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the standard bar complex of a bimodule in homological algebra is a special case of the bar construction of an algebra over a monad. I have added that as an example to bar construction.
I also added the crucial remark (taken from Ginzburg’s lecture notes) that this is where the term “bar” originates from in the first place: the original authors used to write the elements in the bar complex using a notaiton with lots of vertical bars (!).
(That’s a bad undescriptive choice of terminoiogy. But still not as bad as calling something a “triple”. So we have no reason to complain. ;-)
starting some minimum, cross-linking with quaternion-Kähler manifold and Sp(n).Sp(1)
Used unicode subscripts for indices of exceptional Lie groups including title and links. When not linked, usual formulas are used. See discussion here. Links will be re-checked after all titles have been changed. (Removed two redirects for “E10” from the top and added one for “E10” at the bottom of the page.)
I worked on brushing up (infinity,1)-category a little
mostly I added in a section on homotopical categories, using some paragraphs from Andre Joyal's message to the CatTheory mailing list.
in this context I also rearranged the order of the subsections
I removed in the introduction the link to the page "Why (oo,1)-categories" and instead expanded the Idea section a bit.
added a paragraph to the beginning of the subsection on model categories
added the new Dugger/Spivak references on the relation between quasi-cats and SSet-cats (added that also to quasi-category and to relation between quasi-categories and simplicial categories)
a bare list of references, to be !include-ed at proof assistant and at machine learning, for ease of synchronizing
this MO comment made me realize that we didn’t have an entry proof assistant, so I started one
Created:
\tableofcontents
Prevalence refers to ideas revolving around associating an enhanced measurable space to a complete space metrizable space topological group.
Suppose G is a complete space metrizable space topological group. A Borel subset S⊂G is shy if there is a compactly supported nonzero Borel measure μ such that μ(xS)=0 for all x∈G.
The triple (G,BG,SG), where BG is the σ-algebra of Borel subsets and SG is the σ-ideal of shy sets is an enhanced measurable space.
We may also want to complete enhanced measurable space (G,BG,SG), extending the notion of shy and prevalent sets to non-Borel sets.
Brian R. Hunt, Tim Sauer, James A. Yorke, Prevalence: a translation-invariant “almost every” on infinite-dimensional spaces, Bull. Amer. Math. Soc. (N.S.) 27 (1992), no. 2, 217–238. doi.
Brian R. Hunt, Tim Sauer, James A. Yorke, Prevalence. An addendum to: “Prevalence: a translation-invariant ‘almost every’ on infinite-dimensional spaces”, Bull. Amer. Math. Soc. (N.S.) 28 (1993), no. 2, 306–307. doi.
Survey:
giving this its own entry, not to bury the material all at braid group
I began to add a definition of conformal field theory using the Wightman resp. Osterwalder-Schrader axiomatic approach. My intention is to define and explain the most common concepts that appear again and again in the physics literature, but are rarely defined, like “primary field” or “operator product expansion”.
(I remember that I asked myself, when I first saw an operator product expansion, if the existence of one is an axiom or a theorem, I don’t remember reading or hearing an answer of that until I looked in the book by Schottenloher).
added pointer to:
added to path space object an Examples-section with some model category-theoretic discussion, leading up to the statement that in a simplicial model category for fibrant X the powering XΔ[1] is always a path space object.
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 f:A→B. The canonical map coker(kerf)→ker(cokerf) 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 ker(cokerf)→B 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 f: 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 f a monic and you get a decomposition into iso iso kernel and for f 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 ℤ→ℤ 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 2:ℤ→ℤ 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 tfAb are the maps whose cokernel (in Ab) is torsion. Thus 2:ℤ→ℤ is monic and epic, so tfAb is not balanced. And since 2:ℤ→ℤ is its own canonical map, that canonical map is monic and epic in tfAb, 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]
Added a “warning” for something that tripped me up: the classifying topos of a classical first-order theory is typically not Boolean, even though the classifying pretopos is Boolean. For a topos to be Boolean is much stronger – as Blass and Scedrov showed, it implies ℵ0-categoricity.
