Links, Stars, and Deletions

One of the simplest operations one can do on an acyclic category is that of a deletion of a set of vertices. Once the vertex is deleted, all the simplices that contained this vertex in the nerve will be deleted as well hence for an arbitrary acyclic category C, and for arbitrary set of objects S C 0(C), we have A C S) = dU(c)S. 

C x C x, C x, and Cyx are all acyclic as well. Furthermore, for posets we simply get the standard notions of the subposets of all elements below or above x. 

It is easy to see that this bijection induces a trisp isomorphism, and hence formula (10.2) is proved. Formula (10.3) is proved in just the same way, with 

The natural generalization of the notion of a lattice would be to require that the acyclic category have all finite products and coproducts. Unfortunately, this does not bring an3d,hing new, as the next proposition shows. 

Proposition 10.9. Let C be an acyclic category that has all products of two elements. Then C is a poset. 

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