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Journal of Applied and Computational Topology (2019) 3:315–358
https://doi.org/10.1007/s41468-019-00038-7
Toward a spectral theory of cellular sheaves
Jakob Hansen1·Robert Ghrist1,2
Received: 4 August 2018 / Accepted: 16 August 2019 / Published online: 30 August 2019
© The Author(s) 2019
Abstract
This paper outlines a program in what one might call spectral sheaf theory—an exten-
sion of spectral graph theory to cellular sheaves. By lifting the combinatorial graph
Laplacian to the Hodge Laplacian on a cellular sheaf of vector spaces over a regular
cell complex, one can relate spectral data to the sheaf cohomology and cell structure in
a manner reminiscent of spectral graph theory. This work gives an exploratory intro-
duction, and includes discussion of eigenvalue interlacing, sparsification, effective
resistance, synchronization, and sheaf approximation. These results and subsequent
applications are prefaced by an introduction to cellular sheaves and Laplacians.
Keywords Cohomology ·Cellular sheaf theory ·Spectral graph theory ·
Effective resistance ·Eigenvalue interlacing
Mathematics Subject Classification MSC 55N30 ·MSC 05C50
1 Introduction
In spectral graph theory, one associates to a combinatorial graph additional algebraic
structures in the form of square matrices whose spectral data is then investigated and
related to the graph. These matrices come in several variants, most particularly degree
and adjacency matrices, Laplacian matrices, and weighted or normalized versions
thereof. In most cases, the size of the implicated matrix is based on the vertex set,
while the structure of the matrix encodes data carried by the edges.
BRobert Ghrist
ghrist@math.upenn.edu
Jakob Hansen
jhansen@math.upenn.edu
1Department of Mathematics, University of Pennsylvania, 209 S. 33rd St., Philadelphia, PA
19104-6395, USA
2Department of Electrical and Systems Engineering, University of Pennsylvania, 200 S. 33rd St.,
Philadelphia, PA 19104-6314, USA
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316 J. Hansen , R. Ghrist
To say that spectral graph theory is useful is an understatement. Spectral methods
are key in such disparate fields as data analysis (Belkin and Niyogi 2003; Coifman
and Lafon 2006), theoretical computer science (Hoory et al. 2006; Cvetcovi´c and
Simi´c2011), probability theory (Lyons and Peres 2016), control theory (Bullo 2018),
numerical linear algebra (Spielman and Teng 2014), coding theory (Spielman 1996),
and graph theory itself (Chung 1992; Brouwer and Haemers. 2012).
Much of spectral graph theory focuses on the Laplacian, leveraging its unique com-
bination of analytic, geometric, and probabilistic interpretations in the discrete setting.
This is not the complete story. Many of the most well-known and well-used results
on the spectrum of the graph Laplacian return features that are neither exclusively
geometric nor even combinatorial in nature, but rather more qualitative. For example,
it is among the first facts of spectral graph theory that the multiplicity of the zero eigen-
value of the graph Laplacian enumerates connected components of the graph, and the
relative size of the smallest nonzero eigenvalue in a connected graph is a measure of
approximate dis-connectivity. Such features are topological.
There is another branch of mathematics in which Laplacians hold sway: Hodge
theory. This is the slice of algebraic and differential geometry that uses Laplacians
on (complex) Riemannian manifolds to characterize global features. The classical
initial result is that one recovers the cohomology of the manifold as the kernel of the
Laplacian on differential forms (Abraham et al. 1988). For example, the dimension
of the kernel of the Laplacian on 0-forms (R-valued functions) is equal to the rank
of H0, the 0-th cohomology group (with coefficients in R), whose dimension is the
number of connected components. In spirit, then, Hodge theory categorifies elements
of spectral graph theory.
Hodge theory, like much of algebraic topology, survives the discretization from
Riemannian manifolds to (weighted) cell complexes (Eckmann 1945; Friedman 1998).
The classical boundary operator for a cell complex and its formal adjoint combine
to yield a generalization of the graph Laplacian which, like the Laplacian of Hodge
theory, acts on higher dimensional objects (cellular cochains, as opposed to differential
forms). The kernel of this discrete Laplacian is isomorphic to the cellular cohomology
of the complex with coefficients in the reals, generalizing the connectivity detection
of the graph Laplacian in grading zero. As such, the spectral theory of the discrete
Laplacian offers a geometric perspective on algebraic-topological features of higher-
dimensional complexes. Laplacians of higher-dimensional complexes have been the
subject of recent investigation (Parzanchevski 2013; Steenbergen 2013; Horak and
Jost 2013).
This is not the end. Our aim is a generalization of both spectral graph theory and
discrete Hodge theory which ties in to recent developments in topological data analysis.
The past two decades have witnessed a burst of activity in computing the homology
of cell complexes (and sequences thereof) to extract robust global features, leading
to the development of specialized tools, such as persistent homology, barcodes, and
more, as descriptors for cell complexes (Carlsson 2012; Edelsbrunner and Harer 2010;
Kaczynski et al. 2004; Otter et al. 2017).
Topological data analysis is evolving rapidly. One particular direction of evolution
concerns a change in perspective from working with cell complexes as topological
spaces in and of themselves to focusing instead on data over a cell complex—viewing
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Toward a spectral theory of cellular sheaves 317
the cell complex as a base on which data to be investigatedresides. For example, one can
consider scalar-valued data over cell complexes, as occurs in weighted networks and
complexes; or sensor data, as occurs in target enumeration problems (Curry et al. 2012).
Richer data involves vector spaces and linear transformations, as with recent work in
cryo-EM (Hadani and Singer 2011) and synchronization problems (Bandeira 2015).
Recent work in TDA points to a generalization of these and related data structures
over topological spaces. This is the theory of sheaves.
We will work exclusively with cellular sheaves (Curry 2014). Fix a (regular, locally
finite) cell complex—a triangulated surface will suffice for purposes of imagination. A
cellular sheaf of vector spaces is, in essence, a data structure on this domain, assigning
local data (in the form of vector spaces) to cells and compatibility relations (linear
transformations) between cells of incident ascending dimension. These structure maps
send data over vertices to data over incident edges, data over edges to data over incident
2-cells, etc. As a trivial example, the constant sheaf assigns a rank-one vector space to
each cell and identity isomorphisms according to boundary faces. More interesting is
the cellular analogue of a vector bundle: a cellular sheaf which assigns a fixed vector
space of dimension nto each cell and isomorphisms as linear transformations (with
specializations to O(n)or SO(n)as desired).
The data assigned to a cellular sheaf naturally arranges into a cochain complex
graded by dimension of cells. As such, cellular sheaves possess a Laplacian that
specializes to the graph Laplacian and the Hodge Laplacian for the constant sheaf.
For cellular sheaves of real vector spaces, a spectral theory—an examination of the
eigenvalues and eigenvectors of the sheaf Laplacian—is natural, motivated, and, to
date, unexamined apart from a few special cases (see Sect. 3.6).
This paper sketches an emerging spectral theory for cellular sheaves. Given the
motivation as a generalization of spectral graph theory, we will often specialize to
cellular sheaves over a 1-dimensional cell complex (that is, a graph, allowing when
necessary multiple edges between a pair of vertices). This is mostly for the sake of
simplicity and initial applications, as zero- and one-dimensional homological invari-
ants are the most readily applicable. However, as the theory is general, we occasionally
point to higher-dimensional side-quests.
The plan of this paper is as follows. In Sect. 2, we cover the necessary topological
and algebraic preliminaries, including definitions of cellular sheaves. Next, Sect. 3
gives definitions of the various matrices involved in the extension of spectral theory to
cellular sheaves. Section 4uses these to explore issues related to harmonic functions
and cochains on sheaves. In Sect. 5, we extend various elementary results from spectral
graph theory to cellular sheaves. The subsequent two sections treat more sophisticated
topics, effective resistance (Sect. 6) and the Cheeger inequality (Sect. 7), for which
we have some preliminary results. We conclude with outlines of potential applications
for the theory in Sect. 8and directions for future inquiry in Sect. 9.
The results and applications we sketch are at the beginnings of the subject, and a
great deal more in way of fundamental and applied work remains.
This paper has been written in order to be readable without particular expertise
in algebraic topology beyond the basic ideas of cellular homology and cohomology.
