Re: Weighted limits

[David Roberts <droberts.65537@gmail.com>]

Dear Jean

[apologies to the moderator for sending the below message from the
wrong email address]

I must correct myself: in the paragraph

>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.

I should not have said '(the inclusion of) a subcategory Ins(f,g)',
but rather 'a faithful functor Ins(f,g) --> X'.
With this change everything proceeds as before.

I can even supply a different, and cleaner, direct construction of the
inverter of the natural transformation a: f => g: X --> Y.

First define the object B of S as the pullback of

Obj(X) -- (g,f) --> Obj(Y) x Obj(Y) <---- (s,t) ---- Arr(Y)  (note
order of f and g)

as before. This gives us the projection map b': B --> Arr(Y) as noted
above, which will eventually give the putative inverse of a, and we
also have the composite map B --> Obj(X) ---a--> Arr(Y), which I will
call a_B. We can define two maps

(1)    (a_B)b': B ---(a_B,b')--> Arr(Y) x_Obj(Y) Arr(Y) --> Arr(Y)
(2)    b'(a_B): B ---(b',a_B)--> Arr(Y) x_Obj(Y) Arr(Y) --> Arr(Y)
(latter arrow is composition in both cases)

(1) gives the component of what will be a natural transformation from
f to itself, and (2) likewise, except from g to itself

Now take the equaliser of (1) and the map

B ---> Obj(X) --f--> Obj(Y) --> Arr(Y)

to get the subobject B_f --> B, and take the equaliser of (2) and the map

B ---> Obj(X) --g--> Obj(Y) --> Arr(Y)

to get the subobject B_g --> B. Now take the pullback of B_f --> B <--
B_g to get the subobject Inv(f,g)_0 ---> B. Now consider the composite
map Inv(f,g)_0 ---> Obj(X): this will be the object component of the
map from the inverter to X. Form the category J = X[ Inv(f,g)_0 ],
which has as objects Inv(f,g)_0 and as arrows the pullback (Inv(f,g)_0
x Inv(f,g)_0) x_{Obj(X) x Obj(X)} Arr(X), and comes equipped with a
fully faithful functor (in the internal sense) J --> X. Let b:
Inv(f,g)_0 ---> B --b'-> Arr(Y) be the obvious composite.

Now we need to build a wide subcategory Inv(f,g) of J and this will be
the inverter, via the given map to X. We have the component map b:
Obj(J) = Inv(f,g)_0 --> Arr(Y), but it is not necessarily natural with
respect to all the arrows of J (considered as eg generalised elements,
or in the internal language). So we consider the subobject Inv(f,g)_1
--> Arr(J), defined equationally (hence by a certain equaliser), so
that naturality squares for b commute, for arrows in Inv(f,g)_1.

Then Inv(f,g) --> J --> X is the equaliser you are looking for, and I
only used finite limits in S.

Apologies for being so long-winded, but you gave us a nice exercise
and I wanted to see it through (modulo the very last bit, I hope it is
obvious)

Best regards,
David

PS one can build all cotensors (=powers) in Cat(S) with all finite
categories using the same pedestrian logic; hence with all conical
strict 2-limits and cotensors one gets all strict weighted limits.


On 15 February 2017 at 21:03, David Roberts <a1078662@adelaide.edu.au> wrote:
> 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.
>>
>> ________________________________
>> From: Jean Benabou <jean.benabou@wanadoo.fr>
>> Sent: 15 February 2017 5:47 AM
>> To: David Roberts; John Power; Ross Street; Categories
>> Subject: Re: categories: Weighted limits

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