The infinite random simplicial complex
نویسندگان
چکیده
We study the Fräıssé limit of the class of all finite simplicial complexes. Whilst the natural model-theoretic setting for this class uses an infinite language, a range of results associated with Fräıssé limits of structures for finite languages carry across to this important example. We introduce the notion of a local class, with the class of finite simplicial complexes as an archetypal example, and in this general context prove the existence of a 0-1 law and other basic model-theoretic results. Constraining to the case where all relations are symmetric, we show that every direct limit of finite groups, and every metrizable profinite group, appears as a subgroup of the automorphism group of the Fräıssé limit. Finally, for the specific case of simplicial complexes, we show that its geometric realisation is topologically surprisingly simple: despite the combinatorial complexity of the Fräıssé limit, its geometric realisation is homeomorphic to the infinite simplex.
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