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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    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. – YCor Nov 8 at 22:49
  • @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? – User371 Nov 9 at 17:32
  • 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). – YCor Nov 9 at 19:43
  • 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. – Karol Szumiło yesterday
  • 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. – Dmitri Pavlov 23 hours ago

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.

  • 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. – Dmitri Pavlov Nov 10 at 4:02
  • 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. – User371 yesterday

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