Functors on the category of finite sets

Dougherty, Randall

doi:10.1090/S0002-9947-1992-1053111-4

Functors on the category of finite sets
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Trans. Amer. Math. Soc. 330 (1992), 859-886 Request permission

Abstract:

Given a covariant or contravariant functor from the category of finite sets to itself, one can define a function from natural numbers to natural numbers by seeing how the functor maps cardinalities. In this paper we answer the question: what numerical functions arise in this way from functors? The sufficiency of the conditions we give is shown by simple constructions of functors. In order to show the necessity, we analyze the way in which functions in the domain category act on members of objects in the range category, and define combinatorial objects describing this action; the permutation groups in the domain category act on these combinatorial objects, and the possible sizes of orbits under this action restrict the values of the numerical function. Most of the arguments are purely combinatorial, but one case is reduced to a statement about permutation groups which is proved by group-theoretic methods.

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