Re: Functoriality of pullbacks of sets

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

On 2014-05-20 00:54, Vaughan Pratt wrote:
> On 5/19/2014 7:03 AM, pjf wrote:
>> Every category with finite limits is equivalent to a
>> tau-category with a functorial choice of finite limits (and the
>> construction is choice-free).
>
> Why merely finite?  Didn't you show this for all \omega-polynomials
> (i.e. less than \omega^\omega), or have I overlooked something?
>
> Vaughan


Vaughan,

\omega^\omega did play a role (40 years ago!) but not the one you
describe.

The full subcategory of (ZF) sets whose objects are the von Neumann
ordinals has an easy tau-structure. The canonical pullbacks, for
example, are those in which the order on the NW corner coincides with
the order that is lexicographically induced by the two maps therefrom.
(So, yes, products are strictly associative and a monic is an
"inclusion" if it's order-preserving.) In section 1.4(12) of "Cats &
Alligators" \omega^\omega appears as the set of objects of a full
subcategory denoted _P_.

    METATHEOREM. A equation is true for all tau-categories iff it
    is true for _P_.

Given a counterexample in an arbitrary tau-category to an equation in
the (essentially algebraic) theory of tau-categories the proof
constructs (yes, constructs) a counterexample in _P_.

     Best thoughts,
       Peter



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