sources,
then one of the following is true:

* $\lower5pt\hbox{\includegraphics[height=.5cm]{graffles/oneportcircuit.pdf}} \semEq \lower4pt\hbox{\includegraphics[height=.8cm]{graffles/v0r.pdf}}$ for some $V_0$ and $R$,

* $\lower5pt\hbox{\includegraphics[height=.5cm]{graffles/oneportcircuit.pdf}} \semEq
\lower4pt\hbox{\includegraphics[height=.8cm]{graffles/i0.pdf}}$ for some $I_0$,

* $\lower5pt\hbox{\includegraphics[height=.5cm]{graffles/oneportcircuit.pdf}}$ denotes the empty relation.

See <ref>.

§ CONCLUSION AND FUTURE WORK

We extended existing compositional treatments of electrical circuit theory by including meters and controlled sources, enabling the analysis of closed circuits. We introduced the impedance calculus and demonstrated its power by solving simple circuits, and by proving a number of results—with the superposition theorem and Thévenin's theorem particularly notable, well-known examples from the corpus of standard electrical circuit theory. We also proved a representation theorem and characterised the expressivity of our circuit language, which is exactly that of circuits built up of standard textbook elements. With the syntactic approach, proofs are pleasingly simple, involving simple invariants and induction.

It is worth to step back and consider the bigger picture. We believe that the results herein are evidence that, while it is uncontroversial to say that electrical circuits are relational entities, they are better understood using relational mathematics. The benefit of not turning everything into matrices is that theorems are more crisp to state, the relationships and symmetries between various results become more immediately apparent, annoying “degenerate” cases are not ignored, and proofs become more satisfying.

There is much future work: while we could not include all of the theorems, we are confident of being able to replicate the majority of standard electrical circuit theory using these techniques. An obvious open question is how far can this approach take us? The passage from Graphical Linear Algebra to Graphical Affine Algebra allows to pass from passive circuits to non-passive circuits. The next, obvious challenge is to go beyond linearity: for example, if we were  to satisfactory capture the diode, we would get transistors, enabling the compositional analysis of arbitrary electronic circuits.

[1]
authorJohn C. Baez &
authorBrendan Fong
(year2015): titleA Compositional Framework for
Passive Linear Circuits.
journalCoRR volumeabs/1504.05625.

[2]
authorFilippo Bonchi,
authorJoshua Holland,
authorDusko Pavlovic &
authorPaweł Sobociński
(year2017): titleRefinement for signal flow
In: booktitleConcurrency Theory - 28th International
Conference, (CONCUR 2017).

[3]
authorFilippo Bonchi,
authorRobin Piedeleu,
authorPaweł Sobociński &
authorFabio Zanasi
(year2019): titleGraphical Affine Algebra.
In: booktitle34th Annual ACM/IEEE Symposium
on Logic in Computer Science (LICS),
publisherIEEE, addressVancouver, BC, Canada, pp.
pages1–12, 10.1109/LICS.2019.8785877.

[4]
authorFilippo Bonchi,
authorPaweł Sobociński &
authorFabio Zanasi
(year2017): titleInteracting Hopf
journalJournal of Pure and Applied Algebra
volume221(number1), pp. pages144–184,

[5]
authorAurelio Carboni &
authorR. F. C. Walters
(year1987): titleCartesian Bicategories I.
journalJournal of Pure and Applied Algebra
volume49, pp. pages11–32.

[6]
authorBrandon Coya (year2018):
titleCircuits, Bond Graphs, and Signal-Flow Diagrams: A
Categorical Perspective.
Ph.D. thesis, schoolUniversity of California Riverside.

[7]
authorCharles A. Desoer &
authorErnest S. Kuh
(year1969): titleBasic Circuit Theory.
publisherMcGraw-Hill, addressNew York.

§ PROOFS

§.§ <Ref>

\[
(i): \lower18pt\hbox{$\includegraphics[height=1.4cm]{graffles/impedancecomp1.pdf}$}
\quad = \quad  \lower20pt\hbox{$\includegraphics[height=1.6cm]{graffles/impedancecomp2.pdf}$}
\quad = \quad  \lower18pt\hbox{$\includegraphics[height=1.4cm]{graffles/impedancecomp3.pdf}$}
\]

\[
(ii): \lower22pt\hbox{$\includegraphics[height=1.8cm]{graffles/impedancepar1.pdf}$}
\quad = \quad \lower22pt\hbox{$\includegraphics[height=1.8cm]{graffles/impedancepar2.pdf}$}
\quad = \quad \lower22pt\hbox{$\includegraphics[height=1.8cm]{graffles/impedancepar3.pdf}$}
\]

\begin{multline*}
(iii): \lower22pt\hbox{$\includegraphics[height=1.8cm]{graffles/S1.pdf}$}
\quad = \quad  \lower22pt\hbox{$\includegraphics[height=1.8cm]{graffles/S2.pdf}$} \\
= \quad  \lower22pt\hbox{$\includegraphics[height=1.8cm]{graffles/S3.pdf}$}
\quad = \quad \lower13pt\hbox{$\includegraphics[height=1.2cm]{graffles/S4.pdf}$}
\end{multline*}

\[
(iv): \lower13pt\hbox{$\includegraphics[height=1.2cm]{graffles/closed1.pdf}$} \quad=\quad \lower4pt\hbox{$\includegraphics[height=.4cm]{graffles/closedRHS.pdf}$}
\]

§.§ Relativity of Potentials (<Ref>)

First, a lemma:

\[
\lower12pt\hbox{\includegraphics[height=1.2cm]{graffles/vsource1.pdf}}
\semEq
\lower12pt\hbox{\includegraphics[height=1.2cm]{graffles/vsource2.pdf}}
\semEq
\lower12pt\hbox{\includegraphics[height=1.2cm]{graffles/vsource3.pdf}}
\semEq
\lower12pt\hbox{\includegraphics[height=1cm]{graffles/vsource4.pdf}}
\semEq\
\lower6pt\hbox{\includegraphics[height=.5cm]{graffles/vsource5.pdf}}
\]

and of course, similarly

\[
\lower12pt\hbox{\includegraphics[height=1.2cm]{graffles/vsource1.pdf}}
\semEq\
\lower6pt\hbox{\includegraphics[height=.5cm]{graffles/vsource5.pdf}}.
\]

From this we can show that the property holds for circuit elements.
The argument, which we show below for resistors, relies on the fact that circuit elements
commute with voltage sources, which can be easily shown using the impedance calculus.

\[
\lower3pt\hbox{\includegraphics[height=.7cm]{graffles/resvsource1.pdf}}
\semEq
\lower8pt\hbox{\includegraphics[height=1cm]{graffles/resvsource2.pdf}}
\semEq
\lower8pt\hbox{\includegraphics[height=1cm]{graffles/resvsource3.pdf}}
\semEq
\lower6pt\hbox{\includegraphics[height=.8cm]{graffles/resvsource4.pdf}}
\semEq
\lower3pt\hbox{\includegraphics[height=.7cm]{graffles/resvsource5.pdf}}
\]

Given this, we have:

\[
\lower8pt\hbox{\includegraphics[height=1cm]{graffles/baseres1.pdf}}
\semEq
\lower8pt\hbox{\includegraphics[height=1cm]{graffles/baseres2.pdf}}
\semEq
\lower6pt\hbox{\includegraphics[height=.7cm]{graffles/baseres3.pdf}}
\]

Voltage sources commute with junctions in the following sense:

\[
\semElec{\lower10pt\hbox{\includegraphics[height=1.2cm]{graffles/vsourcecoj1.pdf}}}
\lower8pt\hbox{\includegraphics[height=1cm]{graffles/vsourcecoj2.pdf}}
\lower8pt\hbox{\includegraphics[height=1cm]{graffles/vsourcecoj3.pdf}}
\semElec{\lower8pt\hbox{\includegraphics[height=1cm]{graffles/vsourcecoj4.pdf}}}
\]

and so

\[
\lower9pt\hbox{\includegraphics[height=1.3cm]{graffles/cojcase1.pdf}}
\semEq
\lower8pt\hbox{\includegraphics[height=1.2cm]{graffles/cojcase2.pdf}}
\semEq
\lower8pt\hbox{\includegraphics[height=1cm]{graffles/cojcase3.pdf}}
\]

The remainder of the structure is similarly easy.
What remains are the inductive cases.
Parallel composition is immediate.
Only sequential composition remains:

\begin{multline*}
\lower9pt\hbox{\includegraphics[height=1.2cm]{graffles/inductive1.pdf}}
\semEq
\lower9pt\hbox{\includegraphics[height=1.3cm]{graffles/inductive2.pdf}}
\semEq
\lower9pt\hbox{\includegraphics[height=1.3cm]{graffles/inductive3.pdf}} \\
\semEq
\lower12pt\hbox{\includegraphics[height=1.5cm]{graffles/inductive4.pdf}}
\semEq
\lower12pt\hbox{\includegraphics[height=.8cm]{graffles/inductive5.pdf}}
\end{multline*}

§.§ Proof of the Representation Theorem (<Ref>)

