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Added a bit to skeleton about skeletons of internal categories
added to exact functor a new subsection “Between abelian categories” and listed there (briefly) the standard characterizations of left/right exact functors in terms of preservation of left/right exact sequences.
Also added a reference by Michael Barr on the relation between exactness and respect for homology in very general contexts.
added to injective object propositions and examples for injective modules and injective abelian groups
P.S. I am checking if I am missing something: Toën on page 48/49 here behaves as if it were clear that there is a model structure on positive cochain complexes of R-modules for all R in which the fibrations are the epis. But from the statements that i am aware of at model structure on chain complexes, in general the fibrations may be taken to be those epis that have injective kernels. For R a field this is an empty condition and we are in business and find the familiar model structures. But for R not a field? Notably simply R=ℤ What am I missing?
I fixed the definition at over quasi-category so it makes the adjointness relationship clearer between overcategories and joins. In particular, Lurie’s notation and definition makes it very hard to see this. It’s much easier to see what’s going on when we look at things as follows: The join with K fixed in the first coordinate, S↦iK⋆SK:K→K⋆S, where iK⋆SK is the canonical inclusion, is a functor SSet→(K↓SSet). Then the undercategory construction gives the adjoint to this functor sending (K↓SSet)→SSet. This makes it substantially clearer to understand what’s going on, since HomSSet(S,XF/):=HomK↓SSet(iK⋆SK,F) is the set of those maps f:K⋆S→X such that f|K=f∘iK⋆SK=F.
Lurie’s notation HomSSet(S,XF/):=HomF(K⋆S,X) is nonstandard and inferior, since it obscures the obvious adjointness property.
The definition for overcategories is “dual” (by looking at the join of K on the right).
started nice simplicial topological space with material provided by David Roberts
stub for simplicial topological space
considerably expanded Lie infinity-groupoid. But still stubby.
Aleks Kissinger has given us sifted colimit. Although I don’t quite understand the definition.
I revisited some old discussion with Mike at sequence. Are you happy now, Mike?
I linked to it, so here is jointly epimorphic family, our newest stub.
started oo-vector bundle on my personal web, following my latest remarks in the thread here on deformation theory.
New page: indecomposable object, following (what I think is) Johnstone's definition. I also found it in some online topos theory lecture notes by Ieke Moerdijk and Jaap van Oosten.
Lambek and Scott give a different definition in Intro. to Higher-order Cat. Log., p. 168. I'm not sure how it relates to Johnstone's.
I've also given a proof that indecomposable <=> connected in an extensive category. I'd be interested to know whether this hypothesis is the weakest possible, if anyone has any ideas (or just likely-looking references).
I could have sworn that we had something for thin category, at least a redirect, but we don’t. Or didn’t. Now we do.
Not much to it, just a note of terminology, like inductive limit or (0,1)-topos.
There’s also a diagram that I can’t to get to work there, if anybody wants to help.
created circle n-bundle with connection.
See the nForum thread on oo-Chern Weil theory for background.
This is mathematically much simpler than the classical Gleason’s Theorem, but I added it to Gleason’s theorem anyway.
In gluing categories from localizations (zoranskoda) the main section
From a family of localizations to a comonad
is fully rewritten in improved notation. In other way, it is explained better how to get a comonad from a cover of a category by not necessarily compatible flat localizations. This generalizes the Sweedler's coring to relative situations. Now from such data one can make a two category, which I will explain in few days.
This is a preliminary to something I am writing at the moment namely to explain in such terms actions of comonads and monoidal categories on such descent categories. This part will be analogous to description of equivariant maps among G-manifolds in pairs of local charts, but because of the distributive laws with coherences, the thing complicates.
started stacked cover
I started quantifier, but I ran out of time to say all that I wanted. I’ll probably get back to it in a couple of hours.
I redid everything that includes contents using the new click-based menu system. This includes HomePage; there didn’t seem to be a need anymore to have two columns, so I put them back in one column. However, those are separate issues; we could put them back in two columns again and still do the click thingy.
I added some stuff about states in statistical physics to state.
I moved states in AQFT and operator algebra to state on an operator algebra, and I also included there a redirect from state on an algebra, which had been requested on quantum state. Hopefully it makes sense that way.
I almost wrote state on an algebra myself, but I’m glad that I didn’t, since everything that I would have said (and more) was already on the extant page.
A couple of new pages have appeared, theory of primes and PrimeDeGold, the latter being the author of the former (and itself). They seem like nonsense, but perhaps someone (Andrew? Toby?) is doing testing? Or else testing for a sort of spamming?
created and expanded infinity-Lie algebra cohomology.
There is now a section on the (∞,1)-topos theoretic interpretation, and one on how to understand ∞-Lie algebra extensions as special cases of the general nonsense on principal ∞-bundles.
The discussion leaves quite a bit of room for polishing, but I don’t feel like spending much more time on this right this moment.
shrinkable map, so I can reference it for Urs’ question on Hurewicz fibrations.
