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I am trying to write up an elementary exposition for how the Hochschild chain complex for a commutative associate algebra is the normalized chains/Moore complex of the simplicial algebra that one gets by tensoring the algebra with the simplicial set :
I would like to get feedback on whether or not my exposition is in fact understandable in an elementary way.
The section that contains this material is the section
at the entry Hochschild cohomology. Just this one section. It’s not long.
It describes first the simplicial set , then discusses how the coproduct in is given by the tensor product over , and deduces from that what the simplicial algebra is like.
After taking the normalized chains of that, the result is Pirashvili’s construction of a chain complex from a simplicial set and a commutative algebra. I just think it is important to amplify that this construction of Pirashvili’s is a categorical tensoring=copower operation. Because that connects the construction to general abstract constructions. That’s what the beginning of the above entry is about. But for the moment I would just like to make the elementary exposition of the tensoring operation itself pretty and understandable.
New page: bicategory of maps.
tried to bring the old neglected entry sSet-category roughly into some kind of stubby shape. Added Porter-Cordier and LurieA.3 as references. The former was my motivation for doing this. Eventually it would be good to have here a detailed discussion of -category models for -category theory. See the discussion with Tim over in the other thread on the -Yoneda lemma.
(I don’t have time for this now. I am saying all this in the hope that somebody else has.)
I’ve cleaned up diffeological space a little. In particular:
I have been advising Herman Stel on his master thesis, which is due out in a few days. I thought it would be nice to have an nLab entry on the topic of the thesis, and so I started one: function algebras on infinity-stacks.
For any abelian Lawvere theory, we establish a simplicial Quillen adjunction between model category structures on cosimplicial -algebras and on simplicial presheaves over duals of -algebras. We find mild general conditions under which this descends to the local model structure that models -stacks over duals of -algebras. In these cases the Quillen adjunction models small objects relative to a choice of a small full subcategory of the localization
of the -topos of -sheaves over duals of -algebras at those morphisms that induce isomorphisms in cohomology with coefficients the canonical -line object. In as far as objects of have the interpretation of ∞-Lie groupoids the objects of have the interpretatin of ∞-Lie algebroids.
For the special case where is the theory of ordinary commutative algebras this reproduces the situation of (Toën) and many statements are straightforward generalizations from that situation. For the case that is the theory of smooth algebras (-rings) we obtain a refinement of this to the context of synthetic differential geometry.
As an application, we show how Anders Kock’s simplicial model for synthetic combinatorial differential forms finds a natural interpretation as the differentiable -stack of infinitesimal paths of a manifold. This construction is an -categorical and synthetic differential resolution of the de Rham space functor introduced by Grothendieck for the cohomological description of flat connections. We observe that also the construction of the -stack of modules lifts to the synthetic differential setup and thus obtain a notion of synthetic -vector bundles with flat connection.
The entry is of course as yet incomplete, as you will see.
I created split coequalizer and absolute coequalizer, the latter including a characterization of all absolute coequalizers via an “-ary splitting.” While I was doing this, I noticed that monadic adjunction included a statement of the monadicity theorem without a link to the corresponding page, so I added one. (The discussion at the bottom of monadic adjunction should probably be merged into the page somehow.) Then I noticed that while we had a page preserved limit, we didn’t have reflected limit or created limit, so I created them. They could use some examples, however.
I would also like to include an example of how to actually use the monadicity theorem to prove that a functor is monadic. Something simpler than the classic example in CWM about compact Hausdorff spaces; maybe monadicity of categories over quivers? Probably not something that you would need the monadicity theorem for in practice, so that it can be simple and easy to understand.
Under the Definitions at topos, I gave definitions of Grothendieck toposes and W-toposes, since these are two very important kinds of toposes that some authors (at least) often call simply ‘toposes’. (Also it gave me a place to redirect W-topos and its synonym topos with NNO.)
On Tim’s suggestion, I have been in contact with Ronnie Brown on some of the history behind “convenient categories”, and received a wealth of information from him. I have made an initial attempt to summarize what I have learned in the Historical Remarks section of convenient category of topological spaces, but it might be somewhat garbled still. Hopefully Ronnie and/or Tim will have a look. I will be adding more references by and by.
I also added in the follow-up discussion with Callot under Counterexamples.
Francois Métayer already has a lab-entry. (NB the acute accent is missing on the new one! That was the cause of the error.)
I added some more details to the section on ultrafilter monad at ultrafilter. Incidentally, it seems to me that the bit on Barr’s observation (“topological spaces = relational -modules”) is too terse. There is a lot of generalized topology via abstract nonsense that deserves more explanation.
I also added some elementary material to cartesian closed category, mainly to indicate to the novice how exponentials deserve to be thought of as function spaces, how internal composition works, and so forth. I left the job somewhat unfinished.
split off model structure on strict omega-categories from the page canonical model structure (that page could do with some cleaning-up)
Krzysztof Worytkiewicz kindly informs me that an old question to Francois Metayer has now been answered: the folk model structure on strict -categories does restrict to the Brown-Golasinski model structure on strict -groupoids. (The latter is indeed the transferred model structure along the forgetful functor to the former).
This is now written up in
Ara, Métayer, The Brown-Golasinki model structure on oo-groupoids revisited (pdf)
As my connection allows, I will insert this into the nLab entry now…
created framed little n-disk operad and added to BV-algebra its relation to higher BV-algebras
stub for multivector field
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.
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.