Using the conclusions of Propositions <ref> and <ref> we have

\[
\lower8pt\hbox{\includegraphics[height=.5cm]{graffles/repcircuit1.pdf}}
\semEq
\lower9pt\hbox{\includegraphics[height=1.2cm]{graffles/repcircuit2.pdf}}
\semEq
\lower9pt\hbox{\includegraphics[height=1.5cm]{graffles/repcircuit3.pdf}}
\]

A simple calculation, therefore, confirms that:

\[
\lower8pt\hbox{\includegraphics[height=.7cm]{graffles/jc1.pdf}}
\lower19pt\hbox{\includegraphics[height=1.7cm]{graffles/jc2.pdf}}.
\]

\begin{multline*}
\semElec{\lower15pt\hbox{\includegraphics[height=1.4cm]{graffles/repfinal1.pdf}}}
\lower19pt\hbox{\includegraphics[height=1.7cm]{graffles/repfinal2.pdf}}
\lower22pt\hbox{\includegraphics[height=1.9cm]{graffles/repfinal3.pdf}} \\
\lower30pt\hbox{\includegraphics[height=2.1cm]{graffles/repfinal4.pdf}}
\lower22pt\hbox{\includegraphics[height=1.7cm]{graffles/jc2.pdf}}
\lower8pt\hbox{\includegraphics[height=.7cm]{graffles/jc1.pdf}}
\semElec{\lower4pt\hbox{\includegraphics[height=.5cm]{graffles/repcircuit1.pdf}}}.
\end{multline*}

§.§ Proof of the Expressivity Theorem (<Ref>)

Take an arbitrary circuit type $\sort{\bullet^m\bluebullet^o}{\bullet^n\bluebullet^p}$.
and let $R$ be an invariant preserving affine relation of the corresponding type

\[
\includegraphics[height=1cm]{graffles/rexpressivity}.
\]

First, it is convenient to capture invariant-preservation directly at the level of the affine relation.
Let $\sigma_p\from 2p \to 2p$ and $\sigma_o\from 2o \to 2o$ be permutations that “collect” all the voltage wires and current wires. Then, for $R$ to be invariant-preserving means simply that the following equality holds:

\[
\lower12pt\hbox{\includegraphics[height=1.2cm]{graffles/invariantLHS}} \quad = \quad
\lower30pt\hbox{\includegraphics[height=2.4cm]{graffles/invariantRHS}}.
\]

Using this fact, we will now show how to obtain $R$ as the semantics of a circuit. We start by constructing $R'$ by rewiring $R$, as shown below:

\[
\lower8pt\hbox{\includegraphics[height=.8cm]{graffles/Rprime}} \quad
\Defeq\quad
\lower20pt\hbox{\includegraphics[height=2cm]{graffles/Rprimedef}}
\]

In Section <ref> we showed that the generators of $\mathsf{GAA}$ can be expressed on the information wires using circuits. Using these circuits, we can construct a circuit for $R'$ using purely circuit components, i.e. without using $\mathsf{GAA}$ generators. With slight abuse of notation, we again refer to it as $R'$ below, where we construct the following, larger circuit:

\[
\lower20pt\hbox{\includegraphics[height=3.5cm]{graffles/Rprimecircuit}}
\]

The following derivation shows that its semantics (image under $\mathcal{I}$) is $R$, indeed:

\begin{multline*}
\lower40pt\hbox{\includegraphics[height=4cm]{graffles/expressivity1}}
\\
= \lower30pt\hbox{\includegraphics[height=2.3cm]{graffles/expressivity2}}
= \lower30pt\hbox{\includegraphics[height=2.3cm]{graffles/expressivity3}} \\
= \lower40pt\hbox{\includegraphics[height=2cm]{graffles/expressivity4}}
= \lower15pt\hbox{\includegraphics[height=1cm]{graffles/expressivity5}}
= \lower15pt\hbox{\includegraphics[height=1cm]{graffles/rexpressivity}}
\end{multline*}

§.§ Proof of Thévenin's Theorem (<Ref>)

If $C$ is a one-port made only of resistors,
then one of the following is true:

* $\lower5pt\hbox{\includegraphics[height=.5cm]{graffles/oneportcircuit.pdf}} \semEq \resistor$ for some $R$

* $\lower5pt\hbox{\includegraphics[height=.5cm]{graffles/oneportcircuit.pdf}} \semEq \circcounit\ \circunit$

This theorem relies on an inductive invariant that we will not show here,
but works as follows:
taking the semantics of a circuit with no info wires,
we can bend wires to have all current wires as inputs and all voltage wires as outputs.
If done carefully, a simple inductive proof shows that
for a circuit with only resistors such a relation is positive semidefinite,
i.e. can be written a $R^T;R$ for some relation $R$.

Let $c$ be the impedance of a one-port $C$ made only of resistors.
The argument above implies that $c$ is positive semidefinite as well.
We know all the possible 1-to-1 linear relations,
which leaves only the following two possibilities:

Now let $C$ be a circuit made of resistors and independent sources.
$C$ has an affine impedance, which we can write as follows, with $c$ linear:

An affine relation is either empty or has the following form:

Now define the circuit $C_0$ as follows.

$C_0$ is $C$ with all sources turned off, and therefore is made only of resistors.
We can thus apply the previous lemma to it:

which give respectively:


# Simplicial pseudohyperplane arrangements
give weak Garside groups

Katherine Goldman

###### Abstract.

In this note we connect the language of Bessis’s Garisde categories with
Salvetti’s metrical-hemisphere complexes in order to find new examples of weak
Garside groups. As our main example, we show that the fundamental group of the
(appropriately defined) complexified complement of a pseudohyperplane
arrangement is a weak Garside group. As a consequence of the Folkman-Lawrence
topological realization theorem, we also show that the fundamental group of
the Salvetti complex of a (“simplicial”) oriented matroid is a weak Garside
group. This provides novel examples of weak Garside groups.

Weak Garside groups were formalized by Bessis in [Bes06]. A traditional
Garside group is a group endowed with a certain combinatorial structure which
provides fruitful information about the group. The structure on a Garside
group generalizes readily to groupoids, giving Bessis’s Garside groupoids. A
weak Garside group is a group which is equivalent (as a category) to a Garside
groupoid111We clarify that a “weak Garside group” is not the same as a “quasi
Garside group”. A quasi Garside group may be thought of as an “infinite-type”
Garside group.. In general, weak Garside groups may not be Garside groups, but
the weak Garside property has shown to be rather fruitful on its own. Many
results for Garside groups can be easily extended to weak Garside groups, and
some recent work has been done to study weak Garside groups in their own
right. Some notable properties for a group $G$ which is a finite-type (weak)
Garside group include

  1. (1)

The word problem for $G$ is solvable [DP99].

  2. (2)

$G$ is biautomatic [DP99]; in particular, the $k$-th order homological Dehn
function and homotopical Dehn function of $G$ are Euclidean [Wen05, BD19].

  3. (3)

$G$ has a finite $K(G,1)$ [Bes06]; in particular, $G$ is torsion-free.

  4. (4)

$G$ is Helly [HO21]; in particular, this implies

     * •

$G$ admits a geometric action on an injective metric space,

     * •

$G$ admits an EZ-boundary and a Tits boundary,

     * •

The Farrell-Jones conjecture with finite wreath products holds for $G$, and

     * •

The coarse Baum-Connes conjecture holds for $G$.

Primary examples of (weak) Garside groups come from complements of _complex
hyperplane arrangements_. A complex hyperplane arrangement is a finite
collection $\mathcal{H}$ of linear hyperplanes (i.e., codimension-1 linear
subspaces) in $\mathbb{C}^{n}$. We denote the complement of this collection by

$M(\mathcal{H})=\mathbb{C}^{n}\setminus\bigcup_{H\in\mathcal{H}}H.$

There are two beautiful results concerning when $\pi_{1}(M(\mathcal{H}))$ is a
weak Garside group. The first, by Deligne [Del72], considers the following
class: suppose $\mathcal{H}$ is a finite collection of _real_ linear
hyperplanes (i.e., codimension-1 linear subspaces of $\mathbb{R}^{n}$). We
call a connected component of
$\mathbb{R}^{n}\setminus\bigcup_{H\in\mathcal{H}}H$ a _chamber_ , and we call
$\mathcal{H}$ _simplicial_ if each chamber is a simplicial cone (i.e., the
$\mathbb{R}^{+}$-span of a simplex). Let
$\mathcal{H}_{\mathbb{C}}=\\{\,H_{\mathbb{C}}:H\in\mathcal{H}\,\\}$, where
$H_{\mathbb{C}}$ denotes the complexification of the linear space $H$. Then if
$\mathcal{H}$ is simplicial, $\pi_{1}(M(\mathcal{H}_{\mathbb{C}}))$ is a weak
Garside group [Del72]. The prototypical example of a Garside group arising in
this way is a braid group on $n$-strands [Gar69], or, more generally, the
spherical-type Artin groups [BS72] (of which the braid groups are examples).

