Poset fibration

The notion of poset fibrations satisfies the following universality property. Theorem 11.9. (Decomposition theorem)... 

For an arbitrary poset fibration B, F), where F = Fx xeB> o,nd an arbitrary poset P, there is a 1-to-l correspondence between... 

Proof. The role of the base space here is played by the poset Q, and the fiber maps gq are given by the acyclic matchings on the subposets ip q). The decomposition theorem tells us that there exists a poset map from P to the total space of the corresponding poset fibration, and that the fibers of this map are the same as the fibers of the fiber maps Qq. Since the latter are given by acyclic matchings, we conclude that we have a poset map from P with small fibers that corresponds precisely to the patching of acyclic matchings on the subposets 95 (g), for q Q. ... 

Our main innovation in Section 11.1 is the equivalent reformulation of acyclic matchings in terms of poset maps with small fibers, as well as the introduction of the universal object connected to each acyclic matching. The patchwork theorem 11.10 is a standard tool, used previously by several authors. We think that the terminology of poset fibrations together with the decomposition theorem 11.9 give the patchworking particular clarity. 

Associated to such a fibration we have a poset E B,T), defined as the union with the order relation given by a > / if either a, / Fx, and... 

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