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?

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    $\begingroup$ There's a notion of cubical complex, and its subclass of CAT(0) cube complexes, studied for about 25 years now, has many recent developments. $\endgroup$ – YCor Nov 8 '18 at 22:49
  • $\begingroup$ @YCor Thank for the comment. The definitions I have found are all akin to defining simplicial complexes as gluing simplices injectively along full faces. Which is a description of the geometric realization of an abstract simplicial complex. In this terminology, the question is: what is an abstract cubical complex? $\endgroup$ – User371 Nov 9 '18 at 17:32
  • $\begingroup$ I'm not sure what you mean by abstract, but a CAT(0) cubical complex is determined by its 1-skeleton. A graph is 1-skeleton of a CAT(0) cube complex iff it's median. Median is a metric condition (on triples inside a given connected component), which is possibly not what you want. But it can be characterized by a pair of conditions. The first is local, namely that the link at every vertex is a flag complex. The second is global, namely that the fundamental group of any component is generated by loops of size 4 (I'm not sure how to formulate this in a more canonical way). $\endgroup$ – YCor Nov 9 '18 at 19:43
  • $\begingroup$ Take a look at Sections 4 and 5 of Jardine's Cubical Homotopy Theory: A Beginning. I doesn't quite give the definition that you want, but maybe it comes close. $\endgroup$ – Karol Szumiło Nov 12 '18 at 8:56
  • $\begingroup$ Your definition of an ordering on a simplicial complex does not allow one to encode as an ordered abstract simplicial complex the circle composed of n vertices, labeled 0, 1, …, n−1, and n (ordered) edges i→i+1, with the last edge being n−1→0. By transitivity, any ordering on {0,1,…,n−1} must satisfy i≤j for all i and j, which contradicts the requirement that the restriction of the global ordering to any simplex must be total, in particular, antisymmetric. One typically wants to avoid introducing a global ordering, instead ordering each simplex separately in a compatible way. $\endgroup$ – Dmitri Pavlov Nov 12 '18 at 21:31

(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!

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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.

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  • $\begingroup$ Your answer describes locally ordered simplicial complexes, not simplicial complexes. In an abstract simplicial complex there is no natural ordering on vertices of a simplex. $\endgroup$ – Dmitri Pavlov Nov 10 '18 at 4:02
  • $\begingroup$ I know Dimitri, oriented is a word appearing in the original question. This type of order is naturally present when thinking of a simplicial complexes as special type of simplicial set. $\endgroup$ – User371 Nov 12 '18 at 15:21
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    $\begingroup$ One could define a (finite) abstract cubical complex to be a finite meet-semilattice, say with minimum element 0, such that every interval [0,x] isomorphic to the face lattice of a cube. $\endgroup$ – Richard Stanley Dec 13 '18 at 2:29

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