The second major example comes from Bessis, who in [Bes15] proved the
following. Suppose $\mathcal{H}$ is the set of reflection hyperplanes for a
well-generated complex reflection group. (A well-generated complex reflection
group is a finite subgroup of $\mathrm{GL}_{n}(\mathbb{C})$ generated by
exactly $n$ complex reflections; complex reflections are finite-order linear
transformations of $\mathbb{C}^{n}$ fixing a hyperplane.) Then
$\pi_{1}(M(\mathcal{H}))$ is a weak Garside group [Bes15].

One may ask if the assumption that the hyperplanes are linear in the previous
examples may be dropped. That is, are there similar results for finite
collections of submanifolds which are simply complex codimension-1 and
intersect at the origin? While it is certainly unfeasable to expect this to be
true of any arbitrary submanifolds, if we restrict ourselves to a certain
class of submanifolds, called pseudohyperplanes, we show that the answer is
yes; i.e., the fundamental group of the complement of a simplicial
pseudohyperplane arrangement is a weak Garside group. We give the precise
statements now.

These definitions are based on those found in [Man82]. Consider an embedding
$\mathbb{S}^{d-1}\hookrightarrow\mathbb{S}^{d}$ with image $S$. The subspace
$S\subseteq\mathbb{S}^{d}$ is called a _pseudosphere_ if the components of
$\mathbb{S}^{d}\setminus S$ are homeomorphic to the unit ball in
$\mathbb{R}^{d}$. In this case, the components of $\mathbb{S}^{d}\setminus S$
are called the (open) halfspaces of $S$. (One can verify that there are always
two such components.) Let $\mathcal{A}=\\{S_{i}\\}_{i\in I}$ be a finite
collection of pseudospheres of $\mathbb{S}^{d}$. For $J\subseteq I$, let
$S_{J}=\bigcap_{j\in J}S_{j}$. We call $\mathcal{A}$ an _arrangement of
pseudospheres (in $\mathbb{S}^{d}$)_ if for each $J\subseteq I$ and $k\in I$
satisfying $S_{J}\not\subset S_{k}$, we have that $S_{J}\cap S_{k}$ is a
pseudosphere in $S_{J}$ with halfspaces coming from the intersection of
$S_{J}$ with the halfspaces of $S_{k}$. (Compare to the intersection of a
hyperplane arrangement with the unit sphere.) If $\mathcal{A}$ is a
pseudosphere arrangement, then the cone

$\displaystyle c\mathcal{A}=\\{\,cS:S\in\mathcal{A}\,\\}$

is called a _pseudohyperplane arrangement_. (The cone $cS$ on a pseudosphere
$S$ is the set $cS=\\{\,\alpha x:\alpha\in[0,\infty),x\in S\,\\}$.) To a
pseudohyperplane arrangement, one can associate a natural “complexified
complement” (even though these are not linear subspaces) by

$M(c\mathcal{A})=\mathbb{C}^{n}\setminus\bigcup_{H\in c\mathcal{A}}(H+iH),$

where $H+iH=\\{\,h_{1}+ih_{2}:h_{i}\in H\,\\}$. One of our main results (part
of Theorem 33) is as follows (comparable to [Del72]).

######  Theorem.

If $c\mathcal{A}$ is a centrally symmetric simplicial pseudohyperplane
arrangement, then $\pi_{1}(M(c\mathcal{A}))$ is a weak Garside group.

In particular, this positively answers a question of Deshpande regarding the
bi-automaticity of such groups [Des16, p. 419], as well as showing that they
possesses the other properties of weak Garside groups given above.

We note that there are many pseudohyperplane arrangements which are not
realizable as hyperplane arrangements. A famous example is the non-Pappus
arrangement seen in Figure 1.

Figure 1. The non-Pappus arrangement, from [Des16, Fig. 1]

Other examples come from pseudoline arrangements in $\mathbb{RP}^{2}$ [Grü67,
Ch. 2]; these are finite collections of simple closed curves in
$\mathbb{RP}^{2}$ such that every two curves intersect (transversely) in
exactly one point. By lifting to $\mathbb{S}^{2}$ and taking cones, we have a
non-realizable arrangement of pseudoplanes in $\mathbb{R}^{3}$. There are (at
least) seven infinite families of simplicial pseudoline arrangements [Grü67,
Ch. 3].

We also note that as a consequence of [DF17], there exist pseudohyperplane
arrangement complements whose fundamental groups are not isomorphic to the
fundamental group of any hyperplane arrangement complement. Although, this is
explicitly demonstrated using the above non-Pappus arrangement, which is not
simplicial. We believe that there exist such simplicial examples, but have not
yet been able to find any in the literature.

The famous Folkman-Lawrence topological realization theorem gives a 1-1
correspondence between pseudohyperplane arrangements222Originally shown for
“pseudohemisphere arrangements” in [FL78], stated and proven in terms of
pseudosphere/pseudohyperplane arrangements in [Man82] and oriented matroids
(see Proposition 35). Oriented matroids are common combinatorial objects which
have broad uses in mathematics, and in particular capture the combinatorics of
pseudohyperplane arrangements. For a given oriented matroid $\mathcal{M}$,
there is an associated simplicial complex, called the _Salvetti complex_
$\mathrm{Sal}(\mathcal{M})$ [Zie87, §5.5]. This complex turns out to be
homotopy equivalent to $M(c\mathcal{A})$ for $c\mathcal{A}$ the
pseudohyperplane arrangement associated to $\mathcal{M}$ [Des16, Thm. 2.30].
In Section 3, we utilize this deep connection of pseudohyperplane arrangements
with oriented matroids to show the following (Theorem 36).

######  Theorem.

If $\mathcal{M}$ is a “simplicial” oriented matroid, then
$\pi_{1}(\mathrm{Sal}(\mathcal{M}))$ is a weak Garside group.

Our main technical tool, defined in Section 1.2, are the metrical-hemisphere
complexes introduced by Salvetti in [Sal93]. These complexes act as a basis
for generalizing the construction of the “Salvetti complex” of a hyperplane
arrangement to the pseudosphere/pseudohyperplane arrangements. It is shown in
[Des16] that, analagous to the hyperplane case, the Salvetti complex of a
pseudohyperplane arrangement is homotopic to its complexified complement. We
may thus adapt many of the arguments involving the Salvetti complex of a
hyperplane arrangement to those of pseudohyperplane arrangements—and in fact,
our arguments may be applied to the Salvetti complex of any (suitable)
metrical-hemisphere complex. This is the content of Section 2. See Theorem 14
and Corollary 24 for the full statement of our most general results.

The paper is structured as follows. In Section 1, we provide the relevant
background information regarding Garside categories and metrical hemisphere
complexes. In Section 2, we show that the Salvetti complex of appropriate
metrical hemisphere complexes gives rise to a natural categorical Garside
structure, and the isotropy group of this Garside structure is isomorphic to
the fundamental group of the Salvetti complex. In Section 3, we apply this
result to simplicial pseudohyperplane arrangements and (simplicial) oriented
matroids.

##  1\. Background

###  1.1. Garside categories

The material in this section originally comes from [Bes06], but our statements
are primarily adapted from [HH22]. We recall some notions from basic category
theory first, mainly to establish notation. Throughout this paper, we assume
all categories are small.

Let $\mathcal{C}$ be a (small) category. The set of objects of $\mathcal{C}$
is denoted $\mathsf{Obj}(\mathcal{C})$ and the set of morphisms is denoted
$\mathsf{Hom}(\mathcal{C})$. We denote the identity functor of $\mathcal{C}$
by $1_{\mathcal{C}}$. For $x,y\in\mathsf{Obj}(\mathcal{C})$, let
$\mathcal{C}_{x\rightarrow}$ denote the set of morphisms starting at $x$,
$\mathcal{C}_{\rightarrow y}$ the set of morphisms ending at $y$, and
$\mathcal{C}_{x\rightarrow y}$ the set of morphisms from $x$ to $y$. If
$f\in\mathcal{C}_{x\rightarrow y}$, we sometimes write $x\xrightarrow{f}y$. If
$x\xrightarrow{f}y\xrightarrow{g}z$ then the composition of $f$ and $g$ is
$x\xrightarrow{fg}z$ (so we adopt the “path” style of the order of composition
rather than the “function” style).

We recall that a natural transformation $\tau$ between functors
$F,G:\mathcal{C}\to\mathcal{D}$, which we denote by $\tau:F\to G$, is a
mapping from $\mathsf{Obj}(\mathcal{C})$ to $\mathsf{Hom}(\mathcal{D})$ which
for $x,y\in\mathsf{Obj}(\mathcal{C})$ and $x\xrightarrow{f}y$ satisfies

  1. (1)

$F(x)\xrightarrow{\tau(x)}G(x)$, and

  2. (2)

$F(f)\tau(y)=\tau(x)G(f)$.

For $f,g\in\mathsf{Hom}(\mathcal{C})$, we write $f\preccurlyeq g$ if there is
an $h\in\mathsf{Hom}(\mathcal{C})$ such that $g=fh$. We write $g\succcurlyeq
f$ if there is an $h\in\mathsf{Hom}(\mathcal{C})$ such that $g=hf$. A
nontrivial $f\in\mathsf{Hom}(\mathcal{C})$ which cannot be factored into two
nontrivial factors is called an atom.

