# Meaning of “combinatorial data”

I saw several times that often some data describing certain algebraic objects, eg the set of cells of a simplical complex or a Cech cycle of a chosen coving of a variety are called "combinatorial data" or "encoding combinatorial data".

My question is simply what is the justification of the name "combinatorial" here? what is combinatorial on these data?

More generally can it be precised when a certain data describing an algebraical object is called "combinatorial"?

A short UPDATE to precise what I mean (literally that's the comments below):

The meaning of "combinatorial" for abstract simplicial complexes is pretty easy to see: indeed, a data consisting of set of vertices $$V=\{ v_1,v_2,...,v_n \}$$ and a $$m$$-simplex of $$S$$ is defined as a subset $$\{ v_{i_1},...,v_{i_m} \}$$ of $$V$$. then a subset $$S$$ of power set $$P(V)$$ of $$V$$ is called abstract simplicial complex if for every $$m$$-simplex $$\{ v_{i_1},...,v_{i_m}\}$$ contained in $$S$$ every subset $$\{ v_{i_{j_1}},...,v_{i_{j_d}} \}$$ is (as a $$d$$-simplex) is contained in $$S$$ as well.

Therefore obviously not every subset of the power set $$P(V)$$ of $$V$$ is a abstract simplicial complex. So to determine which subsets of $$P(V)$$ can occure as abstract simplicial complexes is a combinatorial problem. That's where I see the "combinatorial flavour" here. So I think that this is precisely the justification for the word "combinatorial" if one can associate to an algebraic or topological object an abstract simplical complex encoding sometimes a lot of information about the original object. (the most prominent example is surely the Nerve theorem which desides when this "combinatorial data" essentially suffice to reconstruct the original object up to homotopy).

But how to draw the same analogy to Cech cycles isn't clear to me. Is it possible to associate abstractly a abstract simplicial complex to a Cech cycle in order to "make" it combinatorial?

• I don't think there's a precise formal meaning. I understand it to mean something Kronecker would approve of, i.e., an essentially finistic mathematical object, something you could encode on a computer (e.g., a finite graph or hypergraph, as opposed to something like a differentiable manifold). – Gro-Tsen Jul 11 '20 at 21:35
• the origin of the name as well as a general definition is discussed in Wikipedia --- the combinatorial map tells you how to combine simplices to form a simplicial complex. – Carlo Beenakker Jul 11 '20 at 21:38
• yes I thought that orginally this arose from the concept of abstract simplicial complex: a data $S$ consisting of vetrices $V={v_1,v_2,...,v_n}$ an a $m$-simplex of $S$ is a subset ${v_{i_1},..., v_{i_m}}$ of $V$. $S$ is called abstract simplicial complex if for every $m$-simplex ${v_{i_1},..., v_{i_m}}$ contained in $K$ every subset ${v_{i_{j_1}},..., v_{i_{j_d}}}$ is (as a $d$-simplex) is contained in $S$. – user7391733 Jul 11 '20 at 22:42
• Therefore not every subset of the power set $P(V)$ of $V$ is a abstract simplicial complex. So to determine which subsets of $P(V)$ can occure as abstract simplicial complexes is a combinatorial problem. But how to draw the same analogy to cech cycles isn't clear to me. Is it possible to associate abstractly a abstract simplicial complex to a Cech cycle in order to "make" it combinatorial? – user7391733 Jul 11 '20 at 22:57

## 1 Answer

I think this is substantially an issue of tradition. Some decades ago (e.g. 1970), when an otherwise finitistic thing seemed to admit no simpler description than ... itself (e.g., in terms of Kolmogorov-Solomonov-Chaitin complexity, for example), it was referred to as "combinatorial".

Yes, that sense of the word is not closely related to usage in 2020.

In particular, in older work, there is probably no reward in looking for significant subtler meaning, other than "it is what it is"... as opposed to being "compressible" in any way in the larger context of mathematics.