Want to take part in these discussions? Sign in if you have an account, or apply for one below
Vanilla 1.1.10 is a product of Lussumo. More Information: Documentation, Community Support.
created directed homotopy type theory
To support mentioning weak wreath product in a parallel discussion with Urs, I created a stub for weak bialgebra with redirect weak Hopf algebra.
in analogy to what I just did at classical mechanics, I have now added some basic but central content to quantum mechanics:
Quantum mechanical systems
States and observables
Spaces of states
Flows and time evolution
Still incomplete and rough. But I have to quit now.
a stub, for the moment just to make some cross-links work (such as with topological order and topological entanglement entropy)
added pointer to:
created worldline formalism to go with this Physics.SE answer
I expanded perfect infinity-stack a little. But this deserves further polishing and expanding.
I noticed that the entry classifying space is in bad shape. I have added a table of contents and tried to structure it slightly, but much more needs to be done here.
I have added a paragraph on standard classifying spaces for topological principal bundles via the geometric realization of the simplicial space associated to the given topological group.
In the section “For crossed complexes” there is material that had been provided by Ronnie Brown which needs to be harmonized with the existing Idea-section. It proposes something like a general axiomatics on the notion of “classifying space” more than giving details on the geometric realization of crossed complexes
The Idea-section at quasi-Hopf algebra had been confused and wrong. I have removed it and written a new one.
one more from Meissner & Nicolai, last week:
I moved the definition of promonoidal categories from Day convolution to promonoidal category, and expanded on it a bit.
New entry representable morphism, in the sense of Grothendieck school. The notion is used at closed immersion of schemes where I just made some changes.
stub for jet bundle
An edit to normed ring is being discussed in another thread, here.
added pointer to:
added to Cartan connection
definition
a standard reference
a standard example
I see super-Cartan geometry is taking shape. Will Clifford algebras make an appearance in the The super-Klein geometry: super-Minkowski spacetime section?
Is there a higher super-Cartan way of thinking about what is at 3-category of fermionic conformal nets, about the String 2-group and superstrings, as here about the spin group and fermions.
added ISBN:9783540125464
I am changing the page title – this used to be “A first idea of quantum field theory”, which of course still redirects. The “A first idea…” seemed a good title for when this was an ongoing lecture that was being posted to PhysicsForums. I enjoyed the double meaning one could read into it, but it’s a bad idea to carve such jokes into stone. And now that the material takes its place among the other chapters of geometry of physics, with the web of cross-links becoming thicker, the canonical page name clearly is “perturbative quantum field theory”.
added pointer to these two recent references, identifying further -algebra structure in Feynman amplitudes/S-matrices of perturbative quantum field theory:
Markus B. Fröb, Anomalies in time-ordered products and applications to the BV-BRST formulation of quantum gauge theories (arXiv:1803.10235)
Alex Arvanitakis, The -algebra of the S-matrix (arXiv:1903.05643)
finally added the actual definition, !include-ed from Knizhnik-Zamolodchikov-Kontsevich construction – definition (as per the discussion here)
renaming page to the one used in the Stacks reference, parallel with Artinian ring.
Anonymous
Created W-type.
Adding reference
Anonymouse
this is a bare list of references which used to be (and still is) at entanglement entropy. But since the same references are now also needed at long-range entanglement, I am putting them in a separate page here, to be !include-ed into both these entries
I toiuched the formatting and the hyperlinking of the paragraphs on compatibility of limits with other universal constructions.
Merged the previous tiny subsections on this to a single one, now Compatibility with universal constructions.
added the hyperlink to the stand-alone entry adjoints preserve (co-)limits.
Will create an analogous stand-alone entry for limits commute with limits.
Balancing doesn’t mention duals anywhere, and makes sense even without duals. I removed an incorrect statement and replaced it with the correct one. Not sure if it needs a reference, but the correct result appears as Lemma 4.20 in https://arxiv.org/pdf/0908.3347.pdf (where it’s attributed to Deligne, but the citation is to Yetter).
I have touched the formatting at pre-abelian category.
I am hereby moving an old query-box discussion from abelian category to here. I suggest that to the extent this reached a conclusion, that conclusion should be moved to the Properties-section of the entry
[begin forwarded discussion]
The following discussion is about whether a pre-abelian category in which (epi,mono) is a factorization system is necessarily abelian.
+–{: .query} Mike: In Categories Work, and on Wikipedia, an abelian category is defined to be (in the terms above) a pre-abelian category such that every monic is a kernel and every epi is a cokernel. This implies that (epi, mono) is an orthogonal factorization system, but I don’t see why the converse should hold, as this seems to assert.
Zoran Skoda It is very late night here in Bonn, so check on my reasoning, but I think that the answer is simple. Let . The canonical map exists as long as we have additive category admitting kernels and cokernels. The arrow from A to coker (ker f) is epi as every cokernel arrow, and the arrow of is mono. Now canonical arrow in between the two is automatically both mono and epi. For all that reasoning I did not yet assume the axiom on uniquely unique factorization. Now assume it and you get that the canonical map must be isomorphism because it is the unique iso between the two decompositions of : one in which you take epi followed by (the composition of) two monics and another in which you have (the composition of) two epis followed by one monic. Right ?
