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.
started adding to (infinity,1)-topos a section on the (oo,1)-category of (oo,1)-toposes.
New stub hom-connection. I should figure it out once. While tensor product is involved in many constructions in algebra, some are dual with Hom instead, for example there are contramodules in addition to comodules over a coring. In similar vain hom-connections were devised, but there are some really intriguing examples (including superconnections, right connections of Manin etc.) and there are relations to examples of noncommutative integration of various kind.
Created homotopy equivalence of toposes.
created stub for Gerstenhaber algebra
stub for En-algebra
I created a stub for spectral theory. It is related to the deficiency of the entries in functional analysis.
I have added some discussion to the page on orientals (in the sense of Ross Street), regarding the link to the convex geometry of cyclic polytopes (as discussed by Kapranov and Voevodsky).
My selfish motive for doing so is that I am curious if my recent work with Steffen Oppermann which includes a new description of the triangulations of (even-dimensional) cyclic polytopes, has any relevance to the study of orientals, or higher category theory more broadly. (In particular, if there are explicit questions about the internal structure of orientals which are of interest, I would like to hear about them.)
A particularly speculative version of my question, would be whether there is a natural connection between orientals and the representation theory which we are studying in that paper (which necessitated a detour into convex geometry). We biject triangulations of an even-dimensional cyclic polytope to (a nice class of) tilting objects for a certain algebra. The simplest version of this (which was already known) is that triangulations of an -gon correspond to tilting objects for the path algebra of the quiver consisting of a directed path with vertices. (These tilting objects then give derived equivalences between the derived category of this path algebra, and the derived category of the endomorphism ring of this tilting object.)
Questions, speculations, or suggestions would be very welcome.
Hugh
to the functional analysis crew of the Lab: where should operator spectrum point to? Do we have any suitable entry?
created hyper-derived functor
a little acyclic object
New stubs Oka principle, Oka manifold (with redirect Oka map) and Franc Forstnerič. Jardine has shown that one can use the Toen-Vezzosi like engineering with his intermediate model structure on the category of simplicial presheaves on a simplicial version of the Stein site. The -stacks/fibrants will be Oka maps; those cofibrants which are represented by complex manifolds are in fact Stein manifolds.
created a stub future cone. This was in the context of directed homotopy theory, but clearly could be developed and linked to other topics.
New: choice function
expanded Vopenka’s principle
tried to brush-up tensor power a little
I expanded some entries related to the Café-discussion:
at over-(infinity,1)-topos I expanded the Idea-section, added a few remarks on proofs and polished a bit,
and added the equivalence to the Examples-section
at base change geometric morphism I restructured the entry a little and then included the proof of the existence of the base change geometric morphism
wrote out the proof of the expected statement at limits in over-(oo,1)-categories
added to adjunct the description in terms of units and counits.
created (infinity,1)-algebraic theory.
I tried to adapt Rosicky’s and Lurie’s terminology such as to match that at algebraic theory, but Mike, Toby, Todd and whoever else feels expert should please check if I did it right.
added the equivalence
added some propositions to Pr(infinity,1)Cat that support the analogy to linear algebra, as described there and at integral transforms on sheaves
I satisfied a few requests for ordered field. Pretty basic.
Kevin Walker was so kind to add a bit of material to blob homology. Notably he added a link to a set of notes now available that has more details.
I added formatting and some hyperlinks.
stub for Boardman-Vogt resolution
edited E-infinity algebra a little. Still all very stubby
I added to loop space a reference to Jim’s classic article, which was only linked to from H-space and put pointers indicating that his delooping result in is a special case of a general statement in any -topos.
By the way: it seems we have slight collision of terminology convention here: at “loop space” it says that H-spaces are homotopy associative, but at “H-space” only a homotopy-unital binary composition is required, no associativity. I think this is the standard use. I’d think we need to modify the wording at loop space a little.
I reworked A-infinity algebra so as to apply to algebras over any -operad in any ambient category. So I created subsections “In chain complexes”, “In topological spaces”.
