The n-Category Café

Urs Schreiber: Apart from the seminar notes on $(\infty,1)$-topos theory that I keep writing (but I think $(\infty,1)$-category theory - contents and $(\infty,1)$-topos theory - contents are beginning to look good) I am mainly trying to bring on my personal web the entry differential cohomology in an $(\infty,1)$-topos, and the entries relating to it, into shape.

Progress is unfortunately much slower than I would hope. But some things improve, slowly but surely. For instance I am glad that I finally cleaned up and started to expand Lie infinity-algebroid with some serious indications of and examples for the incarnation as infinitesimal $\infty$-Lie groupoids. In that context I also started telling more details of the story at Lie infinity-groupoid, but that is still pretty stubby.

In the course of this I keep adding bits and pieces here and there. For instance this morning I posted some proofs at model structure on simplicial presheaves on the Localization of $(\infty,1)$-presheaves at a coverage (instead of at a full Grothendieck topology). This I am using in the discussion at locally n-connected (n,1)-topos as a tool for constructing examples of those. The central motivating example is the category of $(\infty,1)$-sheaves on the site CartSp equipped with its good open cover coverage – and finally I had recently found the time to leave some comments at CartSp as to the relevance of this innocent-looking little category.

All this, of course, is still an outgrowth of the development we had a while ago at homotopy groups in an $\infty$-topos. After some back and forth it looks like the following generalization of the picture there is beginning to stabilize finally: $\infty$-Lie theory is about geometric homotopy theory in a locally $\infty$-connected $\infty$-topos, but not with respect to the global geometric morphisms $\mathbf{H} \stackrel{\overset{\Pi}{\to}}{\stackrel{\leftarrow}{\to}} \infty Grd$, but with respect to a relative such morphisms $\mathbf{H} \stackrel{\overset{\Pi_{inf}}{\to}}{\stackrel{\leftarrow}{\to}} \mathbf{H}_{red} \stackrel{\overset{\Pi}{\to}}{\stackrel{\leftarrow}{\to}} \infty Grd$, where $\mathbf{H}$ is an infinitesimal thickening of $\mathbf{H}_{red}$, which in turn is a finite thickening of the point.

The picture emerging here is still being drawn at structures in an $\infty$-topos.

I just wish the day had more hours. It’s shame how long I have been elaborating on all this already, and still counting.

Andrew Stacey: My recent n-Activity can be roughly split into three areas. The first of these is differential topology of mapping spaces, the second is related to topological vector spaces, whilst the third is not so much nLab as nForum.

Sparked by a project with David Roberts to mollify his construction of the topological bigroupoid, I started transferring my old notes on the differential topology of loop spaces to the nLab. This is something I’ve been threatening to do for a while, and finally the overwhelming urge of sheer laziness prompted me to do something about it. (To explain, I found that I was writing out essentially the same argument for the bigroupoid as I had for the loop space; as any good mathematician knows, there are only three numbers: 0, 1, and ∞. Having started writing the details twice, I was worried that I would end up writing them out ∞ number of times. So writing them into the nLab means that I can accomplish that ∞ number of times in one go.) In order to make it look something like active research, I’ve extended it from loop spaces to mapping spaces. The case where the mapping space is a compact (smooth) manifold is known, so I’ve had to go to sequentially compact Froelicher spaces to find something new to say.

(NB For those unfamiliar with terms from functional analysis, a mollifier is a function that takes non-smooth maps and makes them smooth.)

This led me into the demesne of topological vector spaces, as the linear models of mapping spaces can be fairly arbitrary locally convex topological vector spaces. Again, there’s a fair amount of material that I’ve been meaning to import to the nLab but not gotten round to. I thought I’d start with Greg Kuperberg’s amazing graph. With Tim van Beek, I’ve been having fun reminding myself of the subtleties and differences between barreled spaces, bornological spaces and lots of other variants.