I added new links to my nLab page. I have links to my ’table of categories’ and ’database of categories’ here
Baues-Wirsching cohomology of small categories got a stub. Updates to Hans-Joachim Baues.
New entry Masoud Khalkhali related to the quite recent cyclic cohomology entry (which still lacks the basic material).
In my personal nlab a bibliography for an interesting topic, which some of my peers in Zagreb got recently interested in: Feynman proof of the Lorentz force equations (zoranskoda). It is funny – deriving gauge theories and even gravity just from commutation relations for the generators (coordinates and “covariant momenta”) without any action principle, that is without assuming Lagrange or Hamilton formalism to hold. It has nice extensions and it may be important for the philosophy of gauge theories aka connections on vector bundles. Any ideas of categorification may be interesting…
When talking personal nlab I wrote a long general advice page for students who may ask for my future mentorship in Zagreb of some sort. Well, if I stay in Zagreb. The things are getting rough for science here, and I am not sure of my own future.
Hisham Sati in January posted a survey Geometric and topological structures related to M-branes. In a hard effort of several hours of intense work I created an entry containing hyperlinked bibliography of that article (I took LaTeX source, scraped off various LaTeX commands like bf, it, bibitem etc. and then started creating various hyperlinks). Most of the hyperlinks to the arxiv and few to the project euclid are created so far. Many items still do not have proper external links which would be very welcome. This is a very nice bibliography for something of much interest to Urs, me and some other nlabizants, and I would like to have it practical for our systematic online study.
Added a proof of the pasting lemma to pullback, and the corresponding lemma to comma object (also added the construction by pullbacks and cotensors there).
I’ve started infinitary Lawvere theory following this n-Forum discussion.
I have added a stub entry to the lab on Dominique Bourn. There are quite a few links that need developing there as the protomodular category stuff is quite rudimentary. I would need to learn more about it to fill things up so if anyone does feel they can help, please charge ahead.
am starting to create stubs
and am heading for
On variety of algebras appears the sentence “(This paragraph may be original research. Probably the concept does appear in the literature but under a different name.)”. The paragraph in question is about typed varieties of algebras. Looking at the history, this sentence (and indeed, the whole page!) appears to be due to Toby (Bartels).
I’m curious as to what part that sentence refers to, in particular due to my interest in what I call graded varieties of algebras (nomenclature coming from algebraic topology and graded cohomology theories), which I thought was just an example of a heterogeneous variety of algebras, a term that I’ve come across in the literature. Certainly the concepts feel closely related, and it took a fair amount of paper chasing to find the term “heterogeneous” (though “many-sorted” theories seemed a bit more of a common term), but despite my interest, I’m no expert and am sure I’m missing something. Problem is: I don’t know what and I don’t know how to properly formulate my question!
(Added in edit): Actually, I see that the term “multisorted” is in use on Lawvere theory.
The definition of “subnet” under net looked wrong to me (part of the wrongness was obvious), so I changed it so that it looks correct to me. Could someone please give independent verification?
I found a good constructive definition of proper subset and put it in there. Also I wrote improper subset.
Edit: also family of subsets; see below.
started floating TOC differential cohomology - contents
A semester ago I announced a possible mentorship in Zagreb if a physics student would like to take to digest and write a diploma on the basis of Baez-Schreiber work on higher gauge theory. Nobody chose the topic but the page in my personal nlab is left out from those times, and maybe it will be recycled by a future announcement, though it is questionable as I am likely to leave my present institution in few months. But in order to be functional, it is good to have also the list of literature which i just compiled, including the appended list of very advanced references so that it might serve at all levels. Suggestions and usage for your own purposes are welcome.
added a stub entry for holonomy.
Just the bare definition, and of that even only the most naive one. Don’t have time for more. But created it anyway because I needed the link.
(Sounds a bit like like: I was young and needed the money…)
I wrote about generaliasations of real numbers and managed to follow one link to create characteristic.
started automorphism infinity-Lie algebra with some basics from Sullivan’s old article
started stub for differential 2-crossed module
stub for Lie operad
We talk of a ’homogeneous linear functor’ at Goodwillie calculus, a functor which maps homotopy pushout squares to homotopy pullback squares. There are also higher degree homogeneous functors which map (n+1)-dimensional cubical homotopy pushout diagrams to (n+1)-dimensional cubical homotopy pullback diagrams. These allow polynomial approximation in the functor calculus.
We also have linear functor and polynomial functor. I take it that these latter two are unrelated to each other, and to the functor calculus terms. I think we need some disambiguation.
Does anyone know why in the Goodwillie calculus those functors are called linear? Perhaps this helps:
At the heart of Algebraic Topology is the study of geometric objects via algebraic invariants. One would like such invariants to be subtle enough to capture interesting geometric information, while still being computable in the sense of satisfying some sort of local-to-global properties.