Category-theoretic terminology is used sparingly and for concision. Given the well-
earned reputation of sheaf theory as difficult for the non-specialist, we have provided
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318 J. Hansen , R. Ghrist
an introductory section with terminology and core concepts, noting that much more is
available in the literature (Bredon 1997; Kashiwara and Schapira 1990). Our recourse
to the cellular theory greatly increases simplicity, readability, and applicability, while
resonating with the spirit of spectral graph theory. There are abundant references
available for the reader who requires more information on algebraic topology (Hatcher
2001), applications thereof (Edelsbrunner and Harer 2010;Ghrist2014), and cellular
sheaf theory (Curry 2014;Ghrist2014).
2 Preliminaries
2.1 Cell complexes
Definition 2.1 Aregular cell complex is a topological space Xwith a partition into
subspaces {Xα}α∈PXsatisfying the following conditions:
1. For each x∈X, every sufficiently small neighborhood of xintersects finitely
many Xα.
2. For all α, β,Xα∩Xβ= ∅only if Xβ⊆Xα.
3. Every Xαis homeomorphic to Rnαfor some nα.
4. For every α, there is a homeomorphism of a closed ball in Rnαto Xαthat maps
the interior of the ball homeomorphically onto Xα.
Condition (2) implies that the set PXhas a poset structure, given by β≤αiff
Xβ⊆Xα. This is known as the face poset of X. The regularity condition (4) implies
that all topological information about Xis encoded in the poset structure of PX.For
our purposes, we will identify a regular cell complex with its face poset, writing the
incidence relation βα. The class of posets that arise in this way can be characterized
combinatorially (Björner 1984). For our purposes, a morphism of cell complexes is a
morphism of posets between their face incidence posets that arises from a continuous
map between their associated topological spaces. In particular, morphisms of simplicial
and cubical complexes will qualify as morphisms of regular cell complexes.
The class of regular cell complexes includes simplicial complexes, cubical com-
plexes, and so-called multigraphs (as 1-dimensional cell complexes). As nearly every
space that can be characterized combinatorially can be represented as a regular cell
complex, these will serve well as a default class of spaces over which to develop a
combinatorial spectral theory of sheaves. We note that the spectral theory of complexes
has heretofore been largely restricted to the study of simplicial complexes (Schaub
et al. 2018). A number of our results will specialize to results about the spectra of
Hodge Laplacians of regular cell complexes by restricting to the constant sheaf.
A few notions associated to cell complexes will be useful.
Definition 2.2 The k-skeleton of a cell complex X, denoted X(k), is the subcomplex
of Xconsisting of cells of dimension at most k.
Definition 2.3 Let σbe a cell of a regular cell complex X. The star of σ, denoted
st(σ ),isthesetofcellsτsuch that στ.
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Toward a spectral theory of cellular sheaves 319
Topologically, st(σ ) is the smallest open collection of cells containing σ,arole
we might denote as the “smallest cellular neighborhood” of σ. Stars serve an impor-
tant purpose in giving combinatorial analogues of topological notions for maps. For
instance, a morphism f:X→Yof cell complexes may be locally injective as defined
on the topological spaces. Topologically, the condition for local injectivity is simply
that every point in Xhave a neighborhood on which fis injective. Translating this to
cell complexes, we require that for every cell σ∈X,fis injective on st(σ ).
Topological continuity ensures that the preimage of a star st(σ ) under a cell mor-
phism f:X→Yis a union of stars; if fis locally injective, we see that it must be
a disjoint union of stars. A locally injective map is, further, a covering map if on each
component of f−1(st(σ )),fis an isomorphism. That is, the fiber of a star consists of
a disjoint union of copies of that star.
2.2 Cellular sheaves
Let Xbe a regular cell complex. A cellular sheaf attaches data spaces to the cells
of Xtogether with relations that specify when assignments to these data spaces are
consistent.
Definition 2.4 Acellular sheaf of vector spaces on a regular cell complex Xis an
assignment of a vector space F(σ ) to each cell σof Xtogether with a linear transfor-
mation Fστ:F(σ ) →F(τ ) for each incident cell pair στ. These must satisfy
both an identity relation Fσσ=id and the composition condition:
ρστ⇒Fρτ=Fστ◦Fρσ.
The vector space F(σ ) is called the stalk of Fat σ. The maps Fστare called the
restriction maps.
For experts, this definition at first seems only reminiscent of the notion of sheaves
familiar to topologists. The depth of the relationship is explained in detail in Curry
(2014), but the essence is this: the data of a cellular sheaf on Xspecifies spaces of local
sections on a cover of Xgiven by open stars of cells. This translates in two different
ways into a genuine sheaf on a topological space. One may either take the Alexandrov
topology on the face incidence poset of the complex, or one may view the open stars
of cells and their natural refinements a basis for the topology of X. There then exists a
natural completion of the data specified by the cellular sheaf to a constructible sheaf
on X.
One may compress the definition of a cellular sheaf to the following: If Xis a
regular cell complex with face incidence poset PX, viewed as a category, a cellular
sheaf is a functor F:PX→Vectkto the category of vector spaces over a field k.
Definition 2.5 Let Fbe a cellular sheaf on X.Aglobal section x of Fis a choice
xσ∈F(σ ) for each cell σof Xsuch that xτ=Fστxσfor all στ. The space of
global sections of Fis denoted Γ(X;F).
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320 J. Hansen , R. Ghrist
Perhaps the simplest sheaf on any complex is the constant sheaf with stalk V, which
we will denote V. This is the sheaf with all stalks equal to Vand all restriction maps
equal to the identity.
2.2.1 Cosheaves
In many situations it is more natural to consider a dual construction to a cellular sheaf.
Acellular cosheaf preserves stalk data but reverses the direction of the face poset,
and with it, the restriction maps.
Definition 2.6 A cellular cosheaf of vector spaces on a regular cell complex Xis an
assignment of a vector space F(σ ) to each cell σof Xtogether with linear maps
Fστ:F(τ ) →F(σ ) for each incident cell pair στwhich satisfies the identity
(Fσσ=id) and composition condition:
ρστ⇒Fρτ=Fρσ◦Fστ.
More concisely, a cellular cosheaf is a functor Pop
X→Vectk. The contravariant
functor Hom(•,k):Vectop
k→Vectkgives every cellular sheaf Fa dual cosheaf ˆ
F
whose stalks are Hom(F(σ ), k).
2.2.2 Homology and cohomology
The cells of a regular cell complex have a natural grading by dimension. By regularity
of the cell complex, this grading can be extracted from the face incidence poset as the
height of a cell in the poset. This means that a cellular sheaf has a graded vector space
of cochains
Ck(X;F)=
dim(σ )=k
F(σ ).
To develop this into a chain complex, we need a boundary operator and a notion of
orientation—a signed incidence relation on PX.Thisisamap[•:•]:PX×PX→
{0,±1}satisfying the following conditions:
1. If [σ:τ] = 0, then στand there are no cells between σand τin the incidence
poset.
2. For any στ,γ∈PX[σ:γ][γ:τ]=0.
Given a signed incidence relation on PX, there exist coboundary maps δk:
Ck(X;F)→Ck+1(X;F). These are given by the formula
δk|F(σ ) =
dim(τ )=k+1[σ:τ]Fστ,
or equivalently,
(δkx)τ=
dim(σ )=k[σ:τ]Fστ(xσ).
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Toward a spectral theory of cellular sheaves 321
Here we use subscripts to denote the value of a cochain in a particular stalk; that is,
xσis the value of the cochain xin the stalk F(σ ).
It is a simple consequence of the properties of the incidence relation and the com-
mutativity of the restriction maps that δk◦δk−1=0, so these coboundary maps define
a cochain complex and hence a cohomology theory for cellular sheaves. In particular,
H0(X;F)is naturally isomorphic to Γ(X;F), the space of global sections. An anal-
ogous construction defines a homology theory for cosheaves. Cosheaf homology may
be thought of as dual to sheaf cohomology in a Poincaré-like sense. That is, frequently
the natural analogue of degree zero sheaf cohomology is degree ncosheaf homology.
A deeper formal version of this fact, exploiting an equivalence of derived categories,
may be found in Curry (2014), ch. 12.
