Weighted Cellular Sheaves are a type of discrete sheaves are a central object in computational spectral sheaf theory, analogous to weighted graphs in spectral graph theory. Weighted cellular sheaves are cellular sheaves of real vector spaces. Weighted cellular sheaves provide a generalization to realvalued topological data structures in computer science. Generalized framework applied in computational problems of eigenvalue interlacing, sparsification, effective resistance, synchronization, and approximation.
Definition
A weighted cellular sheaf is a cellular sheaf with values in $K$vector spaces (where $K = \mathbb{R}$ or $\mathbb{C}$), where the stalks have additionally been given an inner product structure. This can be viewed as a functor $P X \rightarrow \mathbf{Hilb}_K$, where $\mathbf{Hilb}_K$ is the category whose objects are Hilbert spaces over field $K$ and whose morphisms are (bounded) linear maps. The inner products on stalks extend by the orthogonal direct sum to inner products on $C^k(X; F)$, making these Hilbert spaces as well.
In Information Geometry
In the context of spectral sheaf theory we typically consider weighted cellular sheaves that are defined over field $K = \mathbb{R}$, that is realvalued weighted cellular sheaves. This is because spectral sheaf theory tends to be useful in close relationship to information geometry and differentiable computing, where objects of study are typically Riemannian, that is equipped with a metric tensor, an additional structure on a manifold that allows defining distances and angles, just as the Hilbert space induces an inner product on a Vector space and allows defining distances and angles there.
Statistical manifolds of information geometry have a Riemannian metric as given by the Fisher information matrix. In these contexts, variation in weights correspond to change of coordinates on the underlying manifold. Just as statistical manifolds conceptually defined as the space of all measures $\mu$ on $X$ with the fixed sigmaalgebra $\Sigma$ are an infinitedimensional space, (commonly taken to be Fréchet), so it is common to work with its finitedimensional submanifolds parametrized by some smooth, continuously varying parameter $\theta$, where dimension of parameter $\theta$ corresponds to the dimension of the submanifold, so in context of weighted cellular sheaves, in statistical computations, we may wish to consider a finitedimensional subspace of the infinitedimensional Hilbertspace that corresponds to our choice of parameters.
Category of Weighted Cellular Sheaves
Let us first discuss $\mathcal{F}$ and $\mathcal{G}$ as cellular sheaves on a regular cell complex $X$ with values in objects of $\mathbf{Vect}_K$ over $K = \mathbb{R}, \mathbb{C}$, just as with categories of ceullar sheaves in general, a sheaf morphism, $\varphi : \mathcal{F} \to \mathcal{G}$ is a collection of maps $\varphi_\sigma : \mathcal{F}(\sigma) \to \mathcal{G}(\sigma)$ for each cell $\sigma$ of $X$, such that for any cells $\sigma, \tau$ in $X$ when $\sigma \trianglelefteq \tau$:
$$\varphi_\tau \circ \mathcal{F}_{\sigma \trianglelefteq \tau} = \mathcal{G}_{\sigma \trianglelefteq \tau} \circ \varphi_\sigma$$
This commutativity assures that a sheaf morphism $\varphi : \mathcal{F} \to \mathcal{G}$ induces maps $\varphi^k : C^k(X; \mathcal{F}) \to C^k(X; \mathcal{G})$ which commute with coboundary maps, inducing maps on cohomology:
$$H^k_\varphi : H^k(X; \mathcal{F}) \to H^k(X; \mathcal{G})$$
Now consider regular cellular sheaves over $\mathbf{Hilb}_K$, with $K = \mathbb{R}, \mathbb{C}$ an actual $K$weighted cellular sheaf category, since the inner stalks on $\mathcal{F}$ extend by the orthogonal direct sum to inner products on chains, $C^k(X; \mathcal{F})$ are in $\mathbf{Hilb}_K$ as well.
Duals
Every morphism $T : V \to W$ in Hilbert spaces admits an adjoint map $T^\ast : W \to V$, determined by the property that for all $v \in V, w \in W$ we have $\langle w, Tv\rangle = \langle T^\ast w, v\rangle$, and $(T^\ast)^\ast$. We call that a dagger structure.
Categorically, the dagger structure means existence of a contravariant endofunctor $\dagger$, (here corresponding to adjoint operation $(\cdot)^\ast$), that acts as the identity on objects and squares to the identity. In a dagger category, the notion of unitary ismorphisms makes sense: they are the invertible morphisms $T$ such that $T^\dagger = T^{1}$.