I think if we speak generally of “algebra over an operad” then we should also speak generally of “-algebra” even if the enriching category is not chain complexes. Otherwise it will become a mess. But I did link to A-infinity space.
added the definition of “coloured operad” to operad in the section “Rough definition”
(by the way, should we not rather call these “pedestrian definition” or so instead of “rough”? The latter seems to suggest that there is something not quite working yet with these definitions, while in fact they are perfectly fine, just not as high-brow as other definitions.)
created stub for model structure on modules over an algebra over an operad
created stub for module over an algebra over an operad
started a section on cofibrant resolutions at model structure on operads. But incomplete for the time being.
added very briefly the monoidal model structure on -objects in a monoidal model category to monoidal model category (deserves expansion)
there is a span of concepts
higher geometry Isbell duality higher algebra
which is a pretty fundamental thing about math, I think (well, this observation is at least to Lawvere, of course).
I put this span of links at the top of these three entries. I am enjoying that, but let me know if it is once again a silly idea of mine.
(maybe it should also be higher Isbell duality )
I just noticed that aparently last week Adam created indexed functor and has a question there
in preparation of the next session of our Seminar on derived differential geometry I am starting
Someone should improve this article so that it gives a definition of ‘algebraic theory’ before considering special cases such as ‘commutative algebraic theory’.
Thus is the current end to the entry on algebraic theory and I agree. Further I needed FP theory or FP sketch for something so looked at sketch. That looks as if it needs a bit of TLC as well, well not this afternoon as I have some other things that need doing. I did add the link to Barr and Wells, to sketch, however as this is now freely available as a TAC reprint.
copy-and-pasted from MO some properties of homotopy groups of simplicial rings into simplicial ring (since Harry will probably forget to do it himself ;-)
I made sure the following list of entries exists and interlinked everything. Some entries are still stubs that need to be filled with content. Am working on it.
I had created coshape of an (infinity,1)-topos
Mike, what should we do? If we rename that entry to something else we would also need to rename shape of an (infinity,1)-topos.
I’d rather suggest that we proceed entirely in parallel to the dual shape theory and instead create now entries global sections of an (infinity,1)-topos, global sections in an (infinity,1)-topos etc, dual to fundamental infinity-groupoid of a locally infinity-connected (infinity,1)-topos, etc.
I have added to universal covering space a discussion of the “fiber of ” definition in terms of little toposes rather than big ones.
I find this definition of the universal cover extremely appealing. It seems that this sort of thing must have been on the tip of Grothendieck’s tongue, and likewise of all the other people who have studied fundamental groups and groupoids of a topos, but it all becomes so much clearer (I think) when you state it in the language of higher toposes. In this case, merely (2,1)-toposes are enough, so no one can argue that the categorical technology wasn’t there – so why didn’t people see this way of stating it until recently? Or did they?
I made the following obvious fact more manifest in the respective Lab entries:
a pregeometry (for structured (infinity,1)-toposes) is a special case of a (multi-sorted) (infinity,1)-algbraic theory.
A structure -sheaf
on is an -algebra over this -algebraic theory in . The extra conditions on it ensure that it indeed looks like a sheaf of function algebras .
(I added a respective remark to the discussion of pre-geometries and added an Example-sectoin with this to the entry of oo-alghebraic theories.)
I have created a stub for primary homotopy operation. At present it just refers to Whitehead products and composition operations and redirects attention to those entries and to Pi-algebras, which will be next on my list to be created. I do not have access to G. W. Whitehead’s book on homotopy theory so have not given a precise definition nor a discussion of what these are, although the entry on -algebras will to some extent cure that. If anyone knows the definition well or has Whitehead’s book, can they provide the details…. otherwise it will remain a stub. :-(
New page: canonical transformation
The page join of simplicial sets is requesting a page titled “Jack Duskin”. We do have a page titled John Duskin. It that supposed to coincide?
In any case, if anyone who created that unsatisfied link to “Jack Duskin” at join of simplicial sets (also one to van Osdol) could do something such as to satisfy the links, that would be nice.
started stub for operadic Dold-Kan correspondence (for simplicial- vs dg-algebras over operads)
with Birgit Richer’s article we’d also have a notion of “monadic DK correspondence” (for simplicial vs dg-algebras over monads)
does anyone know any direct considerations of “T-algebraic DK-correspondence” (for simplicial vs dg-algebras over a Lawvere theory)?
of course this is to some extent implied by the previous versions. But it would be good to have a direct description.
