Acyclic Category of Intervals and Its Structural Functor

The entirety of intervals of the category C itself can be equipped with the structure of the category. 

When C is a poset, the category /(C) is simply the poset of all closed intervals of C ordered by reverse inclusion. 

Proposition 10.17. For an arbitrary acyclic category C, the category of intervals I(C) is acyclic as well. 

Furthermore, a morphism of an object m to itself must have the form (a,/ ) such that a is a morphism of d m to itself, and / is a morphism of d,m to itself. Again, this implies that a, (3, and hence also (a, / ), are identity morphisms in their respective categories.  

The fact that objects of /(C) actually index intervals of C is best expressed by phrasing it in terms of a functor. 

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