There is a relative version of cellular sheaf cohomology. Let Abe a subcomplex of
X. There is a natural subspace of Ck(X;F)consisting of cochains which vanish on
stalks over cells in A. The coboundary of a cochain which vanishes on Aalso vanishes
on A, since any cell in A(k+1)has only cells in A(k)on its boundary. We therefore get
a subcomplex C•(X,A;F)of C•(X;F). The cohomology of this subcomplex is the
relative sheaf cohomology H•(X,A;F). The natural maps between these spaces of
cochains constitute a short exact sequence of complexes
0→C•(X,A;F)→C•(X;F)→C•(A;F)→0,
from which a long exact sequence for relative sheaf cohomology arises:
0→H0(X,A;F)→H0(X;F)→H0(A;F)→H1(X,A;F)→···
2.2.3 Sheaf morphisms
Definition 2.7 If Fand Gare sheaves on a cell complex X, a sheaf morphism ϕ:
F→Gis a collection of maps ϕσ:F(σ ) →G(σ ) for each cell σof X, such that for
any στ,ϕτ◦Fστ=Gστ◦ϕσ. Equivalently, all diagrams of the following form
commute:
F(σ ) G(σ )
F(τ ) G(τ )
Fστ
ϕσ
Gστ
ϕτ
This commutativity condition assures that a sheaf morphism ϕ:F→Ginduces maps
ϕk:Ck(X;F)→Ck(X;G)which commute with the coboundary maps, resulting in
the induced maps on cohomology Hkϕ:Hk(X;F)→Hk(X;G).
2.2.4 Sheaf operations
There are several standard operations that act on sheaves to produce new sheaves.
Definition 2.8 (Direct sum)IfFand Gare sheaves on X, their direct sum F⊕Gis a
sheaf on Xwith (F⊕G)(σ ) =F(σ ) ⊕G(σ ). The restriction maps are (F⊕G)στ=
Fστ⊕Gστ.
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322 J. Hansen , R. Ghrist
Definition 2.9 (Tensor product)IfFand Gare sheaves on X, their tensor product
F⊗Gis a sheaf on Xwith (F⊗G)(σ ) =F(σ ) ⊗G(σ ). The restriction maps are
(F⊗G)στ=Fστ⊗Gστ.
Definition 2.10 (Pullback)If f:X→Yis a morphism of cell complexes and F
is a sheaf on Y,thepullback f ∗Fis a sheaf on Xwith f∗F(σ ) =F(f(σ )) and
(f∗F)στ=Ff(σ )f(τ ).
Definition 2.11 (Pushforward) The full definition of the pushforward of a cellular
sheaf is somewhat more categorically involved than the previous constructions. If
f:X→Yis a morphism of cell complexes and Fis a sheaf on X,thepushforward
f∗Fis a sheaf on Ywith stalks f∗F(σ ) given as the limit limσf(τ ) F(τ ).The
restriction maps are induced by the restriction maps of F, since whenever σσ,the
cone for the limit defining f∗F(σ ) contains the cone for the limit defining f∗F(σ ),
inducing a unique map f∗F(σ ) →f∗F(σ ).
In this paper, we will mainly work with pushforwards over locally injective cell
maps, that is, those whose geometric realizations are locally injective (see Sect. 2.1).
If f:X→Yis locally injective, every cell σ∈Xmaps to a cell of the same
dimension, and for every cell σ∈Y,f−1(st(σ )) is a disjoint union of subcomplexes,
each of which maps injectively to Y. In this case, f∗F(σ ) σ∈f−1(σ ) F(σ ), and
(f∗F)στ=(σ τ)∈f−1(σ τ) Fστ. This computational formula in fact holds
more generally, if the stars of cells in f−1(σ ) are disjoint.
Those familiar with the definitions of pushforward and pullback for sheaves over
topological spaces will note a reversal of fates when we define sheaves over cell
complexes. Here the pullback is simple to define, while the pushforward is more
involved. This complication arises because cellular sheaves are in a sense defined
pointwise rather than over open sets.
3 Definitions
3.1 Weighted cellular sheaves
Let k=Ror C. A weighted cellular sheaf is a cellular sheaf with values in k-
vector spaces where the stalks have additionally been given an inner product structure.
Adding the condition of completeness to the stalks, one may view this as a functor
PX→Hilbk, where Hilbkis the category whose objects are Hilbert spaces over k
and whose morphisms are (bounded) linear maps.
The inner products on stalks of Fextend by the orthogonal direct sum to inner
products on Ck(X;F), making these Hilbert spaces as well. The canonical inner
products on direct sums and subspaces of Hilbert spaces give the direct sum and tensor
product of weighted cellular sheaves weighted structures. Similarly, the pullbacks
and pushforwards (over locally injective maps) of a weighted sheaf have canonical
weighted structures given by their computational formulae in Sect. 2.2.4.
Every morphism T:V→Wbetween Hilbert spaces admits an adjoint map T∗:
W→V, determined by the property that for all v∈V,w ∈W,w, Tv=T∗w, v.
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Toward a spectral theory of cellular sheaves 323
One may readily check that (T∗)∗=T. This fact gives the category Hilbkadagger
structure, that is, a contravariant endofunctor † (here the adjoint operation ∗) which
acts as the identity on objects and squares to the identity. In a dagger category, the
notion of unitary isomorphisms makes sense: they are the invertible morphisms T
such that T†=T−1.
The dagger structure of Hilbkintroduces some categorical subtleties into the study
of weighted cellular sheaves. The space of global sections of a cellular sheaf is defined
in categorical terms as the limit of the functor X→Vect defining the sheaf. This
defines the space of global sections up to unique isomorphism. We might want a
weighted space of global sections to be a sort of limit in Hilbkwhich is defined up
to unique unitary isomorphism. This is the notion of dagger limit, recently studied
in Heunen and Karvonen (2019). Unfortunately, this work showed that Hilbkdoes
not have all dagger limits; in particular, pullbacks over spans of noninjective maps do
not exist. As a result, there is no single canonical way to define an inner product on
the space of global sections of a cellular sheaf F. There are two approaches that seem
most natural, however. One is to view the space of global sections of Fas ker δ0
F
with the natural inner product given by inclusion into C0(X;F). The other is to view
global sections as lying in σF(σ ). We will generally take the view that global
sections are a subspace of C0(X;F); that is, we will weight Γ(X;F)by its canonical
isomorphism with H0(X;F), as defined in Sect. 3.2.
The dagger structure on Hilbkgives a slightly different way to construct a dual
cosheaf from a weighted cellular sheaf F. Taking the adjoint of each restriction map
reverses their directions and hence yields a cosheaf with the same stalks as the original
sheaf. From a categorical perspective, this amounts to composing the functor Fwith
the dagger endofunctor on Hilbk. When stalks are finite dimensional, this dual cosheaf
is isomorphic to the cosheaf ˆ
Fdefined in Sect. 2.2.1 via the dual vector spaces of stalks.
In this situation, we have an isomorphism between the stalks of Fand its dual cosheaf.
This is reminiscent of the bisheaves recently introduced by MacPherson and Patel
(2018). However, the structure maps F(σ ) →ˆ
F(σ ) will rarely commute with the
restriction and extension maps as required by the definition of the bisheaf—this only
holds in general if all restriction maps are unitary. The bisheaf construction is meant
to give a generalization of local systems, and as such fits better with our discussion of
discrete vector bundles in Sect. 3.5.
3.2 The sheaf Laplacian
Given a chain complex of Hilbert spaces C0→C1→ ··· we can construct the
Hodge Laplacian Δ=(δ +δ∗)2=δ∗δ+δδ∗. This operator is naturally graded
into components Δk:Ck→Ck, with Δk=(δk)∗δk+δk−1(δk−1)∗. This operator
can be further separated into up- (coboundary) and down- (boundary) Laplacians
Δk
+=(δk)∗δkand Δk
−=δk−1(δk−1)∗respectively.
A key observation is that on a finite-dimensional Hilbert space, ker δ∗=(im δ)⊥.
For if δ∗x=0, then for all y,0=δ∗x,y=x,δy, so that x⊥im δ. This allows us
to express the kernels and images necessary to compute cohomology purely in terms
of kernels. This is the content of the central theorem of discrete Hodge theory:
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324 J. Hansen , R. Ghrist
Theorem 3.1 Let C0→C1→··· be a chain complex of finite-dimensional Hilbert
spaces, with Hodge Laplacians Δk. Then ker Δk∼
=Hk(C•).