The category $\mathcal{C}$ is cancellative if for
$a,b,f,g\in\mathsf{Hom}(\mathcal{C})$, $afb=agb$ implies $f=g$. It is
homogeneous if there is a function
$\ell:\mathsf{Hom}(\mathcal{C})\to\mathbb{Z}_{\geq 0}$ such that
$\ell(fg)=\ell(f)+\ell(g)$, and $\ell(f)=0$ if and only if $f$ is trivial.
Note that if $\mathcal{C}$ is homogeneous, then
$(\mathcal{C}_{x\rightarrow},\preccurlyeq)$ and $(\mathcal{C}_{\rightarrow
y},\succcurlyeq)$ are posets.

######  Definition 1.

A _Garside category_ is a homogeneous cancellative category $\mathcal{C}$
equipped with an automorphism $\phi:\mathcal{C}\to\mathcal{C}$ and a natural
transformation $\Delta:1_{\mathcal{C}}\to\phi$ satisfying the following
properties.

  1. (1)

For all $x,y\in\mathsf{Obj}(\mathcal{C})$, the posets
$(\mathcal{C}_{x\rightarrow},\preccurlyeq)$ and $(\mathcal{C}_{\rightarrow
y},\succcurlyeq)$ are lattices.

  2. (2)

All atoms in $\mathsf{Hom}(\mathcal{C})$ are _simple_ ($x\xrightarrow{f}y$ is
simple if there exists an $y\xrightarrow{f^{*}}\phi(x)$ so that
$ff^{*}=\Delta(x)$).

Sometimes for $x\in\mathsf{Obj}(\mathcal{C})$, we write $\Delta_{x}=\Delta(x)$
and $\Delta^{x}=\Delta(\phi^{-1}(x))$. If $x$ is clear from context, we may
write $\Delta=\Delta_{x}$. If the set of simple morphisms is finite, we call
$\mathcal{C}$ _finite type_.

Associated to a Garside category $\mathcal{C}$ is a groupoid $\mathcal{G}$
into which $\mathcal{C}$ embeds, obtained by adding formal inverses to all
morphisms in $\mathcal{C}$. We call such a $\mathcal{G}$ a _Garside groupoid_.
For $x\in\mathsf{Obj}(\mathcal{G})$, we sometimes write $\mathcal{G}_{x}$ for
$\mathcal{G}_{x\rightarrow x}$, and call $\mathcal{G}_{x}$ the _isotropy group
at $x$_. A _weak Garside group_ is a group which is isomorphic to the isotropy
group of some object of a Garside groupoid. A _Garside group_ is a Garside
groupoid with one object. A _Garside monoid_ is a Garside category with one
object. Each of these objects will be called _finite type_ if the
corresponding Garside category is finite type.

###  1.2. Metrical Hemisphere Complexes

The following comes from [Sal93].

Let $Q$ be a connected cell complex with poset of (closed) cells
$\mathcal{Q}$. For a cell $e\in\mathcal{Q}$, let $\mathcal{Q}(e)$ denote the
set of cells of $\mathcal{Q}$ contained in $e$ and let $V(e)$ denote the
vertices of $e$. The set of $i$-cells of $Q$ is denoted $\mathcal{Q}^{(i)}$
and the $i$-skeleton of $Q$ is denoted $Q^{(i)}$. Note that $Q^{(i)}$ is the
union of the cells of $\mathcal{Q}^{(i)}$. For vertices $v,w$ of $Q$, we let
$d(v,w)$ denote the distance between $v$ and $w$ in $Q^{(1)}$ with each edge
given length 1 (so $d(v,w)$ is the combinatorial length of the shortest edge
path from $v$ to $w$).

######  Definition 2.

A regular cell complex $Q$ is quasi-metrical-hemisphere (QMH) if there exist
maps
$\underaccent{\rule{2.45004pt}{0.6pt}}{w},\accentset{\rule{2.45004pt}{0.6pt}}{w}:\mathcal{Q}^{(0)}\times\mathcal{Q}\to\mathcal{Q}^{(0)}$
such that

  1. (i)

For each cell $e$, the functions
$\underaccent{\rule{2.45004pt}{0.6pt}}{w}(\cdot,e)$ and
$\accentset{\rule{2.45004pt}{0.6pt}}{w}(\cdot,e)$ are maps from
$\mathcal{Q}^{(0)}$ to $V(e)$,

  2. (ii)

The vertices $\underaccent{\rule{2.45004pt}{0.6pt}}{w}(v,e)$ and
$\accentset{\rule{2.45004pt}{0.6pt}}{w}(v,e)$ are the unique vertices of $e$
satisfying

$\displaystyle d(v,\underaccent{\rule{2.45004pt}{0.6pt}}{w}(v,e))=\min_{w\in
V(e)}d(v,w)\quad\text{ and }\quad
d(v,\accentset{\rule{2.45004pt}{0.6pt}}{w}(v,e))=\max_{w\in V(e)}d(v,w),$

and

  3. (iii)

For all vertices $v$ and cells $e$ of $Q$, and all cells
$e^{\prime}\in\mathcal{Q}(e)$,

$\displaystyle\underaccent{\rule{2.45004pt}{0.6pt}}{w}(v,e^{\prime})$
$\displaystyle=\underaccent{\rule{2.45004pt}{0.6pt}}{w}(\underaccent{\rule{2.45004pt}{0.6pt}}{w}(v,e),e^{\prime})=\accentset{\rule{2.45004pt}{0.6pt}}{w}(\accentset{\rule{2.45004pt}{0.6pt}}{w}(v,e),e^{\prime}),\quad\text{
and}$ (A) $\displaystyle\accentset{\rule{2.45004pt}{0.6pt}}{w}(v,e^{\prime})$
$\displaystyle=\underaccent{\rule{2.45004pt}{0.6pt}}{w}(\accentset{\rule{2.45004pt}{0.6pt}}{w}(v,e),e^{\prime})=\accentset{\rule{2.45004pt}{0.6pt}}{w}(\underaccent{\rule{2.45004pt}{0.6pt}}{w}(v,e),e^{\prime}).$
(B)

######  Proposition 3.

If $Q$ is a regular cell complex satisfying (i) and (ii) above, then (iii)
(i.e., (A) and (B) together) is equivalent to

$\displaystyle d(v,\accentset{\rule{2.45004pt}{0.6pt}}{w}(v,e))$
$\displaystyle=d(v,w)+d(w,\accentset{\rule{2.45004pt}{0.6pt}}{w}(v,e))$

for all $v\in Q^{(0)}$, $e\in\mathcal{Q}$, and $w\in V(e)$. Moreover, if $Q$
is QMH, then

$\displaystyle d(v,w)$
$\displaystyle=d(v,\underaccent{\rule{2.45004pt}{0.6pt}}{w}(v,e))+d(\underaccent{\rule{2.45004pt}{0.6pt}}{w}(v,e),w)$

for all $v\in Q^{(0)}$, $e\in\mathcal{Q}$, and $w\in V(e)$.

######  Definition 4.

A regular cell complex $Q$ is local-metrical-hemisphere (LMH) if every cell
$e\in\mathcal{Q}$ (viewed as a subspace of $Q$) is QMH, with respective
functions denoted $\accentset{\rule{2.45004pt}{0.6pt}}{w}_{e}$ and
$\underaccent{\rule{2.45004pt}{0.6pt}}{w}_{e}$, such that the following
compatibility condition holds: If
$f\in\mathcal{Q}(e)\cap\mathcal{Q}(e^{\prime})$ and $v\in V(e)\cap
V(e^{\prime})$, then

$\displaystyle\underaccent{\rule{2.45004pt}{0.6pt}}{w}_{e}(v,f)$
$\displaystyle=\underaccent{\rule{2.45004pt}{0.6pt}}{w}_{e^{\prime}}(v,f)\quad\text{
and
}\quad\accentset{\rule{2.45004pt}{0.6pt}}{w}_{e}(v,f)=\accentset{\rule{2.45004pt}{0.6pt}}{w}_{e^{\prime}}(v,f)$

There are QMH complexes which are not LMH complexes; see Figure 3 for an
example.

Figure 2. A complex which is QMH but not LMH [Sal93, Fig. 2a]

Figure 3. A complex which is QMH and LMH but not MH [Sal93, Fig. 2b]

######  Definition 5.

A regular cell complex $Q$ is metrical-hemisphere (MH) if it is both QMH and
LMH, and for all $e\in\mathcal{Q}$, $e^{\prime}\in\mathcal{Q}(e)$, and $v\in
V(e)$, we have

$\displaystyle\underaccent{\rule{2.45004pt}{0.6pt}}{w}(v,e^{\prime})$
$\displaystyle=\underaccent{\rule{2.45004pt}{0.6pt}}{w}_{e}(v,e^{\prime}),\quad\text{
and
}\quad\accentset{\rule{2.45004pt}{0.6pt}}{w}(v,e^{\prime})=\accentset{\rule{2.45004pt}{0.6pt}}{w}_{e}(v,e^{\prime}).$

There are complexes which are both QMH and LMH but not MH. See Figure 3 for an
example. (The gray portion is a 2-cell.) Important examples of MH complexes
come from hyperplane arrangements. This is explained in more detail in Section
3.

