Re: Functoriality of pullbacks of sets

[pjf <pjf@seas.upenn.edu>]

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






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