A cellular cosheaf of vector spaces on a regular cell complex $X$ is an assignment of a vector space $\mathcal{F}(\sigma)$ to each cell $\sigma$ of $X$ together with linear maps $\mathcal{F}_{\sigma \trianglelefteq \tau} : \mathcal{F}(\tau) \to \mathcal{F}(\sigma)$ for each incident cell pair $\sigma \trianglelefteq \tau$ which satisfies $F_{\sigma \trianglelefteq \sigma} = \mathrm{Id}$ and the composition condition:
$$\rho \triangleleft \sigma \triangleleft \tau \implies \mathcal{F}_{\rho \trianglelefteq \tau} = \mathcal{F}_{\rho \trianglelefteq \sigma} \circ \mathcal{F}_{\sigma \trianglelefteq \tau}$$
Concisely, a cosheaf is a functor $P^\mathrm{op}_X \to \mathbf{Vect}_K$. The contravariant functor $\mathrm{Hom}(\bullet, K) : \mathbf{Vect}^\mathrm{op}_K \to \mathbf{Vect}^\mathrm{op}_K$ gives every $\mathcal{F}$ a dual $\hat{\mathcal{F}}$ whose stalks are $\mathrm{Hom}(\mathcal{F}(\sigma), K)$.
The dagger structure on $\mathbf{Hilb}_K$ gives a slightly different way to construct a dual cosheaf $\hat{\mathcal{F}}$ from a weighted cellular sheaf $\mathcal{F}$. Taking the adjoint of each restriction map $\mathcal{F}_{\sigma \trianglelefteq \tau}$ reverses their directions and hence yields a cosheaf with the same stalks as the original sheaf. From the categorical perspective, this amounts to composing the functor $\mathcal{F}$ with the dagger endofunctor on $\mathbf{Hilb}_K$, which, when stalks are finitedimensional, produces a dual cosheaf isomorphic to the cosheaf $\hat{\mathcal{F}}$ defined via dual vector spaces of stalks. In this situation, we have an isomorphism between stalks of $\mathcal{F}$ and its dual cosheaf, but not the restriction maps in general. Structure maps $\mathcal{F}(\sigma) \to \hat{\mathcal{F}}(\sigma)$ will rarely commute with the restriction and extension maps.
If we further restrict all restriction maps to be unitary, however, the structure maps will commute with restriction maps and $\mathcal{F}(\sigma) \to \hat{\mathcal{F}}(\sigma)$ will induce a bisheaf structure. The bisheaves can generalize local structures, and are closely related to discrete vector bundles.
Operations
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 and pushforwards (over locally injective maps) of a weighted sheaf have canonical weighted structures given by their computational formulae. These structures are essential in defining the Hodge Laplacian and other spectral properties of cellular sheaves.
Direct Sum
Given sheaves $\mathcal{F}$ and $\mathcal{G}$ on $X$, their direct sum $\mathcal{F} \oplus \mathcal{G}$ is a sheaf on $X$ with $(\mathcal{F} \oplus \mathcal{G})(\sigma) = \mathcal{F}(\sigma)\oplus\mathcal{G}(\sigma)$ and restriction maps $(\mathcal{F}\oplus\mathcal{G})_{\sigma \trianglelefteq \tau} = \mathcal{F}\oplus\mathcal{G}_{\sigma \trianglelefteq \tau}$
Tensor Product
Given sheaves $\mathcal{F}$ and $\mathcal{G}$ on $X$, their tensor product $\mathcal{F} \otimes \mathcal{G}$ is a sheaf on $X$ with $(\mathcal{F} \otimes \mathcal{G})(\sigma) = \mathcal{F}(\sigma)\otimes\mathcal{G}(\sigma)$ and restriction maps $(\mathcal{F}\otimes\mathcal{G})_{\sigma \trianglelefteq \tau} = \mathcal{F}\otimes\mathcal{G}_{\sigma \trianglelefteq \tau}$
Pullbacks
If $f : X \to Y$ is a morphism of cell complexes and $\mathcal{F}$ is a sheaf on $Y$, the pullback $f^\ast \mathcal{F}$ is a sheaf on $X$ with $f^\ast \mathcal{F}(\sigma) = \mathcal{F}(f(\sigma))$ and restriction maps $(f^\ast \mathcal{F})_{\sigma \trianglelefteq \tau} = \mathcal{F}_{f(\sigma) \trianglelefteq f(\tau)}$.
Pushforwards
If $f : X \to Y$ is a morphism of cell complexes and $\mathcal{F}$ is a sheaf on $X$, the pushforward $f_\ast \mathcal{F}$ is a sheaf on $Y$ with stalks $f_\ast\mathcal{F}(\sigma)$ given as
$$\lim_{\sigma \trianglelefteq f(\tau)} \mathcal{F}(\tau)$$
The restriction maps are induced by restriction maps of $\mathcal{F}$, since whenever $\sigma \trianglelefteq \sigma'$, the cone for the limit defining $f_\ast \mathcal{F}(\sigma)$ contains the cone for the limit defining $f_\ast\mathcal{F}(\sigma')$, this incudes a unique map $f_\ast \mathcal{F}(\sigma) \to f_\ast\mathcal{F}(\sigma')$.
