Uniform Functors on Sets
This paper studies uniformity conditions for endofunctors on sets following Aczel , Turi , and others. The “usual” functors on sets are uniform in our sense, and assuming the Anti-Foundation Axiom AFA, a uniform functor H has the property that its greatest fixed point H * is a final coalgebra whose structure is the identity map. We propose a notion of uniformity whose definition involves notions from recent work in coalgebraic recursion theory: completely iterative monads and completely iterative algebras (cias). Among our new results is one which states that for a uniform H, the entire set-theoretic universe V is a cia: the structure is the inclusion of HV into the universe V itself.
KeywordsNatural Transformation Foundation Axiom Equation Morphism Standard Functor Constant Functor
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