Re: Weighted limits

[David Roberts <a1078662@adelaide.edu.au>]

Dear Jean, (apologies for this, and any future, slow replies. The
necessities of life take up a lot of my time at the moment)

My approach below is pedestrian, but I hope clear.

Strict inverters are PIE-limits. Thus they can be computed in Cat(S)
once we know it has each of products, inserters, equifers -- in fact
just the latter two, in a rather straightforward way, using no more
than two of each. To quote the nLab, "first we insert a 2-morphism
b going in the opposite direction from a, then we equify ba and ab
with identities." (this quote may be likewise borrowed from either
Kelly or Street)

Let as assume S has finite limits throughout.  For what it's worth,
products obviously exist in Cat(S).

Note that For X a category in S, and a subobject U >--> Obj(X), we can
build the full subcategory X[U] of X on U (as an object of Cat(S))
using only finite limits in S. To build the equifer of a,b: f => g: X
--> Y, we only need the equaliser E in S of the component maps a,b:
Obj(X) --> Arr(Y), and then the equifer is the inclusion  X[E] --> X
of the full subcategory on the subobject E.

Thus we are reduced to building inserters, which is the real meat of
the problem, as inserters are not equivalent to any conical 2-limit.

Consider a diagram f,g: X --> Y in Cat(S). The inserter of this
diagram is (the inclusion of) a subcategory Ins(f,g) of X. We can
compute the object Ins(f,g)_0 of objects of the inserter as the
pullback of

Obj(X) -- (f,g) --> Obj(Y) x Obj(Y) <---- (s,t) ---- Arr(Y)

in S.

Then the inserter is a wide subcategory of X[ Ins(f,g)_0 ] (itself a
full subcategory of X). Note that there is a map a: Ins(f,g)_0 --->
Arr(Y) which will be the component map of the universal natural
transformation we are inserting.

The arrows of Ins(f,g) are the largest subobject Ins(f,g)_1 --> Arr(X)
such that a is natural with respect to such arrows. This can be
defined by an equaliser in S.

Thus we can construct, using solely finite limits in S, (products,)
equifiers and inserters, and hence inverters, in Cat(S).

One could perhaps examine this proof more closely to see what kind of
internal categories in non-finitely-complete S are necessary for it to
work (eg those such that (s,t) belong to a class of which all
pullbacks exist, and are again in the class etc). This perhaps would
fit with your general philosophy on generalising fibration technology.

I hope this answers your qualms, and apologies for being slightly
telegraphic in my description.

Best regards,

David

PS I regret we did not have the chance to meet at Topos à l'IHÉS in
2015. Perhaps one day...






On 15 February 2017 at 20:09, Jean Benabou <jean.benabou@wanadoo.fr> wrote:
> Dear John,
>
> Thank you for your mail and the precisions you give in it, but I'm not
> interested, for the time being, in general questions about 2-categories.
> Let me repeat precisely my question:  If  S is a category with finite limits
> and Cat(S) is the 2-category of internal categories of S, under which
> condition does Cat(S) have strict inverters?
> Can you, or anybody give a precise answer? (Of course I know that Cat(S) is
> cotensored with 2)
> .
> David Roberts says that finite limits in S suffice. As I I said I don't
> believe that. I'm perhaps wrong. In that case, could he, you, or anybody
> tell me how to construct strict inverters when all I assume is that S has
> finite limits?
>
> It is always a pleasure to hear from you. All the best ,
>
> Jean
>
>
> Le 15 févr. 17 à 08:41, John Power a écrit :
>
> Dear Jean,
>
> Max wrote an expository paper which I believe was called "Elementary
> Observations on 2-Categorical Limits" and was published in the Bulletin of
> the Australian Mathematical Society I think around 1990. He would have had a
> discussion of inverters there.
>
> Strict inverters are a kind of strict weighted limit (see, for instance,
> https://golem.ph.utexas.edu/category/2014/04/elementary_observations_on_2ca.html)
> and a 2-category has all strict weighted limits if it has all strict conical
> limits and all strict cotensors, as a 2-category is a Cat-enriched category.
> So if one can prove that Cat(S) has strict conical limits and strict
> cotensors, one can construct strict inverters by following the procedure in
> the  link above.
>
> For strict cotensors, it suffices to prove that a 2-category has strict
> cotensors with the arrow category. I believe that is straightforward for
> Cat(S) if you follow the case of S = Set.
>
> Once again, it is always lovely to hear from you.
>
> All the best,
>
> John.
>

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