Since we are working over regular cells, (as in topological data analysis in general), pushforwards are usually given over locally injective cell maps, whose geometric realizations are locally injective, that is topological information about $X$ is encoded in the poset structure $P_X$, it is common to work with locally injective maps that respect the poset structure. When morphisms of cell complexes $f : X \to Y$ is locally injective, every cell $\sigma \in X$ maps to a cell of the same dimension, and for every cell $\sigma \in Y$, $f^{1}(\mathrm{st}(\sigma)$ is a disjoint union of subcomplexes, each of which maps injectively to $Y$. In this case,
$$f^\ast \mathcal{F}(\sigma) \simeq \bigoplus_{\sigma' \in f^{1}(\sigma)} \mathcal{F}(\sigma')$$
and for restriction maps,
$$(f^\ast \mathcal{F})_{\sigma \trianglelefteq \tau} = \bigoplus_{(\sigma' \trianglelefteq \tau') \in f^{1}(\sigma \trianglelefteq \tau)} \mathcal{F}_{\sigma' \trianglelefteq \tau'}$$
This holds more generally, even if the stars of the cells in $f^{1}(\sigma)$ are disjoint.
Space of Global Sections
TODO
This section has been left to be completed at a later date. You can help by editing it.
The dagger structure of $\mathbf{Hilb}_K$ introduces further categorical subtleties into the category of weighted cellular sheaves. The space of global sections of a cellular sheaf is defined in categorical terms as the limit of the functor $X \to \mathbf{Vect}_K$ defining the sheaf, which defines the space of global sections up to unique isomorphism. We may want a weighted space to be a sort of limit in $\mathbf{Hilb}_K$ which is defined up to a unique unitary isomorphism, which we refer to as the dagger limit. Unfortunately as work by Haunen, Karvonen (2019) has shown, $\mathbf{Hilb}_K$ does 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 space on the space of global sections of a cellular sheaf $\mathcal{F}$. Instead, there are two apporaches:
 Harmonic View: Space of global sections of $\mathcal{F}$ is understood to be $\mathrm{ker} \delta^0_\mathcal{F}$ with the natural inner product given by inclusion into $C^0(X; \mathcal{F})$. Given a chain complex $C^\bullet$ of finitedimensional Hilbert spaces with Hodge Laplacians $\Delta^k$, we have $\mathrm{ker}\Delta^k \cong H^k(C^\bullet)$. This result provides the fact that the kernel of $\Delta^k$ gives a set of canonical representatives for elements of $H^k(C^\bullet)$ as a space of harmonic cochains which we denote $\mathcal{H}^k(C^\bullet)$. In fact there is an orthogonal decomposition $C^k = \mathcal{H}^k \oplus \mathrm{im} \delta^{k1} \oplus \mathrm{im}(\delta^k)^\ast$. Since degree0 Sheaf Laplacian generalizes the graph Laplacian, this view most naturally extends the graph Laplacian approach of spectral graph theory. We then weight $\Gamma(X;\mathcal{F})$ by its canonical isomorphism with $\mathcal{H}^0(X; \mathcal{F})$. This view naturally yields to the study of sheaf diffusion dynamics.
 Graded View: Global sections lie in $\bigoplus_\sigma \mathcal{F}(\sigma)$. This section is incomplete.
Sheaf Laplacian
Main article: Sheaf Laplacian
Since weighted cellular sheaves are, a spectral theory,  an examination of the eigenvalues and eigenvectors of the sheaf Laplacian, is natural and motivated. This extends the combinatorial graph Laplacian to the Hodge Laplacian on a cellular sheaf of vector spaces over a regular cell complex, relating spectral data to the sheaf cohomology and regular cell structure.
In case of weighted cellular sheaves in particular, the sheaf Laplacian can be represented as a symmetric block matrix with blocks indexed by the vertices of the complex. The entries on the diagonal are given by specific formulas, and the entries on the offdiagonal are defined based on the edges between vertices. Laplacians of other degrees have similar block structures.
Constant sheaf
Main article: Harmonic cochains on sheaves
The concept of harmonic extension plays a crucial role in understanding the algebraic properties of the sheaf Laplacian and allows us to define and compute constant subsheaves of a given weighted cellular sheaf. A $k$cochain $x$ of a weighted cellular sheaf $F$ on a regular cell complex $X$ is said to be harmonic on a set $S$ of $k$cells if $(\Delta^k F x)_S = 0$. When $k = 0$ and $F$ is the constant sheaf.
See also
