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Vanilla 1.1.10 is a product of Lussumo. More Information: Documentation, Community Support.
added some comments on history to neutrino.
I have split off complex projective space from projective space and added some basic facts about its cohomology.
added some pointers:
Yves Félix, Stephen Halperin, Jean-Claude Thomas, p. 142 in: Rational Homotopy Theory, Graduate Texts in Mathematics, 205, Springer-Verlag, 2000 (doi:10.1007/978-1-4613-0105-9)
Luc Menichi, Section 1.2 in: Rational homotopy – Sullivan models, IRMA Lect. Math. Theor. Phys., EMS (arXiv:1308.6685)
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 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.
brief category: people-entry for hyperlinking references at higher category theory and physics
brief category: people-entry for hyperlinking references at Calabi-Penrose fibration
stub for Hilbert’s sixth problem
brief category: people-entry for hyperlinking references at higher category theory and physics
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 -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 -model quantum field theories of “-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.
Apart from this I added more references and some cross-links.
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.
for completeness and to satisfy links from Calabi-Penrose fibration
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 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 -categories as a foundation for -categories? What happens when we let ?
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 , a frame , and a frame monomorphism , or equivalently as a set , a locale , and an epimorphism of locales, where is the discrete space on 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 , a frame , and a frame monomorphism from to the power frame of , 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 ]
For completeness, so that we now have this list: