The Notion of Acyclic Category

We start by defining the main character of this chapter. 

Definition 10.1. A small category is called acyclic if only identity mor-phisms have inverses, and any morphism from an object to itself is an identity. 

We shall always assume that both 0 C) and M C) are finite. This makes statements and proofs easier, though many results remain valid in the infinite case as well, either in their original form or with minor alterations. 

Recall from Chapter 4 that any poset P can be viewed as a category in the following way the objects of this category are the elements of P, and for every pair of elements x,y P, the set of morphisms M x,y) has precisely one element if x y, and is empty otherwise. Clearly, this determines the composition rule for the morphisms uniquely. When a poset is viewed as a category in this way, it is of course an acyclic category, and intuitively, if posets appear to be more comfortable gadgets, one may think of acyclic categories as 

It can be shown, and is left to the reader, that a category C is acyclic if and only if 

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