Re: Functoriality of pullbacks of sets

[Robin Cockett <robin@ucalgary.ca>]

Another quite independent way of showing that limits can be chosen
canonically relies on

(a) being able to move to an equivalent category in which products are
canonical.
(b) being able to move to an equivalent category which has a "canonical
inclusion system"

Step (b) has received attention as it is related to subtype systems for CS
applications.  Essentially in the equivalent category each equivalence
class of monics  has a canonically chosen one: these have been studied
by  Grigore
Rosu (who called them "Weak inclusion systems"), by Andree Ehresmann (who
called them ss-admissible systems).  Recently they were linked to
restriction categories by Hilberdink ("Inclusions for Partiality" to appear
in MSCS).  There is an axiom of choice free way of doing (b) -- which does,
however, heavily use equivalence classes.  One moves to the partial map
category, this is a restriction category. One then splits the restriction
idempotents: they already split but one uses the formal splittings to give
one a canonical inclusion system then one moves back to the total map
category and !!bingo!! one gets an equivalent category to the original with
a canonical inclusion system.

To get canonical limits of all stripes one then just needs products to be
canonical ....

Of course, obtaining canonical limits in this manner does NOT make pulling
back functorial.   However, what it does immediately do is to make pulling
back of inclusions canonical.

-robin


On Mon, May 19, 2014 at 10:54 PM, Vaughan Pratt <pratt@cs.stanford.edu>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
>
>

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