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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