Intervals in Acyclic Categories

In particular, as an immediate corollary of (10.5) we see that an arbitrary regular CW complex is homeomorphic to the order complex of some poset. We also obtain an alternative way to define the barycentric subdivision of 

Example 10.12. Let P be a totally ordered set with n elements. Clearly, its barycentric subdivision is the Boolean algebra with the minimal element removed BdP = 6. Equation (10.7) implies that 

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The next proposition is an important observation connecting intervals in acyclic categories to the arrow categories. 

Finally, let us remark that it is possible to take one of the functors F and G in Definition 4.36 contravariant. This will also yield a category, which we denote by 0(F°p G) if F is taken to be contravariant, and 0 F J. G°p) if G is taken to be contravariant. This construction will come in handy when we define the analogue of intervals for acyclic categories. 

In the context of acyclic categories, the intervals correspond to the following objects. 

The notion of incidence algebra has its analogue for acyclic categories as well. As is to be expected, the role of intervals in a poset is taken over by morphisms of a category. 

The main difference in the acyclic category case is that the notion of the interval is replaced with the notion of the morphism. Accordingly, a maximal chain in the interval is replaced with the composable sequence of indecomposable morphisms, which together compose to yield the corresponding morphism. To abbreviate our language, we say that a maximal composable sequence of morphisms (mi,..., nik) is in m if it composes to m. 

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