Complexes of Combinatorial Properties

Another prominent combinatorial procedure to describe abstract simplicial complexes is the following. Let 1 be a set of combinatorial objects of some kind, equipped with an equivalence relation of isomorphism, and assume that we have a fimction F that associates to each object from 12 a subset of some universe set V. Any collection C of the isomorphism classes in 12 gives rise to an abstract simplicial complex A C) as follows  

Naturally, the examples are endless, and easy to make up. One instance, which has appeared in knot theory, comes from the graph property of being disconnected. It can be shown that the complex of disconnected graphs on n vertices is homotopy equivalent to the order complex of the partition lattice iT see Proposition 13.15. 

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Also, for the complexes of combinatorial properties, the description can be more succinct when it uses the forbidden patterns instead of the allowed ones. For example, for the complex of disconnected graphs, the minimal non-simplices correspond to spanning trees. For the complex of directed forests, the minimal nonsimplices are all pairs of directed edges that have the same end vertex, together with all directed cycles. 

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