Prodsimplicial Flag Complexes

Definition 9.16. Let G he an arbitrary graph. We define the prodsimplicial complex PF[G) as follows the graph G is taken to be the 1-dimensional skeleton of PF[G), and the higher-dimensional cells are taken to be all those products of simplices whose 1-dimensional skeleton is contained in the graph G. The complex PF G) is called prodsimplicial flag complex ofG. 

The prodsimplicial flag construction allows one to specify a prodsimplicial complex by a relatively compact set of data. The reader should note that while a simplicial complex is always also a prodsimplicial complex, a simplicial flag complex is usually not a prodsimplicial flag complex. An example of that is provided by a hollow square. 

Clearly, these complexes are prodsimplicial. It will be shown in Section 18.1.1 that complexes of morphisms Horn, —) are in fact prodsimplicial flag complexes. 

Proposition 18.1 comes in handy when we need to show that certain prod-simplicial complexes caimot be represented as Horn complexes, since it imposes the rather rigid restriction of being a prodsimplicial flag complex. 

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