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This entry defines Grothendieck topologies using sieves.
However, in the original definition (Michael Artin’s seminar notes “Grothendieck topologies”), a Grothendieck topology on a category is defined as a set of coverings.
More precisely (to cite from Artin’s notes), a Grothendieck topology is defined as families of maps such that
for any isomorphism we have ;
if and for each , then ;
if and is a morphism, then exist and .
This is almost identical to the current definition of Grothendieck pretopology, except that in Artin’s definition only the relevant pullbacks are required to exist.
It seems to me that the original definition by Artin is the one used most often in algebraic geometry.
added to simplicial ring the definition of the model catgeory structure on modules over simplicial rings.
I suppose stronger statements are possible, but this I see stated in Toen's lectures and thus briefly noted it down
Beginning a page for algebraic patterns, as they are becoming relatively prominent within the work of Barkan-Chu-Haugseng-Steinebrunner, and many equivariant homotopy theorists are beginning to recognize them as a suitable foundation for burgeoning work concerning equivariant operads.
Natalie Stewart
finally a stub for Segal condition. Just for completeness (and to have a sensible place to put the references about Segal conditions in terms of sheaf conditions).
Added Clark’s comment that many of their bases are EI (∞,1)-categories. I guess many of these are in addition inverse EI (∞,1)-categories.
Is a stratification always well-founded?
I am slowly creating a bunch of entries on basic concepts of equivariant stable homotopy theory, such as
At the moment I am mostly just indexing Stefan Schwede’s
added to multivector field further references on how the divergenc/BV operator is the dual of the de Rham differential.
Domenico, could you tell me if you think that the following statement is correct?
in full abstractness, the content of Lagrangian BV is this:
we
start with a configuration space of sorts
and assume we have a fixed isomorphism between its Hochschild cohomology and Hochshcild homology, which we think of as an iso between its differential forms and its multivector fields induced by a volume form (which it is for finite dimensional spaces);
then we think of an action functional times a volume form on our configuration space as a closed differential form , hence as an element in the Hochschild homology that is also in the cyclic homology
and then use the above isomorphism to think of this equivalently an element in Hochschild cohomology, being a cocycle in cyclic cohomology.
the cyclic differential is the BV-operator and the closure condition is the “master equation” ;
the fact that Lagrangian BV is controled by BV-algebra and hence, by Getzler’s theorem, by algebra over the homology of the little framed disk operad now follows from the fact that Hochschild homology of our space is given by the derived loop space.
Is that right? Is that the NiceStoryAboutLagrangianBV™? If so, is this written out in this fashion explicitly somewhere?
An old query at twisting cochain which was reaction to somebody putting that the motivation is the homological perturbation lemma:
It is equally true that it is related to 20 more areas like that one (which is not the central). Brown’s paper on twisting cochains, is much earlier than homological perturbation theory. basic idea was to give algebraic models for fibrations. Nowedays you have these things in deformation theory, A-infty, gluing of complexes on varieties, Grothendieck duality on complex manifolds (Toledo-Tong), rational homotopy theory etc. One should either give a fairly balanced view to all applications or not list anything, otherwise it is not fair. This should be done together with massive expansion of Maurer-Cartan equation what is almost the same topic. The same with literature: Smirnov’s book on simplicial and operadic methods in algebraic topology is the most wide reference for twisting cochains and related issues in algebraic topology setup; Keller wrote much and well about this and Lefèvre-Hasegawa thesis (pdf) is very good, and the first reference is E. Brown’s paper from 1959. For applications in deformation theory there are many references, pretty good one from dg point of view and using 2-categorical picture of def functors is a trilogy of Efimov, Lunts and orlov on the arXiv. Few days ago Sharygin wrote a long article on twisting cochains on the arXiv, with more specific purposes in index theory. Interesting is the application of Baranovsky on constructing universal enveloping of L infty algebra. – Zoran
Urs: concerning the “either give a fairly balanced view to all applications or not list anything”, I can see where you are coming from, Zoran, but I would still prefer here to have a little bit of material than to have none. The Lb is imperfect almost everywhere, we’ll have to improve it incrementally as we find time, leisure and energy. But it’s good that you point out further aspects in a query box, so that we remember to fill them in later.
Zoran Skoda My experience is that correcting a rambling and unbalanced entries takes more time than writing a new one at a stage when you really work on it. Plus all the communication explaining to others who made original entry which is hastily written. When it becomes very random and biased I stopped enjoying it at all to work on it.
Ronnie Brown It may not possible for one person to give a “balanced entry” and is certainly not possible for me in this area. On the other hand, this may be endemic to the description of an area of maths for students and research workers.
An advantage of the Homological Perturbation Lemma (HPL) is that it is an explicit formula, and this has been exploited by various writers, especially Gugenheim, Larry Lambe and collaborators, Huebschmann, and others, for symbolic computations in homological algebra. It is good of course to have the wide breadth of applications of twisting cochains explained.