To a QMH or LMH complex $Q$ there is a regular cell complex
$X=\mathrm{Sal}(Q)$, which we call the _Salvetti complex_ of $Q$ built in the
following way.

Set $X^{(0)}=Q^{(0)}$. For each edge $e$ of $Q$ with vertices $v_{1},v_{2}$,
attach two 1-cells $\langle e;v_{1}\rangle$, $\langle e;v_{2}\rangle$ to
$v_{1},v_{2}$ in $X$ (so that $\partial\langle
e;v_{i}\rangle=\\{v_{1},v_{2}\\}$). We choose an orientation on $\langle
e,v_{i}\rangle$ so that $v_{i}$ is the source of this edge. This fully
describes the 1-skeleton of $X$ as an oriented graph.

The $i$-skeleton is constructed by induction. Suppose we have already
constructed the $j$-skeleton of $X$ for $1\leq j\leq i-1$. A cell of $X^{(j)}$
will be denoted by $\langle e;v\rangle$ for $e\in\mathcal{Q}^{(j)}$ and
$v\in\mathcal{Q}(e)$. Then for each $i$-cell $e$ of $Q$ and $v\in V(e)$, let
$\phi(v,e):\mathcal{Q}(e)\setminus\\{e\\}\to X^{(i-1)}$ be the map taking a
cell $e^{\prime}$ of $\partial e$ to $\langle
e^{\prime},\underaccent{\rule{2.45004pt}{0.6pt}}{w}(v,e^{\prime})\rangle$. By
the definition of a QMH complex (specifically, (A) and (B)), we have that

$\displaystyle\partial
e\cong\bigcup_{e^{\prime}\in\mathcal{Q}(e)\setminus\\{e\\}}\langle
e^{\prime};\underaccent{\rule{2.45004pt}{0.6pt}}{w}(v,e^{\prime})\rangle.$

So for $i$-cells $e$ of $Q$ and vertices $v$ of $e$, we may attach an
$i$-cell, denoted $\langle e;v\rangle$, via $\phi(e,v)$.

Some properties of $\mathrm{Sal}(Q)$ are summarized in the following

######  Proposition 6.

Let $Q$ be QMH (respectively LMH, MH). Then $\mathrm{Sal}(Q)$ is QMH (LMH, MH,
resp.). There is a natural map $\psi$ from cells of $\mathrm{Sal}(Q)$ to cells
of $Q$ given by $\psi(\langle e;v\rangle)=e$. We also have

$\displaystyle\partial\langle
e;v\rangle=\bigcup_{e^{\prime}\in\mathcal{Q}(e)\setminus\\{e\\}}\langle
e^{\prime};\underaccent{\rule{2.45004pt}{0.6pt}}{w}(v,e^{\prime})\rangle.$

The restriction of $\psi$ to $\mathrm{Sal}(Q)^{(0)}$ is an isomorphism onto
$Q^{(0)}$. In general, for an $i$-cell $e$ of $Q$, the preimage $\psi^{-1}(e)$
consists of $|V(e)|$ $i$-cells of $\mathrm{Sal}(Q)$, i.e., there is one
distinct $i$-cell for every vertex of $e$.

###  1.3. A categorical structure for MH complexes

In this section, let $Q$ be a QMH, LMH, or MH complex with maps
$\accentset{\rule{2.45004pt}{0.6pt}}{w}$ and
$\underaccent{\rule{2.45004pt}{0.6pt}}{w}$ (or
$\accentset{\rule{2.45004pt}{0.6pt}}{w}_{e}$ and
$\underaccent{\rule{2.45004pt}{0.6pt}}{w}_{e}$) as in Definition 2 (resp.
Definition 4), and let $X=\mathrm{Sal}(Q)$ be the Salvetti complex for $Q$. In
this section we define a category $\mathcal{C}=\mathcal{C}(Q)$ based on $Q$
(or more specifically, on $X$).

######  Definition 7.

A _positive (resp. negative) path_ in $X$ is an edge path
$\gamma=(e_{1},\dots,e_{k})$ which is a positive (resp. negative) path in the
1-skeleton of $X$ when $X^{(1)}$ is viewed as an oriented graph with
orientation described above. A positive path $\gamma$ has an associated
negative path $\gamma^{-1}$ formed by traversing $\gamma$ in the reverse
direction (so $\gamma^{-1}=(e_{k}^{-1},\dots,e_{1}^{-1})$ where $e^{-1}$
denotes the edge $e$ with its opposite orientation). The length of $\gamma$ is
defined to be $\ell(\gamma)=k$. A _minimal positive path_ in $X$ is a positive
path $\gamma$ between $v_{1}$ and $v_{2}$ so that
$\ell(\gamma)=d(v_{1},v_{2})$.

Note that there is at least one minimal positive path between any two
vertices.

The following definition will be of use later.

######  Definition 8.

Let $\gamma$ be a positive path in $X$, say $\gamma=(\langle
v_{1},e_{1}\rangle,\dots,\langle v_{k},e_{k}\rangle)$, where each $e_{i}$ is
an edge of $Q$ from $v_{i}$ to $v_{i+1}$, then the path $\gamma^{\circ}$
_opposite to_ $\gamma$ is given by $\gamma^{\circ}(\langle
v_{k+1},e_{k}\rangle,\langle v_{k},e_{k-1}\rangle,\dots,\langle
v_{2},e_{1}\rangle)$.

######  Definition 9.

Let $\gamma_{1},\gamma_{2}$ be positive paths in $X$ with the same initial and
terminal vertices. We say $\gamma_{1}$ and $\gamma_{2}$ are _elementarily
equivalent_ if we can write

$\displaystyle\gamma_{1}$ $\displaystyle=\alpha\hat{\gamma}_{1}\beta,\text{
and}$ $\displaystyle\gamma_{2}$ $\displaystyle=\alpha\hat{\gamma}_{2}\beta,$

where

  1. (i)

$\alpha$ and $\beta$ are positive paths, and

  2. (ii)

$\hat{\gamma}_{1}$ and $\hat{\gamma}_{2}$ are minimal positive paths from a
vertex $v$ of $X$ to the vertex $\accentset{\rule{2.45004pt}{0.6pt}}{w}(v,e)$,
where $e$ is a cell of $Q$ containing $v$.

That is, we can derive $\gamma_{2}$ from $\gamma_{1}$ by replacing a minimal
positive subpath (from $v$ to $\accentset{\rule{2.45004pt}{0.6pt}}{w}(v,e)$)
with another minimal positive path. We say $\gamma_{1}$ and $\gamma_{2}$ are
_equivalent_ , and write $\gamma_{1}\sim\gamma_{2}$, if there is a finite
sequence of elementary equivalences taking $\gamma_{1}$ to $\gamma_{2}$. The
$\sim$ equivalence class of a positive path $\gamma$ is denoted $[\gamma]$.

We note that elementary equivalence clearly does not change lengths, so if
$\gamma_{1}\sim\gamma_{2}$, we know that $\ell(\gamma_{1})=\ell(\gamma_{2})$.
In other words, the length of an equivalence class of paths is well defined,
which by abuse of notation we still call $\ell$.

We can now define a category for an MH complex $Q$.

######  Definition 10.

Let $Q$ be a QMH, LMH, or MH complex. We define a category $\mathcal{C}(Q)$ as
follows: the objects of $\mathcal{C}(Q)$ are the vertices of $Q$, and the set
of morphisms from $x$ to $y$ is the set of equivalence classes of positive
edge paths from $x$ to $y$. Composition is given by concatenation of paths.

An immediate consequence of the definition is the following

######  Proposition 11.

For any $x\in\mathsf{Obj}(\mathcal{C}(Q))$, the map
$\gamma\mapsto\gamma^{\circ}$ is an (order reversing) isomorphism of
$(\mathcal{C}_{x\rightarrow},\preccurlyeq)$ with $(\mathcal{C}_{\rightarrow
x},\succcurlyeq)$.

We will study further properties of $\mathcal{C}(Q)$ in the following section.

##  2\. Simplicial MH complexes

We now turn our attention to the main objects of study, which generalize
simplicial hyperplane arrangements.

######  Definition 12.

Let $Q$ be an MH complex. We call $Q$ _simplicial_ if $Q$ is dual to a
simplicial triangulation of a manifold.

Our following terminology departs from [Sal93] (his “MH∗-complex” is our “flat
involutive MH complex”).

######  Definition 13.

Let $Q$ be a simply connected MH (or LMH) complex. We call $Q$ _flat_ if every
pair $\gamma_{1}$, $\gamma_{2}$ of minimal positive paths in $\mathrm{Sal}(Q)$
with the same initial and terminal vertices are equivalent under $\sim$. If
$Q$ is flat and $x,y$ are vertices of $Q$, then the (unique) equivalence class
of positive minimal paths from $x$ to $y$ will be denoted $u(x,y)$.

