### Journal Club – Geometric Infinity-Function Theory

#### Posted by Urs Schreiber

We decided here that it might be fun to go together through the recent Ben-Zvi/Francis/Nadler work on

in a kind of online “journal club”. The idea would be to go slowly but surely, step-by-step through the material, discuss it, ask questions about it, understand it, and, crucially, distill the joint insight into entries on the $n$Lab.

To start with, I have created in parallel to this blog entry here an $n$Lab entry. The idea is that this blog entry here is the base for the *discussion* part of our “Journal club”, while that $n$Lab entry is the base for the *write-up* part of the undertaking.

We want to sort out questions, strategy and other things that require discussion here on the blog, but want to be sure that all stable insights that we gain in the process will eventually be distilled into $n$Lab entries, lest all the effort will leave no useful traces.

Therefore, in a combined manner similar to what I tentatively started doing for the book Categories and Sheaves and the book Higher Topos Theory and for a possibly similar joint seminar at Journal Club – $(\infty,1)$-categories, the idea is that at the entry geometric $\infty$-function theory we have a bit of background information, literature, summary, overviews, and then crucially a section-by-section list of links to $n$Lab entries which concern themselves with discussing the relavent keywords.

All this will be very incomplete and preliminary in the beginning, but it may be fun to eventually and incrementrally fill in substance and thereby eventually create a useful web of linked entries that become a useful resource for reading, learning and thinking about this stuff.

Currently, there is some first bit of content at that entry, as much as I could come up with using a bit of virtual time that I didn’t really have. As with the rest of the $n$Lab, the stuff that is there is not meant to imply to be complete or even necessarily good. The maxime is: *incrementally improve*.

If you see anything that makes you raise an eyebrow – or if your eyebrows raise because you *don’t* see something – then you are in the right state for contributing: either add a query box as described at HowTo and drop the rest of us a note on why you are unhappy and what you think should be improved, or – much better – hit the “edit” button on the bottom of the page and implement an improvement. But then, please, if it is anything nontrivial, drop the rest of us a brief note indicating what you did, either in the comment section below or at latest changes.

Maybe recall that we had a bit of discussion on the first of the Ben-Zvi/Francis/Nadler articles at Ben-Zvi on geometric function theory.

There is also an $n$Lab entry on geometric function theory.

The title of this “journal club” here is inspired from this, but you should know that the term *geometric $\infty$-function theory* is one that I made up when I thought a bit about how to name this project here.

I am not sure how exactly we should start. Personally I’d find it helpful if we could try to disentangle general concepts here from concretely algebraic constructions. For instance the notions of “derived loop spaces” and the like make very general sense in every $(\infty,1)$-categorical context, while the “collections of $\infty$-functions” realized as derived quasi-coherent sheaves and D-moules make recourse to specific algebraic realizations. I feel like it would be good to separate conceptually what can be be separated, but then, maybe that’s not a helpful attitude here.

Apart from that, I should maybe point out that both the articles listed at the $n$Lab entry start with pretty long summary sections, that go through the entire respective article in a heuristic manner. I can recommend strongly to read these summary sections first to get a rough idea for what is going on. The usual kind of question that will arise at various points is: “How on earth might that work in detail?” Such questions might be good starting points for discussion here, and then eventually for distilled answers over at the $n$Lab.

## size issues and compact objects

Since the claim is that

it seems sensible to start with discussing these. It’s a bit of a pure technicality which may feel like just a nuisance. In fact, the concept of

compact objects sits inside a web of closely related concepts revolving around size issues: accessible categories, presentable categories, categories of ind-objects, all that stuff.This size issue business is maybe something we don’t want to waste the main part of our energy on, but it gives a good feeling to get this out of the way once, lest we’ll continuously be bother by it.

I don’t feel in the position to give a competent bird’s eye perspective on the entire problem complex yet, but in the process of learning this stuff myself I started filling some material into entries that now contain some crucial ideas and some crucial definitions.

Roughly in logical order, check out

cardinal number

compact object in an $(\infty,1)$-category

accessible $(\infinity,1)$-category

presentable $(\infinity,1)$-category

Still lots of room to improve and expand on these entries. But maybe a start. In any case, I need a break now.