Poset Fibrations and the Patchwork Theorem

Beyond the encoding of all allowed collapsing orders as the set of linear extensions of the universal object U (P, M), viewing the posets with small fibers as the central notion of the combinatorial part of discrete Morse theory is also invaluable for the structural explanation of a standard way to construct acyclic matchings as unions of acyclic matchings on fibers of a poset map. 

The following construction generalizes Definition 10.7 of the stack of acyclic categories. Since we will need this only for posets, we satisfy ourselves here with formulating the special case. The generalization to acyclic categories is straightforward. 

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 

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 

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