A Class of Filtrations

Let now A be a finite acyclic category. We would like to describe a special class of filtrations on the cochain complex C A K)). In order to construct such 

Having chosen the subcategory J and the function /, let us now first define an increasing filtration on the regular trisp A K). Let F = ai 

By definition of the nerve of an acyclic category, the differential skips one of the objects of the composable morphism chain. Omitting an object other than the pivot does not alter the weight of this chain, while omitting the pivot turns another element into the new pivot, on which / takes a lower value than on piv(F). Thus the resulting chain has a strictly lower weight. Hence 9 (Z j) C At, i.e., the differential operator respects the filtration. 

Assume that X is a regular trisp, and that Y is a subtrisp of X such that the vertices in X Y have no edges in between them. Then we have 

For a J, we let Sa be the induced subcategory of C with the set of objects (/C J) U 6 JI fib) /(a). Since the objects of J that are added to the complex in the same filtration step form an antichain, we can use the general fact above and derive 

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