I am in the process of reproducing the proof of the main theorem in Schwede-Shipley’s “Equivalence of monoidal model categories” at monoidal Quillen adjunction (see the references and pointers given there).
I find that there are some intermediate steps that need to be filled in and which require a tad more thinking than just copying what they write.
This mainly concerns some pure category-theoretic arguments about adjunctions, which is entirely independent of the model category theoretic argument that is later built on it. I am saying this in case you are an expert eager to help on some pure category theory issues but maybe not so much into model category theory.
I think I can figure things out myself eventually, but since I am a bit time pressured and since working toghether is fun anyway, I thought I’d just highlight here what I am doing and where there is still things remaining to be done.
So I am working on the section Lift to Quillen adjunction on monoids. This breaks up the Schwede-Shipley argument into a bunch of small lemmas and propositions and aims to write out the proofs. Partly this is spelled out. Whenever there is a gap in the argument that still needs to be written up or even figured out, I put ellipses
(...)
for the moment. I’ll be working now on filling these ellipses with content, so where exactly you see them may change over time. But if you feel you can easily help fill some of them, you are kindly invited to do so!
added to oplax monoidal functor the statement how an oplax monoidal structure is induced on a functor from a lax monoidal structure on a right adjoint.
After getting myself confused about the distinction between the various notions of basis in infinite dimensions, I wrote up my attempt to disentangle myself at basis in functional analysis (also redirects from Hamel basis, topological basis, and Schauder basis. Hmm, now I think about it, maybe “topological basis” is too close to “basis of a topology”). I may still be confused about stuff, of course.
created stub for braided monoidal functor – but still too lazy to type the diagrams in the axioms
created bilax monoidal functor
created Frobenius monoidal functor
I have separated Eilenberg-Watts theorem from abelian category and added the references and MR links. One of the queries from the abelian category is moved here with backpointer there. I cleaned up some typoi.
The following discussion is about to which extent abelian categories are a general context for homological algebra.
Zoran: I strongly disagree with the first sentence, particularly with THE (it is THE general context for linear algebra and homological algebra). MacLane was (according to Janelidze) looking whole life for what is the general context for homological algebra, and the current answer of expert are semi-abelian categories of Borceux and Janelidze, and homological categories…Linear algebra as well makes sense in many other contexts. This “idea’ is to me very misleading. MacLane in 1950 was lead by the idea to axiomatize the categories which behave like abelian groups. Grothendieck wanted to unify on the obsrervation that the categories of abelian sheaves and categories of R-modules have the same setup for homological algebra as in Tohoku.
There is much linear algebra you can do with cokernels, for example, as well as much linear algebra which you can not do if you are not over a field for example. So, saying that abelian categories are distinguished is only among categories which have closest properties to abelian sheaves and R-modules, not among principles for homlogical algebra and linear algebra that uniquely (although the strong motivation was ever there).
Mike: I changed it to “a” general context; is that satisfactory? Once we have pages about those other notions, there can be links from here to there.
Toby: I've made the phrasing even weaker. Abelian categories are pretty cool, but (if you don't already have the examples that make it so useful) the definition is a fairly arbitrary place to draw the line.
Tim : I note that sometimes we (collectively) take parts of a discussion and turn it into part of an entry, because of that I would like to note two points here. The first is that the accepted first definition of semi-abelian category is in the Janelidze, Marki, and Tholen (JPAA, but we have a link on the semi-abelian entry.)
The other point is that Tim Van der Linden’s thesis does a lot of stuff that could be useful. It is available online http://arxiv.org/abs/math/0607100
I expanded Street fibration a bit, and created equivalence of 2-categories to discuss the notion classically called “biequivalence.” I also noticed that for a long time the page equivalence of categories has claimed at the end that Bicat is (weakly) equivalent to Str2Cat, which I’m pretty sure is not true, so I fixed it.
I tried to brush-up Warsaw circle and in the process created a bunch of simple stubs:
At flat functor there is a statement that a functor on a complete category to a cocomplete category is left exact precisely if its Yoneda extension is.
I know this for . There must be some extra conditions on that have to be mentioned here.