Now do this for a monic and you get a decomposition into iso iso kernel and for an epi and you get the cokernel iso iso as required.
Mike: Why is the canonical comparison map mono and epi? It’s late for me too right now, but I think that maybe a counterexample is the “multiplication by 2” map in the category of torsion-free abelian groups.
However, if you assume explicitly that that comparison map is always an isomorphism, then I believe it for the reasons that you gave.
Zoran Skoda I do not see this as a counterexample, as this is not a pre-abelian category, you do not have cokernels in this category ? In a pre-abelian category always the canonical map from coker ker to ker coker has its own kernel 0 and cokernel 0.
Mike: Torsion-free abelian groups are reflective in abelian groups, and therefore cocomplete. In particular, they have cokernels, although those cokernels are not computed as in Ab. In particular, the cokernel of is 0.
Zoran Skoda Yes, I was thinking of this reflection argument (equivalence of torsion and localization argument), that is why I put question mark above. Now I tried to prove the assertion that in preabelian cat the canonical map has kernel 0 and cokernel 0 and I can’t for more than an hour. But that would mean that for example Gelfand-Manin book is wrong – it has the discussion on A4 axiom and it says exactly this. Popescu makes an example of preabelian category where canonical map is not iso, but emphasises in his example that it is bimorphism. On the other hand, later, he says that preabelian category is abelian iff it is balanced and the canonical map is bimorphism, hence he requires it explicitly. Let me think more…
Zoran Skoda I have rewritten in minimalistic way, leaving just what I can prove, and assuming that you are right and Gelfand-Manin book has one wrong statement (that the canonical map in preabelian category is mono and epi). But let us leave the discussion here for some time, maybe we can improve the question of the difference between preabelian with factorization and abelian.
Mike: I refactored the page to make clear what we know and what we don’t, and include some examples. Maybe someone will come along and give us a counterexample or a proof. I wonder what the epimorphisms are in the category of torsion-free abelian groups, and in particular whether it is balanced (since if so, it would be a counterexample).
Mike: Okay, it’s obvious: the epimorphisms in are the maps whose cokernel (in ) is torsion. Thus is monic and epic, so is not balanced. And since is its own canonical map, that canonical map is monic and epic in , so this isn’t a counterexample.
Zoran: http://www.uni-trier.de/fileadmin/fb4/INF/TechReports/semi-abelian_categories.pdf says at one place that Palamodov’s version of semi-abelian category is preabelian + canonical morphism is epi and mono. =–
[end forwarded discussion]
stub for dark matter
I have added to M5-brane a fairly detailed discussion of the issue with the fractional quadratic form on differential cohomology for the dual 7d-Chern-Simons theory action (from Witten (1996) with help of Hopkins-Singer (2005)).
In the new section Conformal blocks and 7d Chern-Simons dual.
added to references at Higgs bundle.
Created an entry on Sylvain Douteau, who works on stratified spaces and their homotopy theory.
I have created stratified space in order to collect some references
New page displayed category.
Mike Stay kindly added the standard QM story to path integral.
I changed the section titles a bit and added the reference to the Baer-Pfaeffle article on the QM path integral. Probably the best reference there is on this matter.
I split the material so far at "Bousfield localization" into
and made Bousfield localization itself a dismbiguation page.
Then I edited Bousfield localization of triangulated categories a bit and added some references.
I expect it should be true that the (left) Bousfield localization of a model category whose homotopy category is a triangulated category corresponds to the Bousfield localization of that triangulated homotopy category, but I don't discuss a statement to that extent yet.
I'd be grateful for explicit pointers to the literature on this. I haven't had time yet to look much even at the literature that I do list as BLoc for triang cats. Am too busy at the moment with BLoc for mod cats.
somebody created Freyd category
Somebody named Adam left a comment box a while ago at premonoidal category saying that naturality of the associator requires three naturality squares. I believe that this is true when phrased explicitly in terms of one-variable functors, but the slick approach using the “funny tensor product” allows us to rephrase it as a single natural transformation between functors . I’ve edited the page accordingly. I also added the motivating example (the Kleisli category of a strong monad) and a link to sesquicategory.
There is a comment on the page that “It may be possible to weaken the above make a symmetric monoidal 2-category, in which a monoid object is precisely a premonoidal category”. However, the Power-Robinson paper says that “We remark that is not a 2-functor,” which seems to throw some cold water on the obvious approach to that idea. Was the thought to define a different 2-categorical structure on than the usual one, e.g. using unnatural transformations? It seems that at least one would still have to explicitly require centrality of the coherence isomorphisms.
Just enough to have the definition and main references, for now. Will create cartesian restriction category soon.
Linked to from partial function.
copied article from the michaelshulman wiki at functor comprehension principle (michaelshulman) to the nlab wiki.
C. Silva
mentioned two basic properties at Hodge star operator (namely those needed at holographic principle ;-)