The number of abstract simplicial complexes

On MathOverflow, François G. Dorais pointed out that the partially ordered set of abstract simplicial complexes with vertices drawn from the set { 0, …, n } is a free distributive lattice (with free top element but without free bottom element). This is not hard to see in hindsight: the generators are the (n – 1)-simplices. This poset is of interest to topos theorists because it is precisely the poset of simplicial subsets of the standard n-simplex; hence, we obtain a nice description of the subobject classifier of the topos of simplicial sets.

Amusingly, this observation implies it is very difficult to count the number of abstract simplicial complexes: apparently, the number of elements in a free distributive lattice is not known in general. This is Dedekind’s problem, and the known numbers are enumerated as OEIS sequence A000372.

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