Simplicial set are to cubical sets what simplicial complexes are to ...? Simplicial sets and cubical sets (with or without connections) are defined as presheaves over some indexing categories. There is a full subcategory of simplicial sets that we can identify with the category of (oriented and abstract) simplicial complexes. An object in this category consists of a set together with a collection of its subsets, called simplices, each equipped with a total order. The collection of simplices contains all singletons, is close under taking subsets, and the order of a simplex agrees with the one induced from any simplex containing it. 
Is there an interesting combinatorially defined subcategory of cubical sets? Maybe one analogue to simplicial complexes? 
 A: (This is rather an answer to the question "what is an abstract cubical complex" asked in the comments:)
In this paper, Farley defines an abstract cubical complex $C$ as a collection of subsets of a given vertex $V$ with:


*

*$C$ covers $V$,

*If $\sigma, \tau \in C$, then $\sigma\cap\tau \in C$,

*For each $\sigma\in C$, there exists a bijection from $\sigma$ to some n-cube $\{0,1\}^n$ satisfying the property that any subset of $\sigma$ is in $C$ iff it is mapped to a face of the n-cube.


I'm not sure if that is the best translation of geometric cube complex to an "abstract" setting but it seems to fit the bill!
A: What about this? 
A hypergraph consists of a pair $(V,E)$ with $V$ a set and $E$ a collection of subsets of $V$. Morphisms of hypergraphs are functions $V \to V'$ s.t. the image of each $e \in E$ is in $E'$.
Denote the objects of the Simplex and Cube categories by $[n]$ and $I^n$ respectively.
A hypergraph $(V,E)$ such that all singletons of $V$ are in $E$ is an (abstract) simplicial complex resp. cubical complex if is equipped with a bijection $f_e : [n] \to e$ resp. $f_e : I^n \to e$ for each $e \in E$ such that:
1) For $e' \subset e \in E$ we have $e' \in E\,$ iff there exists an injective morphism $\varphi$ in the category such that the image of $f_e \circ \varphi$ is $e'$.
2) In the case above, $f_e \circ \varphi = i \circ f_{e'}$ where $i$ is the inclusion.  
Morphisms between these are hypergraph morphisms preserving the corresponding structure.
Claim: The categories of simplicial and cubical complexes embed full and faithfully into the categories of simplicial and cubical sets. The geometric realization functor takes the image of a simplicial or cubical complex, as define here, into a simplicial or cubical complex as defined geometrically.
