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Vanilla 1.1.10 is a product of Lussumo. More Information: Documentation, Community Support.

• created a “category: reference”-page The Stacks Project

I have only now had a closer look at this and am impressed by the scope this has. Currently a total of 2288 pages. It starts with all the basics, category theory, commutative algebra and works its way through all the details to arrive at algebraic stacks.

So besides my usual complaint (Why behave as if there are not sites besides the usual suspects on $CRing^{op}$ and either give a general account or call this The Algebraic Stacks Project ? ) I am enjoying seeing this. We should have lots of occasion to link to this. Too bad that this did not start out as a wiki.

• The page collects the various networks and communities of category theorists around the world. As far as I have seen, such page was missing from the nLab!

Feel free to continue the list

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• I have been further working on the entry higher category theory and physics. There is still a huge gap between the current state of the entry and the situation that I am hoping to eventually reach, but at least now I have a version that I no longer feel ashamed of.

Here is what i did:

• Partitioned the entry in two pieces: 1. “Survey”, and 2. “More details”.

• The survey bit is supposed to give a quick idea of what the set of the scene of fundamental physics is. It starts with a kind of creation story of physics from $\infty$-topos theory, which – I think – serves to provide a solid route from just the general abstract concept of space and process to the existence and nature of all $\sigma$-model quantum field theories of “$\infty$-Chern-Simons theory”-type (which includes quite a few) and moreover – by invoking the “holographic principle of higher category theory” – all their boundary theories, which includes all classical phase space physics.

The Survey-bit continues with indicating the formalization of the result of quantizing all these to full extended quantum field theories. It ends with a section meant to indicate what is and what is not yet known about the quantization step itself. This is currently the largest gap in the mathematical (and necessarily higher categorical) formalization of physics: we have a fairly good idea of the mathematics that describes geometric background structure for physics and a fairly good idea of the axioms satisfied by the quantum theories obtained from these, but the step which takes the former to the latter is not yet well understood.

• The “More details”-bit is stubby. I mainly added one fairly long subsection on the topic of “Gauge theory”, where I roughly follow the historical route that eventually led to the understanding that gauge fields are modeled by cocycles in higher (nonabelian) differential cohomology.

I know that the entry is still very imperfect. If you feel like pointing out all the stuff that is still missing, consider adding at least some keywords directly into the entry.

• At the old entry cohomotopy used to be a section on how it may be thought of as a special case of non-abelian cohomology. While I (still) think this is an excellent point to highlight, re-reading this old paragraph now made me feel that it was rather clumsily expressed. Therefore I have rewritten (and shortened) it, now the third paragraph of the Idea-section.

(We had had long discussion about this entry back in the days, but it must have been before we switched to nForum discussion, because on the nForum there seems to be no trace of it.)

• brief category: people-entry for hyperlinking references at twistor fibration

• brief category:people-entry for hyperlinking references at twistor fibration

• I had given it an $n$Lab page already a while back, so that I could stably link to it without it being already there:

Now it’s even “there” in the sense of being incarnated as a pdf.

• at foundation of mathematics I have tried to start an Idea-section.

Also, I am hereby moving a bunch of old discussion boxes from there to here:

[ begin forwarded discussion ]

+– {: .query} Urs asks: Concerning the last parenthetical remark: I suppose in this manner one could imagine $(n+1)$-categories as a foundation for $n$-categories? What happens when we let $n \to \infty$?

Toby answers: That goes in the last, as yet unwritten, section. =–

+– {: .query} Urs asks: Can you say what the problem is?

Toby answers: I'd say that it proved to be overkill; ETCS is simpler and no less conceptual. In ETCC (or whatever you call it), you can neatly define a group (for example) as a category with certain properties rather than as a set with certain structure. But then you still have to define a topological space (for example) as a set with certain structure (where a set is defined to be a discrete category, of course). I think that Lawvere himself still wants an ETCC, but everybody else seems to have decided to stick with ETCS.

Roger Witte asks: Surely in ETCC, you define complete Heyting algebras as particular kinds of category and then work with Frames and Locales (ie follow Paul Taylor’s leaf and apply Stone Duality). You should be able to get to Top by examining relationships between Loc and Set. I thought Top might be the the comma category of forgetful functor from loc to set op and the contravariant powerset functor. Thus a Topological space would consist of a triple S, L, f where S is a set, L is a locale and f is a function from the objects of the locale to the powerset of S. A continuous function from S, L, f to S’, L’, f’ is a pair g, h where g is a function from the powerset of S’ to the powerset of S and g is a frame homomorphism from L’ to L and (I don’t know how to draw the commutation square). However I think this has too many spaces since lattice structures other than the inclusion lattice can be used to define open sets.

Toby: It's straightforward to define a topological space as a set equipped with a subframe of its power set. So you can define it as a set $S$, a frame $F$, and a frame monomorphism $f\colon F \to P(S)$, or equivalently as a set $S$, a locale $L$, and an epimorphism $f\colon L \to Disc(S)$ of locales, where $Disc(S)$ is the discrete space on $S$ as a locale. (Your ’However, […]’ sentence is because you didn't specify epimorphism/monomorphism.) This is a good perspective, but I don't think that it's any cleaner in ETCC than in ETCS.

Roger Witte says Thanks, Toby. I agree with your last sentence but my point is that this approach is equally clean and easy in both systems. The clean thing about ETCC is the uniformity of meta theory and model theory as category theory. The clean thing about ETCS is that we have just been studying sets for 150 years, so we have a good intuition for them.

I was responding to your point ’ETCC is less clean because you have to define some things (eg topological spaces) as sets with a structure’. But you can define and study the structure without referring to the sets and then ’bolt on’ the sets (almost like an afterthought).