I gave the stub-entry Hopf algebroid a paragraph in the Idea-section that points out that already in commutative geometry there are two different kinds of Hopf algebroids associated with a groupoid (just as there are two versions of Hopf algebras associated with a group):
The commutative but non-co-commutative structure obtained by forming ordinary function algebras on objects and morphisms;
The non-commutative but co-commutative structure obtained by forming the groupoid convolution algebra.
For the moment I left the rest of the entry (which vaguely mentions commutative and non-commutative versions without putting them in relation) untouched, but I labelled the whole entry “under constructions”, since I think this issue needs to be discussed more for the entry not to be misleading.
I may find time to get back to this later…
starting an entry on the integer Heisenberg group.
For the moment it remains telegraphic as far as the text is concerned (no Idea-section)
but it contains a slick (I find) computation of the modular transformation of Chern-Simons/WZW states from the manifest modular automorphy of certain integer Heisenberg groups.
Hope to beautify this entry a little more tomorrow (but won’t have much time, being on an intercontinental flight) or else the days after (where I am however at a conference, but we’ll see).
as disucssed in another thread, Todd kindly added some text to arithmetic and to number.
Now I tried to update the cross-links in general, and the list at
in particular.
But most entries listed there are still just stubs.
the term modular group used to redirect to Moebius transformation, which has a subsection of that name, talking about PSL(2,ℤ).
But since also SL(2,ℤ) is called “the modular group” a better disambiguation is desireable.
added to modular form a brief paragraph with a minimum of information on modular forms As automorphic forms. Needs to be expanded.
I added the definition and several references on higher dimensional knots under knot.
created a brief entry IKKT matrix model to record some references. Cross-linked with string field theory, and with BFSS matrix model
I added a reference at differential form to the wiki-textbook Geometry of differential forms, written for a physics audience in mind.
Added:
Thomas Holder has been working on Aufhebung. I have edited the formatting a little (added hyperlinks and more Definition-environments, added another subsection header and some more cross-references, cross-linked with duality of opposites).
Added:
stub for Cartan calculus
I gave chromatic homotopy theory an Idea-section.
To be expanded eventually…
entry with a bare list of references, to be !include-ed into the References-list of related entries, such as membrane and M2-brane and Green-Schwarz sigma-model
Did anyone ever write out on the nLab the proof that for X locally compact and Hausdorff, then Map(X,Y) with the compact-open topology is an exponential object? (Many entries mention this, but I don’t find any that gets into details.)
I have tried to at least add a pointer in the entry to places where the proof is given. There is prop. 1.3.1 in
but of course there are more canonical references. I also added pointer to
Added to derivator the explanation that Denis-Charles Cisinski had posted to the blog.
Zoran, I have made the material you had here the section "References", as this was mainly pointers to the literature. Please move material that you think you should go into other sections.
started a section on the homotopy type of the diffeomorphism group and recorded the case for closed orientable surfaces
added a tiny bit of basics to complex oriented cohomology theory
This week I am at a workshop in Bristol titled Applying homotopy type theory to physics, funded by James Ladyman’s “Homotopy Type Theory project”. David Corfield is also here. The program does not seem to be available publically, but among the other speakers that the nLab community knows is also Jamie Vicary.
Myself, I will give a survey talk titled “Modern physics formalized in Modal homotopy type theory” (which maybe should rather have “to be formalized” in the title, depending on how formal you take formal to be). I am preparing expanded notes to go with this talk, which I am keeping at
This is still a bit rough at some points, but that’s how it goes.
I currently also have a copy of the core of this material in one section at Science of Logic, replacing the puny previous section on formalization that was there. While it’s not puny anymore, now maybe it’s too long and should be split off. But just for the time being I’ll keep it there.
If you look at it, you’ll recognize a few points that I tried to discuss here lately, more or less successfully. This here is not meant to force more discussion about this – we may all be happier with leaving it as it is – it’s just to announce edits, in case anyone watching the RecentlyRevised charts is wondering.