Vol. 63, No. 2, 1976

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Exact functors and measurable cardinals

Andreas Blass

Vol. 63 (1976), No. 2, 335–346

The purpose of this paper is to prove that all exact functors from the category 𝒮 of sets to itself are naturally isomorphic to the identity if and only if there are no measurable cardinals. The first step in the proof is to approximate arbitrary left-exact endofunctors F of 𝒮 with endofunctors of a special sort, reduced powers, and to characterize reduced powers in terms of category-theoretic properties. The next step is to determine the effect, on the approximating reduced powers, of the additional assumption that F preserves coproducts or coequalizers. It turns out that preservation of coequalizers is an extremely strong condition implying preservation of many infinite coproducts. From this fact, the main theorem follows easily.

Mathematical Subject Classification 2000
Primary: 02K35, 02K35
Secondary: 18A15, 02K15
Received: 3 June 1975
Revised: 9 December 1975
Published: 1 April 1976
Andreas Blass
University of Michigan, Ann Arbor
Ann Arbor MI
United States