Finally, I’ve been working on improving the nForum. There, we now have a browser-savvy mathematics capability that will serve you your mathematics to your taste: the chef’s special is, of course, MathML but if you prefer it raw we can do Math Tartar (aka, the raw itex source), or if you prefer it flame-grilled to avoid any risk of Mad-Mathematician Disease, we can also do PNG. For those with exotic tastes, we can also offer it as SVG. We also have a host of new features: tags, a separate area for discussions that aren’t particularly nLab focussed, a notice board, and one day soon we may even get our own favicon.

Tim van Beek: Urs has created a template page.

There was some activity concerning general relativity and Lorentzian geometry, starting with smooth Lorentzian space.

A remark of Zoran Skoda let me to study hyperfunctions a little bit, which is about a very neat alternative approach to generalized functions, more general than Schwartz distributions, and at least in one dimension simpler, too. The link to D-modules is still missing, however.

My quest to get some classical AQFT content to the nLab has led to Haag-Kastler axioms, Haag-Kastler vacuum representation, Wightman axioms, the Osterwalder-Schrader theorem and a definition in terms of these of conformal field theories. This definition includes operator product expansions, which provide the link to vertex operator algebras. As a side effect some operator theory got added, too, for example on the pages von Neumann algebra, operator algebra and unbounded operator. The ultimate goal is of course to join the dots with category theory, for example it would be nice to have the proof of Müger of the Doplicher-Roberts reconstruction theorem on the nLab.

Todd Trimble: Most of my recent efforts have been small scale, but two of the larger ones were AT category and composition algebra. I’ve also been adding at times to the document on Surface Diagrams on my personal web.

Mike Shulman: Recently I’ve been working mostly on two pages. One is codiscrete cofibration, which is about a way to canonically recover a proarrow equipment from a 2-category with sufficient structure. Essentially, one can define in 2-categorical terms which cospans ought to be the cographs of profunctors, and then construct a proarrow equipment out of those. The literature doesn’t seem to contain a version of this which works for arbitrary enriched categories (only for Cauchy-complete ones), but I think that by a judicious choice of a factorization system (now described at codiscrete cofibration it can be made to work. A neat side benefit of this is that there is actually an intrinsic characterization of the $V$-fully-faithful functors in 2-categorical terms in $V Cat$ (they are not just the representably fully faithful ones).

I have also been thinking about how functorial this construction is, which also doesn’t seem to be in the literature. It seems right now to be very functorial; in fact it may even be a full embedding of some 3-category of 2-categories into some 3-category of proarrow equipments. This is particularly interesting because a number of monads on equipments relative to which one might want to define generalized multicategories can be obtained in this way from 2-monads on 2-categories, which require less work to construct.

The other circle of pages I’ve been working on is derivators, along with pointed derivators and stable derivators. I haven’t been able to find a really satisfying introductory treatment of derivators anywhere, so I’m gradually trying to create one at the nLab as I puzzle things out myself. Working with derivators is an interesting experience; it feels kind of like being a new category theory student again and having to learn a whole new set of tools. Statements that are “obvious” for 1-categories require entirely new proofs in the world of derivators, and I’m only gradually getting comfortable with the tools used in those proofs. Hopefully writing slow and detailed proofs on the nLab pages will help me and others learn to manipulate those tools as easily as we do the more familiar toolbox of 1-categories.

David Roberts: I’ve been thinking a little bit about partitions of unity. Covers with subordinate partitions of unity (numerable covers) are particularly nice, but are only guaranteed to exist for paracompact spaces, or normal spaces if you are handed a locally finite cover. In fact numerable covers form a Grothendieck pretopology, and it is using this site that all the classical theorems for algebraic topology should be stated, instead of just for paracompact spaces with the usual open cover pretoplogy.

Actually it is enough, instead of having a partition of unity, to have an arbitrary collection of functions $u_\alpha:X \to \mathbb{R}_{\geq 0}$ such that $\sum_\alpha u_\alpha(x)$ is finite for all $x$, because this can be further refined to end up with a subordinate partition of unity. This is essentially an old result from a PhD thesis from the 60s. This can be interpreted as requiring $X$ having enough coprobes with values in the rig $\mathbb{R}_{\geq 0}$

Zoran Skoda has also been busy.

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