On 2014-05-18 10:59, Colin McLarty wrote:
It seems to me that Peter Freyd remarked it is easy to define pullbacks in ZF (maybe with with global choice?) so that pullback along one side is functorial, but hard to make it functorial on both sides. In other words we can easily make base change functorial in the bases, but not easily make it functorial in the bases at the same time as in the total spaces.
Can anyone direct me to a reference to that work?
thanks, Colin
The subject of "Tau-Categories" was first exposed in my 1974 mimeographed "Pamphlet," and more accessibly in my 1990 book with Andre Scedrov, "Categories, Allegories" (often called "Cats and Alligators") starting at 1.49 (p54). Every category with finite limits is equivalent to a tau-category with a functorial choice of finite limits (and the construction is choice-free). There is, indeed, a necessary asymmetry: we can have canonical pullbacks so that if both interior rectangles in the diagram .-.-. | | | .-.-. are canonical pullbacks then so is the exterior rectangle, but then it will not be the case that such holds for the rectangles in diagrams of the form .-. | | .-. | | .-. (unless, of course, the category is just a semi-lattice). By using tau-categories one can remove the use of the axiom of choice from the constructions of various representation theorems for categories. At the end of my of my 2003 Foreword to the TAC "reprinting" of "Abelian Categories" (http://www.tac.mta.ca/tac/reprints/articles/3/tr3.pdf) I remarked on how one thus gains added "functoriality" for the theorems. Best thoughts, Peter [For admin and other information see: http://www.mta.ca/~cat-dist/ ]