Mike Shulman: In particular cases, yes. I thought the point Toby was trying to make is that only some kinds of structure lend themselves to this naturally. Groups obviously do. Perhaps topological spaces were a poorly chosen example of something that doesn’t, since as you point out they can naturally be defined via frames. But consider, for instance, a metric space. Or a graph. Or a uniform space. Or a semigroup. All of these structures can be easily defined in terms of sets, but I don’t see a natural way to define them in terms of categories without going through discrete categories = sets.

Toby: Roger, I don't understand how you intend to bolt on sets at the end. If I define a topological space as a set $S$, a frame $F$, and a frame monomorphism from $F$ to the power frame of $S$, how do I remove the set from this to get something that I can bolt the set onto afterwards? With semigroups, I can see how, from a certain perspective, it's just as well to study the Lawvere theory of semigroups as a cartesian category, but I don't see what to do with topological spaces.

Roger Witte says If we want to found mathematics in ETCC we want to work on nice categories rather than nice objects. Nice objects in not nice categories are hard work (and probably ’evil’ to somke extent). Thus the answer to Toby is that to do topology in ETCC you do as much as possible in Locale theory (ie pointless topology) and then when you finally need to do stuff with points, you create Top as a comma like construction (ie you never take away the points but you avoid introducing them as long as possible). Is it not true that the only reason you want to introduce points is so that you can test them for equality/inequality (as opposed to topological separation)?

Mike, I spent about two weeks trying to figure out how to get around Toby’s objection ’topology’ and now you chuck four more examples at me. My gut feeling is that the category of directed graphs is found by taking the skeleton of CAT, that metric locales are regular locales with some extra condition to ensure a finite basis, that Toby can mak

[ to be continued in next comment ]

• Updating reference to cubical type theory. This page need more work.

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Egbert Rijke

• For completeness, so that we now have this list:

• this used to be inside 4-sphere. Am giving it its stand-alone entry for ease of listing references

• added to principal 2-bundle in a new Properties-section the classification results by Baez-Stevenson, Stevenson-Roberts (for the topological case) and Nikolaus-Waldorf (for the smooth case).

• added references by Pronk-Scull and by Schwede, and wrote an Idea-section that tries to highlight the expected relation to global equivariant homotopy theory. Right now it reads like so:

On general grounds, since orbifolds $\mathcal{G}$ are special cases of stacks, there is an evident definition of cohomology of orbifolds, given by forming (stable) homotopy groups of derived hom-spaces

$H^\bullet(\mathcal{G}, E) \;\coloneqq\; \pi_\bullet \mathbf{H}( \mathcal{G}, E )$

into any desired coefficient ∞-stack (or sheaf of spectra) $E$.

More specifically, often one is interested in viewing orbifold cohomology as a variant of Bredon equivariant cohomology, based on the idea that the cohomology of a global homotopy quotient orbifold

$\mathcal{G} \;\simeq\; X \sslash G \phantom{AAAA} (1)$

for a given $G$-action on some manifold $X$, should coincide with the $G$-equivariant cohomology of $X$. However, such an identification (1) is not unique: For $G \subset K$ any closed subgroup, we have

$X \sslash G \;\simeq\; \big( X \times_G K\big) \sslash K \,.$

This means that if one is to regard orbifold cohomology as a variant of equivariant cohomology, then one needs to work “globally” in terms of global equivariant homotopy theory, where one considers equivariance with respect to “all compact Lie groups at once”, in a suitable sense.

Concretely, in global equivariant homotopy theory the plain orbit category $Orb_G$ of $G$-equivariant Bredon cohomology is replaced by the global orbit category $Orb_{glb}$ whose objects are the delooping stacks $\mathbf{B}G \coloneqq \ast\sslash G$, and then any orbifold $\mathcal{G}$ becomes an (∞,1)-presheaf $y \mathcal{G}$ over $Orb_{glb}$ by the evident “external Yoneda embedding

$y \mathcal{G} \;\coloneqq\; \mathbf{H}( \mathbf{B}G, \mathcal{G} ) \,.$

More generally, this makes sense for $\mathcal{G}$ any orbispace. In fact, as a construction of an (∞,1)-presheaf on $Orb_{glb}$ it makes sense for $\mathcal{G}$ any ∞-stack, but supposedly precisely if $\mathcal{G}$ is an orbispace among all ∞-stacks does the cohomology of $y \mathcal{G}$ in the sense of global equivariant homotopy theory coincide the cohomology of $\mathcal{G}$ in the intended sense of ∞-stacks, in particular reproducing the intended sense of orbifold cohomology.

At least for topological orbifolds this is indicated in (Schwede 17, Introduction, Schwede 18, p. ix-x, see also Pronk-Scull 07)

• This is a bare list of references, to be !include-ed into relevant entries (such as string phenomenology, heterotic string and GUT), for ease of keeping these entry’s bibliographies in sync

• I used to point to Theorem 5.1.3.6 in http://www.math.harvard.edu/~lurie/papers/higheralgebra.pdf for the May recognition theorem. Now that file is gone, superceded by http://www.math.harvard.edu/~lurie/papers/HA.pdf and the numbering changed. Where in the new file is the May recognition theorem? (It’s not referred to under this name, unfortunately.)

• There is some bug with the display of this page. Some maths doesn’t get rendered and theorems appear in the toc, as if sections. Probably some closing dollar sign is missing somewhere, but I haven’t found it.

• Jim Gates, Yangrui Hu, S.-N. Hazel Mak, Adinkra Foundation of Component Decomposition and the Scan for Superconformal Multiplets in 11D, $\mathcal{N} = 1$ Superspace (arXiv:2002.08502)