Proof By definition, Hk(C•)=ker δk/im δk−1. In a finite dimensional Hilbert space,
ker δk/im δk−1is isomorphic to the orthogonal complement of im δk−1in ker δk,
which we may write (ker δk)∩(im δk−1)⊥=(ker δk)∩(ker(δ k−1)∗). So it suffices to
show that ker Δk=(ker δk)∩(ker(δk−1)∗). Note that ker δk=ker(δk)∗δk=ker Δk
+
and similarly for Δk
−. So we need to show that ker(Δk
++Δk
−)=ker Δk
+∩ker Δk
−,
which will be true if im Δk
+∩im Δk
−=0. But this is true because im Δk
+=im(δk)∗=
(ker δk)⊥and im Δk
−=im δk−1⊆ker δk.
The upshot of this theorem is that the kernel of Δkgives a set of canonical represen-
tatives for elements of Hk(C•). This is commonly known as the space of harmonic
cochains, denoted Hk(C•). In particular, the proof above implies that there is an
orthogonal decomposition Ck=Hk⊕im δk−1⊕im(δk)∗.
When the chain complex in question is the complex of cochains for a weighted
cellular sheaf F, the Hodge construction produces the sheaf Laplacians. The Laplacian
which is easiest to study and most immediately interesting is the degree-0 Laplacian,
which is a generalization of the graph Laplacian. We can represent it as a symmetric
block matrix with blocks indexed by the vertices of the complex. The entries on
the diagonal are Δ0
v,v =veF∗
veFveand the entries on the off-diagonal are
Δ0
u,v =−F∗
ueFve, where eis the edge between vand u. Laplacians of other
degrees have similar block structures.
The majority of results in combinatorial spectral theory have to do with up-
Laplacians. We will frequently denote these Lkby analogy with spectral graph theory,
where Ltypically denotes the (non-normalized) graph Laplacian. In particular, we
will further elide the index kwhen k=0, denoting the graph sheaf Laplacian by
simply L. A subscript will be added when necessary to identify the sheaf, e.g. LFor
Δk
F.
Weighted labeled graphs are in one-to-one correspondence with graph Laplacians.
The analogous statement is not true of sheaves on a graph. For instance, the sheaves
in Fig. 1have coboundary maps with matrix representations
⎡
⎣
1−1
10
01
⎤
⎦and ⎡
⎣
1
√2
1
√2
3
2−3
2⎤
⎦,
which means that the Laplacian for each is equal to
2−1
−12
.
However, these sheaves are not unitarily isomorphic, as can be seen immediately by
checking the stalk dimensions. More pithily, one cannot hear the shape of a sheaf. One
source of the lossiness in the sheaf Laplacian representation is that restriction maps
may be the zero morphism, effectively allowing for edges that are only attached to one
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Toward a spectral theory of cellular sheaves 325
Fig. 1 Two nonisomorphic sheaves with the same Laplacian
vertex. More generally, restriction maps may fail to be full rank, which means that it
is impossible to identify the dimensions of edge stalks from the Laplacian.
3.2.1 Harmonic cochains
The elements of ker Δk=Hkare known as harmonic k-cochains. More generally, a
k-cochain may be harmonic on a subcomplex:
Definition 3.2 Ak-cochain xof a sheaf Fon a cell complex Xis harmonic onaset
Sof k-cells if (Δk
Fx)|S=0.
When k=0 and Fis the constant sheaf (i.e., in spectral graph theory), this can be
expressed as a local averaging property: For each v∈S,xv=1
dvu∼vxu, where ∼
indicates adjacency and dvis the degree of the vertex v.
3.2.2 Identifying sheaf Laplacians
Given a regular cell complex Xand a choice of dimension for each stalk, one can
identify the collection of matrices which arise as coboundary maps for a sheaf on X
as those matrices satisfying a particular block sparsity pattern. This sparsity pattern
controls the number of nonzero blocks in each row of the matrix. Restricting to δ0,
we get a matrix whose rows have at most two nonzero blocks. The space of matrices
which arise as sheaf Laplacians is then the space of matrices which have a factorization
L=δ∗δ, where δis a matrix satisfying this block sparsity condition. Boman et al.
studied this class of matrices when the blocks have size 1×1, calling them matrices of
factor width two (Boman et al. 2005). They showed that this class coincides with the
class of symmetric generalized diagonally dominant matrices, those matrices Lfor
which there exists a positive diagonal matrix Dsuch that DLD is diagonally dominant.
Indeed, the fact that sheaves on graphs are not in general determined by their Laplacians
is in part a consequence of the nonuniqueness of width-two factorizations.
123
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326 J. Hansen , R. Ghrist
3.3 Approaching infinite-dimensional Laplacians
The definitions given in this paper are adapted to the case of sheaves of finite dimen-
sional Hilbert spaces over finite cell complexes. Relaxing these finiteness constraints
introduces new complications.
The spaces of cochains naturally acquire inner products by taking the Hilbert space
direct sum. These are not the same as taking the algebraic direct sum or product of
stalks. However, there is a sequence of inclusions of complexes
C•
c(X;F)⊆L2C•(X;F)⊆C•(X;F)
inducing algebraic maps between the corresponding compactly supported, L2, and
standard sheaf cohomology theories.
The theory of abstract complexes of possibly infinite-dimensional Hilbert spaces
has been developed in Brüning and Lesch (1992). This paper explains conditions for
the spaces of harmonic cochains of a complex to be isomorphic with the algebraic
cohomology of the complex. A particularly nice condition is that the complex have
finitely generated cohomology, which implies that the total coboundary map is a
Fredholm operator. More generally, if the images of the coboundary and its adjoint
are closed, the spaces of harmonic cochains will be isomorphic to the cohomology.
Further issues arise when we consider the coboundary maps δk. For spectral pur-
poses, it is in general desirable for these to be bounded linear maps, for which we
must make some further stipulations. Sufficient conditions for coboundary maps to be
bounded are as follows:
Proposition 3.3 Let Fbe a sheaf of Hilbert spaces on a cell complex X . Suppose
that there exists some Mksuch that for every pair of cells στwith dim σ=k and
dim τ=k+1,Fστ≤Mk. Further suppose that every k -cell in X has at most
dkcofaces of dimension k +1, and every (k+1)-cell in X has at most dk+1faces of
dimension k. Then δk
Fis a bounded linear operator.
Proof We compute:
δkx2=
dim τ=k+1(δkx)τ2≤
dim τ=k+1⎛
⎝
στFστxσ⎞
⎠
2
≤
dim τ=k+1⎛
⎝
στ
Mkxσ⎞
⎠
2
≤M2
k
dim τ=k+1
dk+1
στxσ2
=M2
kdk+1
dim σ=k
στxσ2≤M2
kdk+1dk
dim σ=kxσ2=M2
kdk+1dkx2.
If δkis bounded, its associated Laplacians Δk
+=(δk)∗δkand Δk+1
−=δk(δk)∗
are also bounded. As bounded self-adjoint operators, their spectral theory is relatively
123
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Toward a spectral theory of cellular sheaves 327
unproblematic. Their spectra consist entirely of approximate eigenvalues, those λfor
which there exists a sequence of unit vectors {xk}such that Δk
+xk−λxk→0.
If δkis not just bounded, but compact, the Laplacian spectral theory becomes even
nicer. In this situation, the spectrum of Δk
+has no continuous part, and hence consists
purely of eigenvalues. An appropriate decay condition on norms of restriction maps
ensures compactness.
Proposition 3.4 Let Fbe a sheaf of Hilbert spaces on a cell complex X . Suppose
that for all στwith dim σ=k and dim τ=k+1, the restriction map Fστfor is
compact, and further that στFστ<∞. Then δk
Fis a compact linear operator.
Proof It is clear that δkcannot be compact if any one of its component restriction
maps fails to be compact. Suppose first that all restriction maps are finite rank, and
fix an ordering of (k+1)-cells of X, defining the orthogonal projection operators
Pi:Ck+1(X;F)→Ck+1(X;F)sending stalks over (k+1)cells of index greater
than ito zero. Then Piδkis a finite-rank operator and
Piδk−δk≤
j>i
στj
Fστj,
which goes to zero as i→∞. In the case that the restriction maps are compact but not
finite rank, pick an approximating sequence for each by finite rank maps and combine
the two approximations.
An important note is that when Ck(X;F)is infinite dimensional and δkis compact
with finite dimensional kernel, the eigenvalues of Δk
+will accumulate at zero. This
means that there will be no smallest nontrivial eigenvalue for such Laplacians.
Most of the difficulties considered here already arise in the study of spectra of
infinite graphs. The standard Laplacian associated to an infinite graph is bounded but
not compact, while a proper choice of weights decaying at infinity makes it compact.
The study of sheaves of arbitrary Hilbert spaces on not-necessarily-finite cell com-
plexes is interesting, and indeed suggests itself in certain applications. However, for
the initial development and exposition of the theory, we have elected to focus on the
(still quite interesting) finite-dimensional case. This is sufficient for most applications
we have envisioned, and avoids the need for repeated qualifications and restrictions.
For the balance of this paper, we will assume that all cell complexes are finite and
all vector spaces are finite dimensional, giving where possible proofs that generalize
in some way to the infinite-dimensional setting. Most results that do not explicitly
require a finite complex will extend quite directly to the case of sheaves with com-
pact coboundary operators. Proofs not relying on the Courant-Fischer theorem will
typically apply even to situations where coboundary operators are merely bounded,
although their conclusions may be somewhat weakened.
3.4 The normalized Laplacian and weights
Many results in spectral graph theory rely on a normalized version of the standard
graph Laplacian, which is typically defined in terms of a rescaling of the standard
123
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328 J. Hansen , R. Ghrist
Laplacian. Let Dbe the diagonal matrix whose nonzero entries are the degrees of ver-
tices; then the normalized Laplacian is L=D−1/2LD
−1/2. This definition preserves
the Laplacian as a symmetric matrix, but it obscures the true meaning of the normal-
ization. The normalized Laplacian is the standard Laplacian with a different choice of
weights for the vertices. The matrix D−1/2LD
−1/2is similar to D−1L, which is self
adjoint with respect to the inner product x,y=xTDy. In this interpretation, each
vertex is weighted proportionally to its degree. Viewing the normalization process
as a reweighting of cells leads to the natural definition of normalized Laplacians for
simplicial complexes given by Horak and Jost (2013).
Indeed, following Horak and Jost’s definition for simplicial complexes, we propose
the following extension to sheaves.
Definition 3.5 Let Fbe a weighted cellular sheaf defined on a regular cell complex
X.WesayFis normalized if for every cell σof Xand every x,y∈F(σ ) ∩(ker δ)⊥,
δx,δy=x,y.
Lemma 3.6 Given a weighted sheaf Fon a finite-dimensional cell complex X , it is
always possible to reweight Fto a normalized version.
Proof Note that if Xhas dimension k, the normalization condition is trivially satisfied
for all cells σof dimension k. Thus, starting at cells of dimension k−1, we recursively
redefine the inner products on stalks. If σis a cell of dimension k−1, let Πσbe the
orthogonal projection F(σ ) →F(σ ) ∩ker δ. Then define the normalized inner product
•,•N
σon F(σ ) to be given by x,yN
σ=δ(id −Πσ)x,δ(id −Πσ)y+Πσx,Π
σy.
It is clear that this reweighted sheaf satisfies the condition of Definition 3.5 for cells of
dimension kand k−1. We may then perform this operation on cells of progressively
lower dimension to obtain a fully normalized sheaf.
Note that there is an important change of perspective here: we do not normalize the
Laplacian of a sheaf, but instead normalize the sheaf itself, or more specifically, the
inner products associated with each stalk of the sheaf.
If we apply this process to a sheaf Fon a graph G, there is an immediate inter-
pretation in terms of the original sheaf Laplacian. Let Dbe the block diagonal
of the standard degree 0 sheaf Laplacian, and note that for x⊥ker L,x,Dx
is the reweighted inner product on C0(G;F). In particular, the adjoint of δwith
respect to this inner product has the form D†δT, where D†is the Moore-Penrose
pseudoinverse of D, so that the matrix form of the reweighted Laplacian with
respect to this inner product is D†L. Changing to the standard basis then gives
L=D†/2LD
†/2.
3.5 Discrete vector bundles
A subclass of sheaves of particular interest are those where all restriction maps
are invertible.These sheaves have been the subject of significantly more study than
the general case, since they extend to locally constant sheaves on the geometric
realization of the cell complex. The Riemann-Hilbert correspondence describes an
equivalence between locally constant sheaves (or cosheaves) on X, local systems on
123
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Toward a spectral theory of cellular sheaves 329
X, vector bundles on Xwith a flat connection, and representations of the funda-
mental groupoid of X. (See, e.g., Davis and Kirk 2001, ch. 5 or Zein and Snoussi
2009 for a discussion of some aspects of this correspondence.) When we represent
a local system by a cellular sheaf or cosheaf, we will call it a discrete vector bun-
dle.
One way to understand the space of 0-cochains of a discrete vector bundle is as
representing a subspace of the sections of a geometric realization of the associated
flat vector bundle, defined by linear interpolation over higher-dimensional cells. The
coboundary map can be seen as a sort of discretization of the connection, whose
flatness is manifest in the fact that δ2=0.
Discrete vector bundles have some subtleties when we study their Laplacians. The
sheaf-cosheaf duality corresponding to a local system, given by taking inverses of
restriction maps, is not in general the same as the duality induced by an inner product
on stalks. Indeed, these duals are only the same when the restriction maps are unitary—
their adjoints must be their inverses.
The inner product on stalks of a cellular sheaf has two roles: it gives a rel-
ative weight to vectors in each stalk, but via the restriction maps also gives a
relative weight to cells in the complex. This second role complicates our inter-
pretation of certain sorts of vector bundles. For instance, one might wish to
define an O(n)discrete vector bundle on a graph to be a cellular sheaf of real
vector spaces where all restriction maps are orthogonal. However, from the per-
spective of the degree-0 Laplacian, a uniform scaling of the inner product on
an edge does not change the orthogonality of the bundle, but instead in some
sense changes the length of the edge, or perhaps the degree of emphasis we
give to discrepancies over that edge. So a discrete O(n)-bundle should be one
where the restriction maps on each cell are scalar multiples of orthonormal
maps.
That is, for each cell σ, we have a positive scalar ασ, such that for every στ,the
restriction map Fστis an orthonormal map times ατ/ασ. One way to think of this
is as a scaling of the inner product on each stalk of F. Frequently, especially when
dealing with graphs, we set ασ=1 when dim(σ ) =0, but this is not necessary.
(Indeed, when dealing with the normalized Laplacian of a graph, we have αv=
√dv.)
The rationale for this particular definition is that in the absence of a basis, inner
products are not absolutely defined, but only in relation to maps in or out of a space.
Scaling the inner product on a vector space is meaningless except in relation to a given
collection of maps, which it transforms in a uniform way.
As a special case of this definition, it will be useful to think about weighted ver-
sions of the constant sheaf. These are isomorphic to the ‘true’ constant sheaf, but
not unitarily so. Weighted constant sheaves on a graph are analogous to weighted
graphs. The distinction between the true constant sheaf and weighted versions arises
because it is often convenient to think of the sections of a cellular sheaf as a sub-
space of C0(X;F). As a result, we often only want our sections to be constant on
0-cells, allowing for variation up to a scalar multiple on higher-dimensional cells.
This notion will be necessary in Sect. 8.6 when we discuss approximations of cellular
sheaves.
123
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330 J. Hansen , R. Ghrist
3.6 Comparison with previous constructions
Friedman, in Friedman (2015), gave a definition of a sheaf1on a graph, developed a
homology theory, and suggested constructing sheaf Laplacians and adjacency matri-
ces. The suggestion that one might develop a spectral theory of sheaves on graphs has
remained until now merely a suggestion.
The graph connection Laplacian, introduced by Singer and Wu in Singer and Wu
(2012), is simply the sheaf Laplacian of an O(n)-vector bundle over a graph. This
construction has attracted significant interest from a spectral graph theory perspec-
tive, including the development of a Cheeger-type inequality (Bandeira et al. 2013)
and a study of random walks and sparsification (Chung and Zhao 2012). Connection
Laplacian methods have proven enlightening in the study of synchronization problems.
Others have approached the study of vector bundles, and in particular line bundles, over
graphs without reference to the connection Laplacian, studying analogues of spanning
trees and the Kirchhoff theorems (Kenyon 2011; Catanzaro et al. 2013). Other work on
discrete approximations to connection Laplacians of manifolds has analyzed similar
matrices (Mantuano 2007).
