# 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?

• 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 '18 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 '18 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 '18 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 Nov 12 '18 at 8:56
• 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 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!

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 '18 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 Nov 12 '18 at 15:21
• 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. – Richard Stanley Dec 13 '18 at 2:29