Date: Sat, 31 Dec 1994 11:09:36 -0400 (AST) Subject: G. M. Kelly and anafunctors Date: Fri, 16 Dec 94 14:25:45 EST From: Michael Makkai Some weeks ago, Max Kelly brought to my attention his paper "Complete functors in homology I. Chain maps and endomorphisms", Proc. Cambridge Phil. Soc. 60 (1964), pp. 721-735. In section 2, "Generalities on functors", in the first two (full) paragraphs on page 723, he gives a definition of what I call "anafunctor" in my paper "Avoiding the axiom of choice in general category theory" (relatively) recently announced on the Net. He gives both definitions, the elementary one, and the one in the style of spans that I give in my paper. He introduces the concept (that he does not name) as the general form of the data which one is having at one's disposal in many situations when one wants to introduce a functor but the converting of the data into a functor requires a use of the axiom of choice. He does not attempt to develop a theory of such data; however, he does say: "one who will not admit such choices [requiring the axiom of choice] may work with the pair of honest functors S , T [in the span-style definition of "anafunctor"] in place of the dishonest functor ...". Thus, my paper is a working-out of a thirty-year old idea of Max Kelly's. In the paper, I endeavor to show that one can indeed "work with the pair S , T " without giving up the basic structure of category theory. Michael Makkai Date: Fri, 24 Jan 1997 10:55:18 -0400 (AST) Subject: anafunctors Date: Thu, 23 Jan 1997 13:26:33 -0800 From: john baez What is the locus classicus for "anafunctors"? As far as I know, an anafunctor F: C -> D is a presheaf on C x D^{op} such that F(c,.) is representable for any object c of C. Is this how it's normally defined? Where is composition of anafunctors discussed? Are there other names for these things? John Baez Date: Wed, 29 Jan 1997 16:04:36 -0400 (AST) Subject: anafunctors Date: Wed, 29 Jan 1997 14:56:30 -0500 (EST) From: Michael Makkai John Baez asked, on January 24, about anafunctors. As far as I know, the notion was first explicitly introduced in my paper "Avoiding the axiom of choice in general category theory", in JPAA 108 (1996), 109-173. The term was suggested by Dusko Pavlovic. Precursors occur in the work of Max Kelly, and Andre Joyal, as I explain in the paper. The concept John gives is equivalent to "saturated anafunctor" in the paper; plain anafunctor is something that generalizes "functor". The wording of the definition of "saturated anafunctor" is different from John's definition, but the equivalence is fairly straightforward. I should mention that John`s definition is a very useful formulation, especially when one wants to generalize things to higher dimensional categories, as I came to realize some time after I started studying the John Baez/James Dolan announcement on weak n-categories. In addition to the paper mentioned above, there is reference to anafunctors in "First Order Logic with Dependent Sorts", a monograph that will appear in Springer's Lecture Notes in Logic as soon as I manage to complete the necessary revisions; it is available electronically from the TRIPLES and HYPATIA (?) sites.