Gao, Brodski, and Mukherjee developed a formulation in which the graph connec-
tion Laplacian is explicitly associated to a flat vector bundle on the graph and arises
from a twisted coboundary operator (Gao et al. 2016). This coboundary operator is not
a sheaf coboundary map and has some difficulties in its definition. These arise from a
lack of freedom to choose the basis for the space of sections over an edge of the graph.
Further work by Gao uses a sheaf Laplacian-like construction to study noninvertible
correspondences between probability distributions on surfaces (Gao 2016).
Wu et al. (2018) have recently proposed a construction they call a weighted simpli-
cial complex and studied its associated Laplacians. These are cellular cosheaves where
all stalks are equal to a given vector space and restriction maps are scalar multiples
of the identity. Their work discusses the cohomology and Hodge theory of weighted
simplicial complexes, but does not touch on issues related to the Laplacian spectrum.
4 Harmonicity
As a prelude to results about the spectra of sheaf Laplacians, we will discuss issues
related to harmonic cochains on sheaves. While these do not immediately touch on
the spectral properties of the Laplacian, they are closely bound with its algebraic
properties.
4.1 Harmonic extension
Proposition 4.1 Let X be a regular cell complex with a weighted cellular sheaf F. Let
B⊆X be a subcomplex and let x|B∈Ck(B;F)be an F-valued k-cochain specified
on B. If H k(X,B;F)=0, then there exists a unique cochain x ∈Ck(X;F)which
restricts to x|Bon B and is harmonic on S =X\B.
1In our terminology, Friedman’s sheaves are cellular cosheaves.
123
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Toward a spectral theory of cellular sheaves 331
Proof A matrix algebraic formulation suffices. Representing Δk
Fin block form as
partitioned by Band S, the relevant equation is
Δk
F(S,S)Δ
k
F(S,B)
Δk
F(B,S)Δ
k
F(B,B)x|S
x|B=0
y.
Since yis indeterminate, we can ignore the second row of the matrix, giving the
equation Δk
F(S,S)x|S+Δk
F(S,B)x|B=0. We can write Δk
F(S,S)=(δk|S)∗δk|S+
((δk−1)∗|S)∗(δk−1)∗|S, which is very close to the k-th Hodge Laplacian of the relative
cochain complex
···→Ck−1(X,B;F)→Ck(X,B;F)→Ck+1(X,B;F)→···.
Indeed, we can exploit the fact that this is a subcomplex of C•(X;F)to compute its
Hodge Laplacian in terms of the coboundary maps of C•(X;F). The coboundary map
δSof C•(X,B;F)is simply the restriction of the coboundary map δof C•(X;F)to
the subcomplex: δk
S=πk+1
Sδkik
S, where πk
Sis the orthogonal projection Ck(X;F)→
Ck(X,B;F)and ik
Sthe inclusion Ck(X,B;F)→Ck(X;F). Note that πk
Sand ik
Sare
adjoints, and that ik
Sπk
Sis the identity on im δk−1
S. We may therefore write the Hodge
Laplacian of the relative complex as
Δk(X,B;F)=(δk
S)∗δk
S+δk−1
S(δk−1
S)∗
=πk
S(δk)∗ik+1
Sπk+1
Sδkik
S+πk
Sδk−1ik−1
Sπk−1
S(δk−1)∗ik
S
=πk
S(δk)∗δkik
S+πk
Sδk−1ik−1
Sπk−1
S(δk−1)∗ik
S.
Meanwhile, we can write the submatrix
Δk
F(S,S)=πk
S(δk)∗δkik
S+πk
Sδk−1(δk−1)∗ik
S.
It is then immediate that ker(Δk
F(S,S)) ⊆ker Δk(X,B;F), so that Δk
F(S,S)is
invertible if Hk(X,B;F)=0.
If we restrict to up- or down-Laplacians, a harmonic extension always exists, even
if it is not unique. This is because, for instance, im(δk|S)∗δk|B⊆im(δk|S)∗δk|S.In
particular, this implies that harmonic extension is always possible for 0-cochains, with
uniqueness if and only if H0(X,B;F)=0.
4.2 Kron reduction
Kron reduction is one of many names given to a process of simplifying graphs
with respect to the properties of their Laplacian on a boundary. If Gis a con-
nected graph with a distinguished set of vertices B, which we consider as a sort
of boundary of G, Proposition 4.1 shows that there is a harmonic extension map
E:RB→RV(G). It is then possible to construct a graph Gon Bsuch that for
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332 J. Hansen , R. Ghrist
every function xon the vertices of G,wehaveLGx=πBLGE(x), where πBis
the orthogonal projection map RV(G)→RB. Indeed, letting S=V(G)\B,wehave
E(x)|S=−LG(S,S)−1LG(S,B)x,so
LGx=πBLGE(x)=LG(B,B)x−LG(B,S)LG(S,S)−1LG(S,B)x.
Therefore,
LG=LG(B,B)−LG(B,S)LG(S,S)−1LG(S,B),
that is, LGis the Schur complement of the (B,B)block of LG. It is also the Laplacian
of a graph on B:
Theorem 4.2 (see (Dörfler and Bullo 2013)) If LGis the Laplacian of a con-
nected graph G, and B a subset of vertices of G, then L G=LG(B,B)−
LG(B,S)LG(S,S)−1LG(S,B)is the Laplacian of a graph with vertex set B.
A physically-inspired way to understand this result (and a major use of Kron reduc-
tion in practice) is to view it as reducing a network of resistors given by Gto a smaller
network with node set Bthat has the same electrical behavior on Bas the original net-
work. In this guise, Kron reduction is a high-powered version of the Y-Δand star-mesh
transforms familiar from circuit analysis. Further discussion of Kron reduction and its
various implications and applications may be found in Dörfler and Bullo (2013).
Can we perform Kron reduction on sheaves? That is, given a sheaf Fon a graph
Gwith a prescribed boundary B, can we find a sheaf FBon a graph with vertex set B
only such that for every x∈C0(B;FB)we have LFBx=πC0(B;FB)LFE(x), where
E(x)is the harmonic extension of xto G?
The answer is, in general, no. Suppose we want to remove the vertex vfrom our
graph, i.e.,B=G\{v}.Let Dv=veF∗
veFve=Lv,v. To eliminate the vertex
vwe apply the condition (LF(x,E(x)))(v) =0, and take a Schur complement,
replacing L(B,B)with L(B,B)−L(B,v)D−1
vL(v, B). This means that we add to
the entry L(w, w)the map F∗
weFveD−1
vF∗
veFwe, where eis the edge between
vand w, and ethe edge between vand w. This does not in general translate to a change
in the restriction maps for the edge between wand w. In general, Kron reduction is
not possible for sheaves.
In particular, if x∈C0(G;F)is a section of F, its restriction to Bmust be a section
of FB. Conversely, if xis not a section, its restriction to Bcannot be a section of FB.
But we can construct sheaves with a space of sections on the boundary that cannot be
replicated with a sheaf on the boundary vertices only. For instance, take the star graph
with three boundary vertices, with stalks Rover boundary vertices and edges, and R2
over the internal vertex. Take as the restriction maps from the central vertex restriction
onto the first and second components, and addition of the two components. See Fig. 2
for an illustration.
Note that a global section of this sheaf is determined by its value on the central
vertex. If we label the boundary vertices counterclockwise starting at the top, the space
of global sections for FBmust have as a basis
123
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Toward a spectral theory of cellular sheaves 333
Fig. 2 A sheaf illustrating the
general impossibility of Kron
reduction
x1=⎡
⎣
1
1
0⎤
⎦,x2=⎡
⎣
1
0
1⎤
⎦.
But there is no sheaf on a graph with vertex set Bwhich has this space of global sections.
To see this, note that if x1is a section, the map from F(v1)to F(v3)must be the zero
map, and similarly for the map from F(v2)to F(v3). Similarly, if x2is a section, the
maps F(v1)→F(v2)and F(v3)→F(v2)must be zero. But this already shows that
the vector 100
Tmust be a section, giving FBa three-dimensional space of sections.
The problem is that the internal node allows for constraints between boundary nodes
that cannot be expressed by purely pairwise interactions. This fact is a fundamental
obstruction to Kron reduction for general sheaves.
