From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/9121 Path: news.gmane.org!.POSTED!not-for-mail From: David Roberts Newsgroups: gmane.science.mathematics.categories Subject: Re: Weighted limits Date: Thu, 16 Feb 2017 06:09:44 +1030 Message-ID: References: Reply-To: David Roberts NNTP-Posting-Host: blaine.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: quoted-printable X-Trace: blaine.gmane.org 1487264008 20990 195.159.176.226 (16 Feb 2017 16:53:28 GMT) X-Complaints-To: usenet@blaine.gmane.org NNTP-Posting-Date: Thu, 16 Feb 2017 16:53:28 +0000 (UTC) Cc: John Power , Categories To: Jean Benabou Original-X-From: majordomo@mlist.mta.ca Thu Feb 16 17:53:23 2017 Return-path: Envelope-to: gsmc-categories@m.gmane.org Original-Received: from smtp2.mta.ca ([198.164.44.40]) by blaine.gmane.org with esmtp (Exim 4.84_2) (envelope-from ) id 1cePJ4-0004yL-87 for gsmc-categories@m.gmane.org; Thu, 16 Feb 2017 17:53:22 +0100 Original-Received: from mlist.mta.ca ([138.73.1.63]:44524) by smtp2.mta.ca with esmtp (Exim 4.80) (envelope-from ) id 1cePIe-0005fG-E3; Thu, 16 Feb 2017 12:52:56 -0400 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1cePI6-0002Rt-RO for categories-list@mlist.mta.ca; Thu, 16 Feb 2017 12:52:22 -0400 In-Reply-To: Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:9121 Archived-At: 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 =3D> 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 =3D 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) =3D 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 (=3Dpowers) 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 wrot= e: > 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 =3D> 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 =C3=A0 l'IH=C3=89= S in > 2015. Perhaps one day... > > > > > > > On 15 February 2017 at 20:09, Jean Benabou wrot= e: >> 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 li= mits >> 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 ha= s >> finite limits? >> >> It is always a pleasure to hear from you. All the best , >> >> Jean >> >> >> Le 15 f=C3=A9vr. 17 =C3=A0 08:41, John Power a =C3=A9crit : >> >> 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 h= ad 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 con= ical >> limits and all strict cotensors, as a 2-category is a Cat-enriched categ= ory. >> 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 =3D Set. >> >> Once again, it is always lovely to hear from you. >> >> All the best, >> >> John. >> >> ________________________________ >> From: Jean Benabou >> 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/ ]