We say that $Q$ is _involutive_ if there is an automorphism $\phi:Q^{(1)}\to
Q^{(1)}$ of the 1-skeleton of $Q$ of order 2 (i.e., $\phi^{2}=\mathrm{id}$)
such that

  1. (1)

$\phi(v)$ is the unique vertex such that $d(v,\phi(v))=\max_{w\in
Q^{(0)}}d(v,w)$, and

  2. (2)

For all vertices $v,w$ of $Q$, $d(v,\phi(v))=d(v,w)+d(w,\phi(v))$.

For any vertex $x$, we let $\Delta(x)$ denote a (choice of) minimal positive
path from $x$ to $\phi(x)$.

If $Q$ is involutive, the involution $\phi$ induces a map on the set of
positive paths. By Proposition 24 of [Sal93], $\phi$ preserves $\sim$, and
thus induces an automorphism of $\mathcal{C}(Q)$, which by abuse of notation
we still call $\phi$. Again by abuse of notation, the morphism corresponding
to $\Delta(x)$ in $\mathcal{C}$ is denoted $\Delta(x)$. We note that if $Q$ is
also flat, then _every_ minimal positive path from $x$ to $\phi(x)$ in $Q$ is
equivalent to $\Delta(x)$, and thus the morphism $\Delta(x)$ ($=u(x,\phi(x))$)
is determined independent from our choice of path between $x$ and $\phi(x)$ in
$Q$.

We now have the terminology to state our first main theorem.

######  Theorem 14.

Suppose $Q$ is a flat, involutive, simplicial MH complex (or FISMH). Then
$\mathcal{C}(Q)$ is a Garside category under $\Delta$ (i.e., the map
$x\mapsto\Delta(x)$) and $\phi$ as defined above.

We will prove this theorem in a series of lemmas. The first follows
immediately from the definitions.

######  Lemma 15.

Suppose $Q$ is an involutive MH complex, and let $x,y$ be vertices of $Q$. Let
$f$ be a minimal positive path from $x$ to $y$, and let $f^{*}$ be a minimal
positive path from $y$ to $\phi(x)$. Then the concatenation $ff^{*}$ is a
minimal positive path from $x$ to $\phi(x)$. Thus, if $Q$ is also flat,
$ff^{*}\sim\Delta(x)$.

This implies all atoms are simple, since the atoms are paths of length $1$ and
hence necessarily minimal, satisfying (2) of the definition of a Garside
category. The next lemma follows immediately from our previous discussions,
namely that if $\gamma_{1}\sim\gamma_{2}$, then
$\ell(\gamma_{1})=\ell(\gamma_{2})$.

######  Lemma 16.

If $Q$ is an MH complex, then $\mathcal{C}(Q)$ is homogeneous with length
function $\ell$.

The following paraphrases [Sal93, Thm. 31].

######  Lemma 17.

If $Q$ is FISMH, then $\mathcal{C}(Q)$ is cancellative.

We now confirm that $\Delta$ is indeed a natural transformation.

######  Lemma 18.

If $Q$ is a flat involutive MH complex, then $\Delta$ is a natural
transformation from $1_{\mathcal{C}(Q)}$ to $\phi$.

###### Proof.

First, it is clear from the definition that $\Delta(x)$ is a morphism from $x$
to $\phi(x)$. So it remains to verify that for
$x,y\in\mathsf{Obj}(\mathcal{C}(Q))$ and $x\xrightarrow{f}y$, we have

$f\Delta(y)=\Delta(x)\phi(f).$

After translating notation, this is the statement of [Sal93, Prop. 25(i)]. ∎

The last, and least trivial, portion of the argument is to show that
$(\mathcal{C}_{x\rightarrow},\preccurlyeq)$ and $(\mathcal{C}_{\rightarrow
y},\succcurlyeq)$ are lattices. To do this, we adapt arguments of Deligne
[Del72]. By Proposition 11, it suffices to show that
$(\mathcal{C}_{x\rightarrow},\preccurlyeq)$ is a lattice.

In the rest of this section, let $Q$ be FISMH and
$\mathcal{C}=\mathcal{C}(Q)$.

######  Proposition 19.

_[Sal93, Thm. 31(iii)]_ Let $f\in\mathcal{C}_{x\rightarrow}$. Then there is a
$y\in\mathsf{Obj}(\mathcal{C})$ satisfying the following property:
$u(x,z)\preccurlyeq f$ if and only if $u(x,z)\preccurlyeq u(x,y)$.

The following is the main result which we use when showing
$\mathcal{C}_{x\rightarrow}$ is a lattice.

######  Proposition 20.

_[Del72, Prop 1.14, Cor 1.20]_ Let $x\in\mathsf{Obj}(\mathcal{C})$ and
$S\subseteq\mathcal{C}_{x\rightarrow}$ a set of morphisms starting at $x$ with
bounded length (meaning there is some $N$ so that $\ell(f)\leq N$ for all
$f\in S$). Then $S$ satisfies

  1. (1)

$\mathrm{id}_{x}\in S$,

  2. (2)

If $g\in S$ and $f\preccurlyeq g$, then $f\in S$; and

  3. (3)

Suppose $x\xrightarrow{g}y$ and $y_{1},y_{2}$ are vertices adjacent to $y$ in
$Q$. Let $e\in\mathcal{Q}$ be the (unique) 2-cell containing $y$, $y_{1}$, and
$y_{2}$ (this exists since $Q$ is simplicial). Let
$z=\accentset{\rule{2.45004pt}{0.6pt}}{w}(y,e)$. If $gu(y,y_{i})\in S$ for
$i=1,2$, then $gu(y,z)\in S$.

if and only if there is a (unique) morphism $h\in\mathcal{C}_{x\rightarrow}$
such that

$\displaystyle S=\\{\,g\in\mathcal{C}_{x\rightarrow}:g\preccurlyeq h\,\\}$

For the forward implication, we mimic the proof of Proposition 1.14 of
[Del72]; for the reverse, we mimic the proof of [Del72, Cor. 1.20].

###### Proof.

Suppose (1), (2), and (3) are satisfied. Let $h\in S$ with maximal length
(this exists since $S$ has bounded length). We claim this $h$ works. It
suffices to show that

  1. ($*$)

Suppose $x\xrightarrow{g}y$ and $y_{1}\in\mathsf{Obj}(\mathcal{C})$ is
adjacent to $y$ such that (a) $g\preccurlyeq h$ but (b)
$gu(y,y_{1})\not\preccurlyeq h$ and $gu(y,y_{1})\in S$. Then there exists
$g^{\prime}$, $y^{\prime}$, and $y^{\prime}_{1}$ satisfying (a) and (b) with
$g^{\prime}$ strictly longer than $g$.

Since $gu(y,y_{1})\in S$, the maximality of $\ell(h)$ implies $g\not=h$. Thus
there is some $y_{2}\not=y_{1}$ adjacent to $y$ such that
$gu(y,y_{2})\preccurlyeq h$. Letting $e$ be the 2-cell of $Q$ containing the
vertices $y,y_{1},y_{2}$ and $z=\accentset{\rule{2.45004pt}{0.6pt}}{w}(y,e)$,
this implies that $gu(y,z)\in S$ by (3). Note that since
$gu(y,y_{1})\preccurlyeq gu(y,z)$, we must have that $gu(y,z)\not\preccurlyeq
h$.

Consider $u(y,z)$. Let $y_{0}$ be a vertex of $e$ so that
$u(y,y_{2})\preccurlyeq u(y,y_{0})$, and $u(y,y_{0})$ has the maximal length
among paths $p\preccurlyeq u(y,z)$ satisfying $gp\preccurlyeq h$. Let
$g^{\prime}=gu(y,y_{0})$. Let $y^{\prime}$ be the vertex of $u(y^{\prime},z)$
adjacent to $y^{\prime}$. Then $g^{\prime}$ and $y^{\prime}$ satisfy (a) and
(b) above. Uniqueness of $h$ is immediate.

Now suppose $S=\\{\,g\in\mathcal{C}_{x\rightarrow}:g\preccurlyeq h\,\\}$ for
some $h$. (1) and (2) are clear, so we show that $S$ satisfies (3). Suppose
$g,y,y_{i},z$ are as given in (3). We must show that $gu(y,z)\in S$.

Suppose $g\in S$, say $h=gf$ for some morphism $f$ starting at $y$. Then by
cancellation we must have that $u(y,y_{i})\preccurlyeq f$ for $i=1,2$. Let
$y_{0}$ be the vertex guaranteed to exist by 19 for $f$, so that in particular
$u(y,y_{0})\preccurlyeq f$. One may easily verify that because $y_{1}$ and
$y_{2}$ are distinct vertices of $e$ and $u(y,y_{i})\preccurlyeq f$, we must
have that $y_{0}=\accentset{\rule{2.45004pt}{0.6pt}}{w}(y,e)=z$. ∎

From this, the following is immediate.

