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....) 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

[For admin and other information see: http://www.mta.ca/~cat-dist/ ]