For me, an insight of the HPL was the explicit use of the homotopies in a deformation retract situation to lead to new results. This has been developed to calculate resolutions of groups, where one is constructing inductively a universal cover of a with its contracting homotopy.
So let us continue to have various individually “unbalanced” points of view explained in this wiki, to let the readers be informed, and decide.
Toby: Knowing basically nothing about this, I prefer to see various people explain their own perspectives. Even if they don't try to take the work to integrate them.
I added more to idempotent monad, in particular fixing a mistake that had been on there a long time (on the associated idempotent monad). I had wanted to give an example that addresses Mike’s query box at the bottom, but before going further, I wanted to track down the reference of Joyal-Tierney, or perhaps have someone like Zoran fill in some material on classical descent theory for commutative algebras (he wrote an MO answer about this once) to illustrate the associated idempotent monad.
Some of this (condition 2 in the proposition in the section on algebras) was written as a preparatory step for a to-be-written nLab article on Day’s reflection theorem for symmetric monoidal closed categories, which came up in email with Harry and Ross Street.
Created Gamma-space.
created dg-nerve
category: people page for one of the authors of this preprint
Anonymouse
Somebody sent me an email with the following comment on the entry countable set. I am not in position to react to this, maybe some expert here could reply. The sentence being quoted originates from revision 1 of the entry.
Forwarded message:
“We do have, however, that a countable set is either empty or inhabited, which is classically trivial but need not hold constructively for every set.”
(https://ncatlab.org/nlab/show/countable+set)
The set D={n\in N | 2n+6 is not the sum of two odd primes} is decidable, hence countable. However we cannot decide whether it is empty or inhabited. (We could decide it if we assumed LPO, for instance.)
Do you agree?
I added a couple of comments about topos models to principle of omniscience.
added reference to dendroidal version of Dold-Kan correspondence
added to homotopy groups of spheres the table
| 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | ||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
have created a stub for supersymmetric quantum mechanics
Zoran, I see that you once dropped a big query box at quantum mechanics with a complaint. I disagree with the point you make there: we have fundamental definitions of quantum field theory and restricting them to 1 dimension gives quantum mechanics. If you want to turn this around and understand all QFTs as infinite-dimensional quantum mechanics (which, yes, one can do) you are discarding the nice conceptual models and kill the concept of extended QFT.
In any case, I think remarks like this (in the style of “we can also regard this the other way round like this”) are better added into an entry as what they are – remarks – than as query boxes that give the impression that there is something fishy about the rest of the entry.
for completeness, to go with U(ℋ), for the moment mainly in order to record references, such as:
Here and in some more references it is required that the result of hyperoperation is nonempty, that is the values are in the power set without the emptyset . I do not know what other contributors to the page think about it.
It does not allow me to leave !include algebra - contents so I temporarily erased it. It says that it does not exist.
wrote an entry Deligne’s theorem on tensor categories on the statement that every regular tensor category is equivalent to representations of a supergroup. Added brief paragraphs pointing to this to superalgebra and supersymmetry, added cross-links to Tannaka duality, Doplicher-Roberts reconstruction etc. Also created a disambiguation page Deligne’s theorem
I did not change anything, I would not like to do it without Urs’s consent and some opinion. The entry AQFT equates algebraic QFT and axiomatic QFT. In the traditional circle, algebraic quantum field theory meant being based on local nets – local approach of Haag and Araki. This is what the entry now describes. The Weightman axioms are somewhat different, they are based on fields belonging some spaces of distributions, and 30 years ago it was called field axiomatics, unlike the algebraic axiomatics. But these differences are not that important for the main entry on AQFT. What is a bigger drawback is that the third approach to axiomatic QFT if very different and was very strong few decades ago and still has some followers. That is the S-matrix axiomatics which does not believe in physical existence of observables at finite distance, but only in the asymptotic values given by the S-matrix. The first such axiomatics was due Bogoliubov, I think. (Of course he later worked on other approaches, especially on Wightman’s. Both the Wightman’s and Bogoliubov’s formalisms are earlier than the algebraic QFT.)
I would like to say that axiomatic QFT has 3 groups of approaches, and especially to distinguish S-matrix axiomatics from the “algebraic QFT”. Is this disputable ?
added pointer to:
added pointer to:
(I was looking for references which one might point to concerning the term “quantum/classical divide”. For instance, who actually introduced that specific term? People use it to refer to Bohr’s writings, but Bohr never wrote down the specific words “quantum/classical divide”, did he?)
started an entry infinity-action
I have finally split off dependent sum from dependent product. And added a few more paragraphs.
I discovered that there was an ancient stub entry canonical commutation relation. Have given it a bit more content now.
briefly added something to fusion category. See also this blog comment.
added a brief historical comment to Higgs field and added the historical references
fixed link for
added to KK-theory brief remark and reference to relation to stable -categories / triangulated categories