######  Corollary 21.

Every pair in $\mathcal{C}_{x\rightarrow}$ has a unique greatest lower bound
under $\preccurlyeq$.

###### Proof.

Let $f,g\in\mathcal{C}_{x\rightarrow}$ and let

$\displaystyle S_{f}$
$\displaystyle=\\{\,h\in\mathcal{C}_{x\rightarrow}:h\preccurlyeq f\,\\}$
$\displaystyle S_{g}$
$\displaystyle=\\{\,h\in\mathcal{C}_{x\rightarrow}:h\preccurlyeq g\,\\}$
$\displaystyle S$ $\displaystyle=S_{f}\cap S_{g}.$

It is clear that $S$ has bounded length and satisfies (1), (2), and (3) of
Proposition 20 since both $S_{f}$ and $S_{g}$ do. Thus by Proposition 20 there
is a unique element, which we denote $f\wedge g$ such that

$\displaystyle S=\\{\,h\in\mathcal{C}_{x\rightarrow}:h\preccurlyeq f\wedge
g\,\\}.$

This $f\wedge g$ is clearly a unique greatest lower bound for the pair $f,g$
under $\preccurlyeq$. ∎

In order to show that $\mathcal{C}_{x\rightarrow}$ contains joins, we need the
following

######  Proposition 22.

_[Sal93, Prop. 25(ii)]_ If $Q$ is a flat involutive MH complex and
$c\in\mathcal{C}_{x\rightarrow y}$, then if $f\in\mathcal{C}_{x\rightarrow}$
such that

$f=u(x,x_{1})u(x_{1},x_{2})\dots u(x_{n-1},x_{n})$

for some objects $x_{i}$, we have that $f\preccurlyeq c\Delta^{n}$ (where
$\Delta=\Delta_{y}$ or $\Delta^{y}$ depending on its place in the product).

######  Corollary 23.

Every pair in $\mathcal{C}_{x\rightarrow}$ has a unique least upper bound
under $\preccurlyeq$.

###### Proof.

Let $f,g\in\mathcal{C}_{x\rightarrow}$. Proposition 22 shows that there exists
a common upper bound of $f$ and $g$: if $n_{f},n_{g}$ are the integers
satisfying $f\preccurlyeq\Delta^{n_{f}}$ and $g\preccurlyeq\Delta^{n_{g}}$,
and if $n=\max\\{n_{f},n_{g}\\}$ then both $f\preccurlyeq\Delta^{n}$ and
$g\preccurlyeq\Delta^{n}$.

Let $\mathcal{S}$ be the collection of all sets $S$ of bounded length with
$f,g\in S$ and satisfying (1), (2), and (3) of Proposition 20. Note that
$\mathcal{S}$ is non-empty, since

$\\{\,h\in\mathcal{C}_{x\rightarrow}:h\preccurlyeq\Delta^{n}\,\\}$

is contained in $\mathcal{S}$. Let $S_{0}=\bigcap_{S\in\mathcal{S}}S$. Then
$S_{0}$ has bounded length, satisfies (1),(2),(3), and contains $f,g$ since
each $S\in\mathcal{S}$ satisfies each of these. By Proposition 20, there is a
unique morphism, which we call $f\vee g$, such that

$\displaystyle S_{0}=\\{\,h\in\mathcal{C}_{x\rightarrow}:h\preccurlyeq f\vee
g\,\\}.$

Certainly $f\preccurlyeq f\vee g$ and $g\preccurlyeq f\vee g$, so $f\vee g$ is
an upper bound of $f$ and $g$. If $u$ is another upper bound of $f$ and $g$,
then

$\displaystyle
S^{\prime}\coloneqq\\{\,h\in\mathcal{C}_{x\rightarrow}:h\preccurlyeq
u\,\\}\in\mathcal{S},$

implying $S_{0}\subseteq S^{\prime}$, and in particular $f\vee g\in
S^{\prime}$, hence $f\vee g\preccurlyeq u$. ∎

We can now conclude by proving Theorem 14.

###### Proof (of Theorem 14).

We have seen that $\phi$ is an automorphism of $\mathcal{C}(Q)$. Lemma 18
shows that $\Delta$ is a natural transformation from $1_{\mathcal{C}(Q)}$ to
$\phi$. Lemma 16 shows that $\mathcal{C}(Q)$ is homogeneous and Lemma 17 shows
that $\mathcal{C}(Q)$ is cancellative. Lemma 15 implies that all atoms of
$\mathcal{C}(Q)$ are simple. Corollaries 21 and 23 imply that
$(\mathcal{C}(Q)_{x\rightarrow},\preccurlyeq)$ is a lattice, and their dual
statements (and proofs) imply that $(\mathcal{C}(Q)_{\rightarrow
y},\succcurlyeq)$ is a lattice. Therefore $\mathcal{C}(Q)$ is a Garside
category. ∎

The following is the main consequence that we’re interested in.

######  Corollary 24.

If $Q$ is FISMH, then for any vertex $x$,
$\pi_{1}(\mathrm{Sal}(Q),x)\cong\mathcal{G}_{x}$. In particular,
$\pi_{1}(\mathrm{Sal}(Q))$ is a weak Garside group.

This follows from

######  Lemma 25.

_[Sal93, Cor. 18]_ If $Q$ is an MH complex, then all relations for
$\pi_{1}(\mathrm{Sal}(Q))$ come from the set
$\\{\gamma_{1}\gamma_{2}^{-1}:\gamma_{1},\gamma_{2}\text{ are positive paths
with the same endpoints}\,\\}$.

In other words, homotopies between loops in $\mathrm{Sal}(Q)$ are equivalent
to $\sim$ equivalences of positive subpaths. Since $\mathcal{G}_{x}$ consists
of $\sim$ equivalence classes of positive paths and the formal inverses of
these equivalence classes, Corollary 24 follows.

##  3\. Application to arrangements and matroids

We now discuss applications to simplicial pseudohyperplane arrangements and
their associated matroids. First, we will give further terminology regarding
these arrangements.

Let $\mathcal{A}$ be a pseudosphere arrangement. We say that $\mathcal{A}$ is
_proper_ (also called _essential_ in some texts) if
$\bigcap_{S\in\mathcal{A}}S=\varnothing$. We say that $\mathcal{A}$ is
centrally symmetric if each element $S\in\mathcal{A}$ is preserved under the
central symmetry $-1$ of $\mathbb{S}^{d}$, that is, if $-S=S$ for all
$S\in\mathcal{A}$. We call two arrangements $\mathcal{A}_{1},\mathcal{A}_{2}$
_homeomorphic_ , and write $\mathcal{A}_{1}\cong\mathcal{A}_{2}$, if there is
a homeomorphism of $\mathbb{S}^{d}$ which is a bijection between
$\mathcal{A}_{1}$ and $\mathcal{A}_{2}$.

An _(open) chamber_ of $\mathcal{A}$ is a connected component of
$\mathbb{S}^{d}\setminus\bigcup_{S\in\mathcal{A}}S$. The union of the closures
of the chambers gives a cellulation of $\mathbb{S}^{d}$ (where the
pseudospheres of $\mathcal{A}$ are codimension-1 subcomplexes). We call
$\mathcal{A}$ _simplicial_ if this induced cell structure on $\mathbb{S}^{d}$
is. The _dual complex_ $Q(\mathcal{A})$ to $\mathcal{A}$ is the dual of this
cell structure on $\mathbb{S}^{d}$.

We say that the pseudohyperplane arrangement $c\mathcal{A}$ satisfies the
above properties (e.g., proper, simplicial, etc.) if $\mathcal{A}$ does. A
chamber of $c\mathcal{A}$ is the cone on a chamber of $\mathcal{A}$. We define
$Q(c\mathcal{A})=Q(\mathcal{A})$.

We note that centrally symmetric simplicial arrangements are always proper,
since the intersection of centrally symmetric pseudospheres which bound a
simplex must have empty intersection. We also note that two arrangements are
homeomorphic if and only if their dual complexes are, hence homeomorphism is
determined combinatorially.

The following connects arrangements and MH complexes.

######  Proposition 26.

_[Sal93, Prop. 6]_ Suppose $\mathcal{A}$ is an arrangement of pseudospheres in
$\mathbb{S}^{d}$. Then the complex $Q(\mathcal{A})$ dual to $\mathcal{A}$ is
MH.

Thus, as with the other MH complexes we have considered, there is a Salvetti
complex $\mathrm{Sal}(Q(\mathcal{A}))$, which for brevity we denote
$\mathrm{Sal}(\mathcal{A})$. (We define
$\mathrm{Sal}(c\mathcal{A})=\mathrm{Sal}(\mathcal{A})$.) We now determine
those arrangements $\mathcal{A}$ which give FISMH complexes $Q(\mathcal{A})$,
and hence for which Corollary 14 holds.

######  Lemma 27.

_[Sal93, Thm. 20]_ Suppose $\mathcal{A}$ is a proper pseudosphere arrangement.
Then $Q(\mathcal{A})$ is flat.

The following is clear from the definitions.