However, there is a sheaf Kron reduction for sheaves with vertex stalks of dimension
at most 1. This follows from the identification of the Laplacians of such sheaves as
the matrices of factor width two in Sect. 3.2.2.
Theorem 4.3 The class of matrices of factor width at most two is closed under taking
Schur complements.
Proof By Theorems 8 and 9 of Boman et al. (2005), a matrix Lhas factor width at
most two if and only if it is symmetric and generalized weakly diagonally dominant
with nonnegative diagonal, that is, there exists a positive diagonal matrix Dsuch that
DLD is weakly diagonally dominant. Equivalently, these are the symmetric positive
semidefinite generalized weakly diagonally dominant matrices. The class of gener-
alized weakly diagonally dominant matrices coincides with the class of H-matrices,
which are shown to be closed under Schur complements in Johnson and Smith (2005).
Similarly, the class of symmetric positive definite matrices is closed under Schur
complements, so the intersection of the two classes is also closed.
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334 J. Hansen , R. Ghrist
4.3 Maximum modulus theorem
Harmonic 0-cochains of an O(n)-bundle satisfy a local averaging property which leads
directly to a maximum modulus principle.
Lemma 4.4 Let G be a graph with an O(n)-bundle F, with constant vertex weights
αv=1and arbitrary edge weights αe(as defined in Sect. 3.5).Ifx ∈C0(G;F)is
harmonic at a vertex v, then
xv=1
dv
v,we
v=w
F∗
veFwexw,
where dv=veFve2=veα2
e.
Proof The block row of LFcorresponding to vhas entries −F∗
veFweoff the diago-
nal and veF∗
veFve=veFve2idF(v) on the diagonal. The harmonicity
condition is then
dvxv−
v,we
v=w
F∗
veFwexw=0.
Theorem 4.5 (Maximum modulus principle) Let G be a graph, and B be a thin subset
of vertices of G; that is, G\B is connected, and every vertex in B is connected to
a vertex not in B . Let Fbe an O(n)-bundle on G with av=1for all v∈G, and
suppose x ∈C0(G;F)is harmonic on G \B . Then if x attains its maximum stalkwise
norm at a vertex in G\B , it has constant stalkwise norm.
Proof Note that for a given edge e=v∼w,Fveand Fueare both αetimes an
orthogonal map, so F∗
veFweis α2
etimes an orthogonal map. Let v∈G\Band
suppose xv≥xwfor all w∈G. Then this holds in particular for neighbors of v,
so that we have
xv= 1
dv
v,we
v=w
F∗
veFwexw
≤1
dv
v,we
v=w
F∗
veFwexw
=1
dv
v,we
v=w
α2
exw≤ 1
dv
ve
α2
exv=xv,
Equality holds throughout, which, combined with the assumption that xv≥xw
for all w, forces xv=xwfor w∼v. We then apply the same argument to every
vertex in G\Badjacent to v, and, after iterating, the region of constant stalkwise norm
extends to all of G\Bbecause this subgraph is connected. But since every vertex b∈B
123
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Toward a spectral theory of cellular sheaves 335
is adjacent to some vertex w∈G\B, the same argument applied to the neighborhood
of wforces xb=xw. So any harmonic function that attains its maximum modulus
on G\Bhas constant modulus.
Corollary 4.6 Let B be a thin subset of vertices of G, and Fan O(n)-bundle on G as
before. If x ∈C0(G;F)is harmonic on G\B, then it attains its maximum modulus
on B.
The constant sheaf on a graph is an O(n)-bundle, so this result gives a maximum
modulus principle for harmonic functions on the vertices of a graph. A slightly stronger
result in this vein, involving maxima and minima of x, is discussed in Sunada (2008).
The thinness condition for Bis not strictly necessary for the corollary to hold—there
are a number of potential weakenings of the condition. For instance, we might simply
require that there exists some w∈Bsuch that for every vertex v∈G\Bthere exists
a path from vto wnot passing through B.
5 Spectra of sheaf Laplacians
The results in this section are straightforward generalizations and extensions of familiar
results from spectral graph theory. Most are not particularly difficult, but they illustrate
the potential for lifting nontrivial notions from graphs and complexes to sheaves.
It is useful to note a few basic facts about the spectra of Laplacians arising from
Hodge theory.
Proposition 5.1 The nonzero spectrum of Δkis the disjoint union of the nonzero spec-
tra of Δk
+and Δk
−.
Proof We take advantage of the Hodge decomposition, noting that Ck(X;F)=
ker Δk⊕im Δk
−⊕im Δk
+. This is an orthogonal decomposition, and =0aswell
as Δk
−|(im Δk
+)=0. Further, since ker Δk=ker Δ+∩ker Δ−, both restrict to
zero on the kernel of Δk. We therefore see that Δkis the orthogonal direct sum
0|ker Δk⊕Δk
+|(im Δk
+)⊕Δk
−|(im Δk
−), and hence the spectrum of Δkis the union of the
spectra of these three operators.
Proposition 5.2 The nonzero eigenvalues of Δk
+and Δk+1
−are the same.
Proof We hav e Δk
+=(δk)∗δkand Δk+1
−=δk(δk)∗. The eigendecompositions of
these matrices are determined by the singular value decomposition of δk, and the
nonzero eigenvalues are precisely the squares of the nonzero singular values of δk.
One reason for the study of the normalized graph Laplacian is that its spectrum
is bounded above by 2 (Chung 1992), and hence normalized Laplacian spectra of
different graphs can be easily compared. A similar result holds for up-Laplacians of
normalized simplicial complexes (Horak and Jost 2013): the eigenvalues of the degree-
kup-Laplacian of a normalized simplicial complex are bounded above by k+2. This
fact extends to normalized sheaves on simplicial complexes. This result and others in
this paper will rely on the Courant-Fischer theorem, which we state here for reference.
123
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336 J. Hansen , R. Ghrist
Definition 5.3 Let Abe a self-adjoint operator on a Hilbert space V.Ifx∈V,the
Rayleigh quotient corresponding to xand Ais
RA(x)=x,Ax
x,x.
Theorem 5.4 (Courant-Fischer) Let Abeann×n Hermitian matrix with eigenvalues
λ1≤λ2≤···≤λn. Then
λk=min
dim V=kmax
x∈VRA(x)=max
dim V=n−k+1min
x∈VRA(x).
The proof is immediate once one uses the fact that A is unitarily equivalent to a
diagonal matrix.
Proposition 5.5 Suppose Fis a normalized sheaf on a simplicial complex X . The
eigenvalues of the degree k up-Laplacian Lk
Fare bounded above by k +2.
Proof By the Courant-Fischer theorem, the largest eigenvalue of Lk
Fis equal to
max
x∈Ck(X;F)
x,Lk
Fx
x,x=max
x⊥ker δkδkx,δ
kx
dim σ=kδkxσ,δ
kxσ
=max
x⊥ker δkdim τ=k+1σ,στ[σ:τ][σ:τ]Fστxσ,Fστxσ
dim σ=kστFστxσ,Fστxσ.
Note that for σ= σ,
[σ:τ][σ:τ]Fστxσ,Fστxσ≤FστxσFστxσ
≤1
2Fστxσ2+Fστxσ2
by the Cauchy-Schwarz inequality. In particular, then, the term of the numerator cor-
responding to each τof dimension k+1 is bounded above by
στFστxσ2+1
2
σ=στFστxσ2+Fστxσ2
=(k+2)
στFστxσ2,
by counting the number of times each term Fστxσ2appears in the sum. Meanwhile,
the denominator is equal to dim τ=k+1στFστxσ2, so the Rayleigh quotient
is bounded above by k+2.
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Toward a spectral theory of cellular sheaves 337
5.1 Eigenvalue interlacing
Definition 5.6 Let A,Bbe n×nmatrices with real spectra. Let λ1≤λ2≤···≤λn
be the eigenvalues of Aand μ1≤μ2≤ ··· ≤ μnbe the eigenvalues of B.We
say the eigenvalues of Aare (p,q)-interlaced with the eigenvalues of Bif for all k,
λk−p≤μk≤λk+q.(Weletλk=λ1for k<1 and λk=λnfor k>n.)
The eigenvalues of low-rank perturbations of symmetric positive semidefinite matri-
ces are related by interlacing. The following is a standard result:
Theorem 5.7 Let A and B be positive semidefinite matrices, with rank B=t . Then
the eigenvalues of A are (t,0)-interlaced with the eigenvalues of A −B.
