Localisation by cospans redux

In my previous post I described a construction whereby a 2-category of groupoids internal to a lextensive category whose 1-arrows are functors with object component a coproduct inclusion can be localised at the fully faithful essentially -surjective functors. The 1-arrows of are called complemented cofibrations, and the 1-arrows of the localisation were then certain cospans of such functors. This was rather formal and relied on these notes of mine. However, for the case that the ambient category is that of sets, Land, Nikolaus and Szumiło showed that this localisation is equivalent to localising the category of all small groupoids at the fully faithful essentially surjective functors (although they worked in the setting of -categories, the result is still just a -category). I started sketching how one should construct a 2-functor from a 2-category of internal groupoids with more general functors to the localisation by cospans. (The functors need to satisfy a condition related to the pretopology one is using on the lextensive category, which in the case of a pretopos with the canonical topology turns out to be always true, but there are interesting cases where it is a real restriction.) That said, one can use Pronk’s comparison theorem to show the stronger result that , which I will sketch below the fold.

Cofibrations of internal groupoids and localisations by cospans

In my paper ICAL, I dealt extensively with internal categories and groupoids, and how to localise 2-categories of these at the fully faithful essentially ‘surjective’ functors (‘surjective’ in the sense there is some well-behaved class of regular epimorphisms playing the rôle of surjective maps). Even for 1-categories of ordinary small groupoids or categories one can consider this as a variant on the canonical model structure on groupoids or on categories, and formally invert the fully faithful, essentially surjective functors, which aren’t invertible in the 1-category sense. Actually it really only uses the category of fibrant objects structure, and ICAL (and more in generality, these notes) can be seen as a way to make this work in a way that doesn’t use the full category of fibrant objects structure, but the 2-category structure instead. There is another way to think about this process for the case of small groupoids (and more generally from some cofibration category), which I learned from a paper of Land, Nikolaus and Szumiło, Localization of Cofibration Categories and Groupoid C*-algebras (publisher, arXiv) (cofibrations of groupoids are functors that are injective on objects). I want to outline how one can repeat the 2-categorical, extra weak story for internal groupoids (at least; I haven’t thought about the case of internal categories) using this type of approach. In some respects, I can get more general results, in that I don’t require that I’m localising a small cofibration category with good cylinders, but also kinda less general in the sense that the universal property I get is only for bicategories, not more general -categories (presented by something Quillen equivalent to a combinatorial model category). Below the fold I set up some basic assumptions, and then outline the constructions that will go into the hypotheses of the construction.

The elementary construction of formal anafunctors

Some time ago I wrote notes constructing in a purely 2-categorical language a bicategory of anafunctors, starting from a 2-category $K$ equipped with a notion of cover, which in the original setting that Makkai studied, reduces to a trivial isofibration of categories. I always wanted to do something more with the construction, and I still do, but I thought it worthwhile to get the notes into a shape suitable for public consumption (at one point I had changed notational convention, and I found this week that the transition was half-way through a diagram!). So here they are:

The elementary construction of formal anafunctors, arXiv:1808.04552, doi:10.25909/5b6cfd1a73e55

Abstract: These notes give an elementary and formal 2-categorical construction of the bicategory of anafunctors, starting from a 2-category equipped with a family of covering maps that are fully faithful.

As always, comments welcome.

Added 21 August: In my paper Internal Categories, Anafunctors and Localisation (ICAL), published in TAC in 2012, I mentioned a ‘sequel’, which I then called Strict 2-sites, J-spans and localisations, in obvious parallel with ICAL. The notes above are essentially that sequel, but I have more ideas I want to add to the above before it is is finalised and I submit it for publication.

Choice quote from Venkatesh

ICM 2018 | "A lot of the time, when you do Math, you're stuck. But you feel privileged to work with it. You have a feeling of transcendence and feel like you've been part of something really meaningful." Akshay Venkatesh, 2018 Fields Medallist. #ICM2018 #ICM2018Rio

— ICM2018 (@ICM_2018) August 1, 2018

Gowers on Scholastica

Timothy Gowers, as many would know is a Fields medallist and has become somewhat of a spokesperson, or at the least a figurehead, for open access issues in mathematics publishing. He spearheaded the Cost of Knowledge boycott of Elsevier, helped found the open access journals Forum of Mathematics: Pi and Sigma and more recently, Discrete Analysis and the new Advances in Combinatorics. The difference between the former two and the latter is that the FoM journals are published by a commercial publisher and have non-zero article processing charges (APCs), typically paid for out of research funds, or library OA funds etc, whereas DA and AinC is an ‘arXiv-overlay’ journal: the final versions of articles are stored on the arXiv, and the cost to the journal for each article is O($10), which is covered by a grant/donated funding. Needless to say, these journals are part of the Free Journal Network, the aim of which is to be a loose confederation of open access journals that are free to publish in and meet modern standards of ‘openness’. The journal websites for DA and AinC are hosted by Scholastica, and Tim recently did an interesting interview with Scholastica about the whole business.