######  Lemma 28.

If $\mathcal{A}$ is a proper centrally symmetric pseudosphere arrangement,
then $Q(\mathcal{A})$ is involutive with involution $\phi(x)=-x$.

Since centrally symmetric simplicial arrangements are always proper, we have
the following

######  Proposition 29.

If $\mathcal{A}$ is a centrally symmetric simplicial pseudosphere arrangement,
then $\pi_{1}(\mathrm{Sal}(\mathcal{A}))=\pi_{1}(\mathrm{Sal}(c\mathcal{A}))$
is a weak Garside group.

We can connect the topology of $\mathrm{Sal}(c\mathcal{A})$ more directly with
the arrangement $c\mathcal{A}$ itself. This connection is completely analagous
to the connection between a hyperplane arrangement and its Salvetti complex
via the complement of its complexification.

######  Definition 30.

Let $c\mathcal{A}$ be a pseudohyperplane arrangement in $\mathbb{R}^{d}$. For
$H\in c\mathcal{A}$, define

$\displaystyle H_{\mathbb{C}}=H+iH\subseteq\mathbb{C}^{d}$

(where $H+iH=\\{\,h_{1}+ih_{2}:h_{i}\in H\,\\}$). Then, we define

$\displaystyle M(c\mathcal{A})=\mathbb{C}^{d}\setminus\bigcup_{H\in
c\mathcal{A}}H_{\mathbb{C}}.$

We would like to say that $M(c\mathcal{A})$ is homotopy equivalent to
$\mathrm{Sal}(c\mathcal{A})$, but this is not quite true. Instead, we need the
following variation.

Let $\hat{Q}=\hat{Q}(\mathcal{A})=\hat{Q}(c\mathcal{A})$ be the cellulation of
the unit ball $D^{d+1}$ of $\mathbb{R}^{d+1}$ obtained by identifying
$\partial D^{d+1}$ with the cellulation $Q(\mathcal{A})$ of the unit sphere
$\mathbb{S}^{d}$. We have the following

######  Proposition 31.

_[Sal93, Prop. 9]_ The complex $\hat{Q}(\mathcal{A})$ is an MH complex with
exactly one cell (namely, $D^{d+1}$) of top dimension.

Thus we can define
$\widehat{\mathrm{Sal}}(c\mathcal{A})=\widehat{\mathrm{Sal}}(\mathcal{A})\coloneqq\mathrm{Sal}(\hat{Q}(\mathcal{A}))$,
which we call the _completed Salvetti complex_. Note that the $d$-skeletons of
$Q$ and $\hat{Q}$ are the same, so the $d$-skeletons of
$\mathrm{Sal}(\mathcal{A})$ and $\widehat{\mathrm{Sal}}(\mathcal{A})$ are the
same. In particular, since we’re assuming $d\geq 2$, the 2-skeletons are the
same, so
$\pi_{1}(\widehat{\mathrm{Sal}}(\mathcal{A}))\cong\pi_{1}(\mathrm{Sal}(\mathcal{A}))$.
The relevant result for us is the following (translated into our notation)

######  Proposition 32.

_[Des16, Thm. 2.30]_ If $c\mathcal{A}$ is a pseudohyperplane arrangement, then
$M(c\mathcal{A})$ is homotopy equivalent to
$\widehat{\mathrm{Sal}}(c\mathcal{A})$.

In particular,
$\pi_{1}(M(c\mathcal{A}))\cong\pi_{1}(\mathrm{Sal}(c\mathcal{A}))$. Moreover,
[Sal93, §6] implies $\widehat{\mathrm{Sal}}(c\mathcal{A})$ is a $K(\pi,1)$ for
simplicial arrangements, i.e., has contractible universal cover. (This was
also shown in [Des16].) So, we can summarize all of the above results (in
addition to the known result for $d=1$) into the following

######  Theorem 33.

If $c\mathcal{A}$ is a proper simplicial pseudohyperplane arrangement, then
$\pi=\pi_{1}(M(c\mathcal{A}))$ is a weak Garside group and $M(c\mathcal{A})$
is a $K(\pi,1)$.

The combinatorial information contained in a pseudosphere arrangement can be
extracted to objects known as _oriented matroids_ , which have broad uses in
mathematics. We recall this connection here, and explain how our above results
apply to this setting.

######  Definition 34.

An oriented matroid is a triple $\mathcal{M}=(E,\mathcal{E},{}^{*})$ where $E$
is a finite set, $\mathcal{E}$ a collection of non-empty subsets of $E$, and
${}^{*}:E\to E$ a “fixed point-free” involution of $E$ (so $(x^{*})^{*}=x$ and
$x^{*}\not=x$) such that

  1. (1)

If $A,B\in\mathcal{E}$ and $A\subseteq B$, then $A=B$,

  2. (2)

If $S\in\mathcal{E}$ then $S^{*}\in\mathcal{E}$ and $S\cap S^{*}=\varnothing$,
and

  3. (3)

For $S,T\in\mathcal{E}$ with $x\in S\cap T^{*}$ and $S\not=T^{*}$, there is a
$C\in\mathcal{E}$ with $C\subseteq(S\cup T)\setminus\\{x,x^{*}\\}$.

We call $E$ the _ground set_ and the elements of $\mathcal{E}$ the _circuits_
of the oriented matroid $\mathcal{M}$. The _rank_ $\mathrm{rk}(\mathcal{M})$
of $\mathcal{M}$ is the minimal cardinality of a set in $\mathcal{E}$. An
_isomorphism_ between oriented matroids is a bijection between ground sets
which induces a bijection on the circuits and commutes with the involution.

Let $\mathcal{A}$ be a pseudosphere arrangement in $\mathbb{S}^{d}$ with each
element endowed with a choice of orientation. Let

$\displaystyle E=E(\mathcal{A})=\\{\,H:H\text{ a halfspace of
}S\in\mathcal{A}\,\\}.$

Define ∗ on $E$ by exchanging the halfspaces of any given pseudosphere, e.g.,
if $H_{1},H_{2}$ are the halfspaces of $S$ then $H_{1}^{*}=H_{2}$ and
$H_{2}^{*}=H_{1}$. We then let $\mathcal{E}=\mathcal{E}(\mathcal{A})$ be the
collection of minimal subsets $C$ of $E$ with

  1. (1)

$C\cap C^{*}=\varnothing$, and

  2. (2)

$\bigcup_{H\in C}\overline{H}=\mathbb{S}^{d}$,

where $\overline{H}$ is the topological closure of the halfspace $H$ in
$\mathbb{S}^{d}$. By [FL78, Thm. 16],
$\mathcal{M}(\mathcal{A})\coloneqq(E(\mathcal{A}),\mathcal{E}(\mathcal{A}),{}^{*})$
is an oriented matroid of rank $d+1$. ($\mathcal{M}(\mathcal{A})$ is not to be
confused with $M(c\mathcal{A})$.) The content of the Folkman-Lawrence
realization theorem is the following.

######  Proposition 35.

_[FL78, Thm. 20]_ Let $\mathcal{M}=(E,\mathcal{E},{}^{*})$ be an oriented
matroid with $\mathrm{rk}(\mathcal{M})=r$. Then there is a proper central
pseudosphere arrangement $\mathcal{A}(\mathcal{M})$ in $\mathbb{S}^{r-1}$ such
that the set of cells of the corresponding dual complex
$Q(\mathcal{A}(\mathcal{M}))$ are order-isomorphic to the subsets $C$ of
$\mathcal{E}$ with $C\cap C^{*}=\varnothing$, with vertices of
$Q(\mathcal{A}(\mathcal{M}))$ mapping to elements of $\mathcal{E}$. Moreover,
if $\mathcal{M}$ is an oriented matroid, then

$\mathcal{M}(\mathcal{A}(\mathcal{M}))\cong\mathcal{M},$

and if $\mathcal{A}$ is a proper pseudosphere arrangement, then

$\mathcal{A}(\mathcal{M}(\mathcal{A}))\cong\mathcal{A}$

(where $\cong$ denotes isomorphism and homeomorphism, respectively).

In particular, two pseudosphere arrangements are homeomorphic if and only if
their corresponding oriented matroids are isomorphic.

This allows us to sensibly define
$\mathrm{Sal}(\mathcal{M})=\mathrm{Sal}(\mathcal{A}(\mathcal{M}))$ and
$\widehat{\mathrm{Sal}}(\mathcal{M})=\widehat{\mathrm{Sal}}(\mathcal{A}(\mathcal{M}))$
for any oriented matroid $\mathcal{M}$. By unraveling definitions, one sees
that these are equivalent to typical definitions of a Salvetti complex of an
oriented matroid [Zie87, §5.5]. Thus Theorem 33 can be rephrased in terms of
matroids as follows.

######  Theorem 36.

If $\mathcal{M}$ is an oriented matroid and its pseudosphere realization
$\mathcal{A}(\mathcal{M})$ is simplicial, then
$\pi=\pi_{1}(\mathrm{Sal}(\mathcal{M}))$ is a weak Garside group and
$\widehat{\mathrm{Sal}}(\mathcal{M})$ is a $K(\pi,1)$.

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