Proof Let μkbe the k-th largest eigenvalue of A−Band λkthe k-th largest eigenvalue
of A. By the Courant-Fischer theorem, we have
μk=min
dim Y=kmax
y∈Y,y=0y,Ay−y,By
y,y
≥min
dim Y=kmax
y∈Y∩ker B,y=0y,Ay
y,y
≥min
dim Y=k−tmax
y∈Y,y=0y,Ay
y,y=λk−t
and
λk=min
dim Y=kmax
y∈Y,y=0y,Ay
y,y
≥min
dim Y=kmax
y∈Y,y=0y,Ay−y,By
y,y=μk.
This result is immediately applicable to the spectra of sheaf Laplacians under the
deletion of cells from their underlying complexes. The key part is the interpretation
of the difference of the two Laplacians as the Laplacian of a third sheaf.2Let Fbe
a sheaf on X, and let Cbe an upward-closed set of cells of X, with Y=X\C.The
inclusion map i:Y→Xinduces a restriction of Fonto Y, the pullback sheaf i∗F.
Consider the Hodge Laplacians Δk
Fand Δk
i∗F.IfCcontains cells of dimension k,
these matrices are different sizes, but we can derive a relationship by padding Δk
i∗F
with zeroes. Equivalently, this is the degree-kLaplacian of Fwith the restriction maps
incident to cells in Cset to zero.
Proposition 5.8 Let Gbe the sheaf on X with the same stalks as Fbut with all
restriction maps between cells not in C set to zero. The eigenvalues of Δk
i∗F
are (t,0)-interlaced with the eigenvalues of Δk
F, where t =codim Hk(X;G)=
dim Ck(X;F)−dim Hk(X;G).
2Such subtle moves are part and parcel of a sheaf-theoretic perspective.
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338 J. Hansen , R. Ghrist
Similar results can be derived for the up- and down-Laplacians. Specializing to
graphs, interlacing is also possible for the normalized degree 0 sheaf Laplacian. The
Rayleigh quotient for the normalized Laplacian Li∗Fis
x,D−1/2
i∗FLi∗FD−1/2
i∗Fx
x,x=y,Li∗Fy
y,Di∗Fy=y,LFy−y,LGy
y,DFy−y,DGy,
where we let y=D−1/2
i∗Fx. Then if {λk}are the ordered eigenvalues of LFand {μk}
are the ordered eigenvalues of Li∗F,wehave
μk=min
dim Y=kmax
y∈Y,y=0y,LFy−y,LGy
y,DFy−y,DGy
≥min
dim Y=kmax
y∈Y∩H0(X;G),y=0
y,LFy
y,DFy−y,DGy
≥min
dim Y=kmax
y∈Y∩H0(X;G),y=0
y,LFy
y,DFy
≥min
dim Y=k−tmax
y∈Y,y=0y,LFy
y,DFy=λk−t
μk=max
dim Y=n−k+1min
y∈Y,y=0y,LFy−y,LGy
y,DFy−y,DGy
≤max
dim Y=n−k+1min
y∈Y∩H0(X;G),y=0
y,LFy
y,DFy−y,DGy
≤max
dim Y=n−k+1min
y∈Y∩H0(X;G),y=0
y,LFy
y,DFy
≤max
dim Y=n−k−t+1min
y∈Y,y=0y,LFy
y,DFy=λk+t
Therefore, the eigenvalues of the normalized Laplacians are (t,t)-interlaced. This
generalizes interlacing results for normalized graph Laplacians.
5.2 Sheaf morphisms
Proposition 5.9 Suppose ϕ:F→Gis a morphism of weighted sheaves on a regular
cell complex X . If ϕk+1is a unitary map, then Lk
F=(ϕk)∗Lk
Gϕk.
Proof The commutativity condition ϕk+1δF=δGϕkimplies that (δF)∗(ϕk+1)∗ϕk+1
δF=(ϕk)∗(δG)∗δGϕk=(ϕk)∗Lk
Gϕk. Thus if (ϕk+1)∗ϕk+1=idCk+1(X;F),wehave
Lk
F=(ϕk)∗Lk
Gϕk. This condition holds if ϕk+1is unitary.
An analogous result holds for the down-Laplacians of F, and these combine to a result
for the full Hodge Laplacians.
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Toward a spectral theory of cellular sheaves 339
5.3 Cell complex morphisms
The following constructions are restricted to locally injective cellular morphisms, as
discussed in Sect. 2.1. Recall that under these morphisms, cells map to cells of the same
dimension and the preimage of the star of a cell is a disjoint union of subcomplexes, on
each of which the map acts injectively. The sheaf Laplacian is invariant with respect
to pushforwards over such maps:
Proposition 5.10 Let X and Y be cell complexes, and let f :X→Y bealocally
injective cellular morphism. If Fis a sheaf on X , the kth coboundary Laplacian
corresponding to f∗Fon Y is the same (up to a unitary change of basis) as the kth
coboundary Laplacian of Fon X.
Corollary 5.11 The sheaves Fand f∗Fare isospectral for the coboundary Laplacian.
Proof There is a canonical isometry fk:Ck(X,F)→Ck(Y,f∗F), which is given
on stalks by the obvious inclusion fσ:F(σ ) →f∗F(f(σ )) =f(τ )=f(σ ) F(τ ).
For σσ,fσcommutes with the restriction map Fσσand hence fkcommutes with
the coboundary map. But this implies that:
Lk
f∗F=(δk
f∗F)∗δk
f∗F=(δk
f∗F)∗f∗
(k+1)f(k+1)δk
f∗F=f∗
k(δk
F)∗δk
Ffk=f∗
kLk
Ffk.
General locally injective maps behave nicely with sheaf pushforwards, and covering
maps behave well with sheaf pullbacks. Recall that a covering map of cell complexes
is a locally injective map f:C→Xsuch that for every cell σ∈X,fis an
isomorphism on the disjoint components of f−1(st(σ )).
Proposition 5.12 Let f :C→X be a covering map of cell complexes, with Fa sheaf
on X . Then for any k, the spectrum of L k
Fis contained in the spectrum of L k
f∗F.
Proof Consider the lifting map ϕ:Ck(X;F)→Ck(C;f∗F)given by x→ x◦fk.
This map commutes with δand δ∗. The commutativity with δfollows immediately
from the proof of the contravariant functoriality of cochains. The commutativity with
δ∗is more subtle, and relies on the fact that fis a covering map.
For y∈Ck(C;f∗F)and x∈Ck+1(X;F),wehave
y,δ
∗ϕx=δy,ϕx=
σ,τ ∈PC
στ
[σ:τ]f∗Fστ(yσ), (ϕ x)τ
=
σ,τ∈PX
στ
[σ:τ]
σ∈f−1(σ )
Fστ(yσ), xτ
=
σ,τ∈PX
στ
[σ:τ]Fστ(ϕ∗y)σ,xτ
=δϕ∗y,x=y,ϕδ
∗x.
123
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340 J. Hansen , R. Ghrist
Now, if Lk
Fx=λx,wehave Lk
f∗Fϕx=(δk
f∗F)∗δk
f∗Fϕx=ϕ(δk
F)∗δk
Fx=
ϕLk
Fx=λϕx,soλis an eigenvalue of Lk
f∗F.
Even if f:Y→Xis not quite a covering map, it is still possible to get some
information about the spectrum of f∗F. For instance, for dimension-preserving cell
maps with uniform fiber size we have a bound on the smallest nontrivial eigenvalue
of the pullback:
Proposition 5.13 Suppose f :Y→Xisadimension-preserving map of regular
cell complexes such that for dim(σ ) =d, f−1(σ )=dis constant, and let Fbe
a sheaf on X . If λk(F)is the smallest nontrivial eigenvalue of L d
F, then λk(F)≥
d
d+1λk(f∗F).
Proof Let xbe an eigenvector corresponding to λk(F). Note that since every fiber is
the same size, the lift ϕpreserves the inner product up to a scaling. That is, if yand z
are d-cochains, ϕy,ϕz=dy,z. This means that the pullback of xis orthogonal
to the pullback of any cochain in the kernel of LF. Therefore, we have
λk(F)=δx,δx
x,x=dϕδx,ϕδx
d+1ϕx,ϕx=dδϕx,δϕx
d+1ϕx,ϕx≥d
d+1
λk(f∗F).