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fianlly added the details of Dugger’s description of cofibrant objects in the projective model structure on simplicial presheaves in the section Cofibrant objects.
After Urs’ post at the café about “Tricategory of conformal nets” at Oberwolfach I took a look at the paper Conformal nets and local field theory and noted that I would have to ask some trivial and boring questions about nomenclature before I could even try to get to the content.
One example is about “Haag duality”: It seems to me that we need a generalization of net index sets on the nLab that includes the bounded open sets used for the Haag-Kastler vacuum representation and the index sets used in the mentioned paper. One of the concept needed would be “causal index set”:
A relation ⊥ on an index set (poset) I is called a causal disjointness relation (and a,b∈I are called causally disjoint if a⊥b) if the following properties are satisfied:
(i) ⊥ is symmetric
(ii) a⊥b and c<b implies a⊥c
(iii) if M⊂I is bounded from above, then a⊥b for all a∈M implies supM⊥b.
(iv) for every a∈I there is a b∈I with a⊥b
A poset with such a relation is called a causal index set.
Well, that’s not completly true, because in the literature that I know there is the additionally assumtion that I contains an infinite unbounded sequence and hence is not finite (that whould be a poset that is ? what? unbounded?), that is not a condition imposed on posets on the nLab.
After this definition one can go on and define “causal complement”, the “causality condition” for a net and then several notions of duality with respect to causal complements etc. all without reference to Minkowski space or any Lorentzian manifolds.
Should I create a page causal index set or is there something similar on the nLab already that I overlooked?
category: people page for
Canadian bacon
I created Galois module. I also added further references to p-divisible group; in particular section 4.2 of Lurie’s survey of elliptic cohomology gives some generalization of the classical theory. I started also a page with -the somehow unfortunate- title relations of certain classes of group schemes- I intended it to give an overview and examples of the basic kinds of group schemes occurring in classical (algebraic) number theory (the page contains more or less two specific examples; so there is still development potential).
added pointer to:
I have created the entry recollement. Adjointness, cohesiveness etc. lovers should be interested.
I corrected an apparent typo:
A 2-monad T as above is lax-idempotent if and only if for any T-algebra a:TA→A there is a 2-cell θa:1⇒η∘a
to
A 2-monad T as above is lax-idempotent if and only if for any T-algebra a:TA→A there is a 2-cell θa:1⇒ηA∘a
It might be nice to say ηA is the unit of the algebra….
added pointer to
and rewrote the Idea-section to make it clear that these authors require not just existence of left and right adjoints, but in fact an ambidextrous adjoint and satisfying an extract coherence condition.
Added a recent reference on Peirce’s Gamma graphs for modal logic. This describes his first approach via broken cuts rather than the later tinctured sheet approach. I keep meaning to see if there’s anything in the latter close to LSR 2-category of modes approach.
According to the broken-cut method, possibility is broken cut surrounding solid cut, while necessity is solid cut surrounding broken cut. Since solid cut is negation, broken cut signifies not-necessarily. Easy to see □¬=¬◊ as the same pattern of three cuts, etc.
In the Alpha case, we’re to think of negated propositions as though written elsewhere on another sheet (or the back of the sheet). There seems to be a three-dimensionality to the graphs, e.g., the conditional as like a tube from one sheet to another, Wikipedia. I gather his later ideas on tinctured graphs had this idea of being inscribed on different sheets.
added pointer to:
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.
I gave root of unity its own entry (it used to redirect to root), copied over the paragraph on properties of roots of unities in fields, and added a paragraph on the arithmetic geometry description via μn=Spec(ℤ[t](tn−1)) and across-pointer with Kummer sequence.
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 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]
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 -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 .
But since also 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.