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- W2964222779 abstract "Two of the pillars of combinatorics are the notion of choosing an arbitrarysubset of a set with n elements (which can be done in 2 n ways), and the notion of choosinga k-element subset of a set with n elements (which can be done in $$ n choose k $$ ways). In this articleI sketch the beginnings of a theory that would import these notions into the categoryof hedral sets in the sense of Morelli and the category of polyhedral sets in the sense ofSchanuel. Both of these theories can be viewed as extensions of the theory of finite sets andmappings between finite sets, with the concept of cardinality being replaced by the moregeneral notion of Euler measure (sometimes called combinatorial Euler characteristic). Iprove a “functoriality” theorem (Theorem 1) for subset-selection in the context of polyhedralsets, which provides quasi-combinatorial interpretations of assertions such as $$ 2^{-1} = {1 over 2} quadmathrm{and}quad{{1/2} choose {2}} =-{1 over 8} $$ . Furthermore, the operation of forming a power set can be viewed as a specialcase of the operation of forming the set of all mappings from one set to another; I concludethe article with a polyhedral analogue of the set of all mappings between two finite sets,and a restrictive but suggestive result (Theorem 2) that offers a hint of what a generaltheory of exponentiation in the polyhedral category might look like. (Other glimpses intothe theory may be found in [11].)" @default.
- W2964222779 created "2019-07-30" @default.
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- W2964222779 date "2003-10-01" @default.
- W2964222779 modified "2023-10-16" @default.
- W2964222779 title "Exponentiation and Euler measure" @default.
- W2964222779 doi "https://doi.org/10.1007/s00012-003-1817-1" @default.
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