A simple and familiar example of this is the Euler characteristic e(X), where the local-to-global property for good decompositions takes the form e(U∪V)=e(U)+e(V)−e(U∩V). A more sophisticated invariant is homology, where the local-to-global equation above is replaced by the Meyer–Vietoris sequence. Finally one can consider the functor SP∞:Top→Top, assigning to a based topological space, its infinite symmetric product. This functor has the property that it takes homotopy pushout squares (i.e. good decompositions) to homotopy pullback squares. As the Dold-Thom theorem tells us that the homotopy groups π*(SP∞(X))=H*(X), the Meyer-Vietoris sequence for homology is thus a consequence of applying π*(−) to the homotopy pullback square.
It was the insight of Tom Goodwillie in the 1980’s that such “linear” functors F:Top→Top form just the beginning of a hierarchy of polynomial functors, where a polynomial functor of degree n takes appropriate sorts of (n+1)-dimensional cubical homotopy pushout diagrams to (n+1)-dimensional cubical homotopy pullback diagrams. Furthermore, many important functors admit good approximations by a Taylor tower of polynomial approximations.
I am a bit stuck/puzzled with the following. Maybe you have an idea:
I have a group object G and a morphism G→Q. I have a model for the universal G-bundle EG (an object weakly equivalent to the point with a fibration EG→BG).
I have another object EQ weakly equivalent to the point such that I get a commuting diagram
G→Q↓↓EG→EQHere Q is not groupal and i write EQ only for the heck of it and to indicate that this is contractible and the vertical morphisms above are monic (cofibrations if due care is taken).
So I have G acting on EG and the coequalizer of that action exists and is BG
G×EG→→EG→BGI can also consider the colimit K of the diagram
G×EG→→EG→EQ.That gives me a canonical morphism BG→K fitting in total into a diagram
G→Q↓↓EG→EQ↓↓BG→K.Now here comes finally the question: I know that the coequalizer of G×EG→→EG is a model for the homotopy colimit over the diagram
⋯G×G→→→G→→*as you can imagine. But I am stuck: what intrinsic (∞,1)-categorical operation is K a model of?
I must be being dense….
Fiore, Lück and Sauer have a new arXiv preprint, Euler characteristics of categories and homotopy colimits, which covers material from Tom Fiore’s talk at the Utrecht higher category theory day (and at CT2010). I added the link to that page.
created
groupal model for universal principal infinity-bundles
in order to record and link David Roberts’s result.
to go with this, I also created universal principal infinity-bundle.
added a stubby
to the entry Lie infinity-groupoid.
The punchline is that if we pick a groupal model for EG – our favorite one is the Lie 2-group INN(G) – then by the general nonsense of Maurer-Cartan forms on ∞-Lie groups there is a Maurer-Cartan form on EG. This is, I claim, the universal Ehresmann connection on EG.
The key steps are indicated in the section now, but not exposed nicely. I expect this is pretty unreadable for the moment and I tried to mark it clearly as being “under construction”. But tomorrow I hope to polish it .
Over a coring there are not only the left and right comodules, but also the left and right contramodules !
created topological submersion. I’ve seen more than one definition of this, and both could be useful. My natural inclination is to the more general, where each point in the domain has a local section through it.
On a side note I use a related condition in my thesis for a topological groupoid over a space: every object is isomorphic to one in the image of a local section. This was used in conjunction with local triviality of topological bigroupoids to define certain sorts of 2-bundles.
expanding the entry hypercohomology started by Kevin Lin, I wrote an Idea-section that tries to explain the nPOV on this
Edited Lie groupoid a little, and new page: locally trivial category. There is an unsaturated link at the former, to Ehresmann’s notion of internal category, which is different to the default (Grothendieck’s, I believe). The difference only shows up when the ambient category doesn’t have all pullbacks (like Diff, which was Ehresmann’s pretty much default arena). It uses sketches, or something like them. There the object of composable arrows is given as part of the data. I suppose the details don’t make too much difference, but for Lie groupoids, it means that no assumption about source and target maps being submersions.
The latter page is under construction, and extends Ehresmann’s notion of locally trivial category/groupoid to more general concrete sites. I presume his theorem about transitive locally trivial groupoids and principal bundles goes through, it’s pretty well written.
created Bianchi identity.
(gave it the ∞-Lie theory toc, but already with the new CSS code. So as soon as that CSS code is activated on the main nLab, that TOC will hide itself and become a drop-down menu. I think.)
I’m so sick of making mistakes about separable algebras and their relation to Frobenius algebras that I wrote a page separable algebra and added more to the page Frobenius algebra. To make these pages make sense, I needed to create pages called semisimple algebra, simple algebra, and division algebra. Also projective module.
I would love it if some experts on algebraic geometry vastly enhanced the little section about algebraic geometry in separable algebra. There’s a question there, and also a very vague sentence about etale coverings.