
Regular cell complexes are a type of cell complexes that are particularly instrumental in TDA and spectral sheaf theory. Regular cell complexes provide combinatorial and topological insights into various spaces, including simplicial and cubical complexes.
Definition
A regular cell complex is a topological space $X$ partitioned into subspaces $\{X_\alpha\}_{\alpha\in PX}$ satisfying the following conditions:
 For each $x \in X$, every sufficiently small neighborhood of $x$ intersects finitely many $X_\alpha$.
 For all $\alpha, \beta$, $X_\alpha \cap X_\beta \neq \emptyset$ only if $X_\beta \subseteq X_\alpha$.
 Every $X_\alpha$ is homeomorphic to $\mathbb{R}^{n_\alpha}$ for some $n_\alpha$.
 For every $\alpha$, there is a homeomorphism of a closed ball in $\mathbb{R}^{n_\alpha}$ to $X_\alpha$ that maps the interior of the ball homeomorphically onto $X_\alpha$.
Condition (2) implies that the set $PX$ has a poset structure, given by $\beta \leq \alpha$ if $X_\beta \subseteq X_\alpha$. This is known as the face poset of $X$. The regularity condition (4) implies that all topological information about $X$ is 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 $\beta \prec \alpha$. The class of posets that 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.
Spectra
Main article: Spectral Sheaf Theory, Hodge Laplacian
The class of regular cell complexes includes simplicial complexes, cubical complexes, and socalled multigraphs (as 1dimensional 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 a spectral sheaf theoryspectral theory of sheaves can be developed. In practice, spectral theory of complexes has been largely restricted to the study of simplicial complexes. A number of results about the spectra of Hodge Laplacians of regular cell complexes can be obtained restricting to the constant sheaf.
See also
