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February 18, 2011

Structured Ring Spectra 2011

Posted by Urs Schreiber

This summer, Hamburg hosts the meeting

  • Structured Ring Spectra - TNG

    1st to 5th August 2011, Hamburg, Germany.

    website

about structured ring spectra.

This is the next in the informal series of conferences on Structured Ring Spectra in Glasgow 2002, Bonn 2004 and Banff 2008. The conference will begin Monday morning and end Friday evening.

Posted at February 18, 2011 2:09 PM UTC

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Re: Structured Ring Spectra 2011

Currently Eric Malm is speaking about his PhD thesis on string topology operations:

String topology and the based loop space ((full pdf), (arXiv:1103.6198)).

He is using derived Poincaré duality to formulate string topology – which ordinarily involves chains on the free loop space – in terms of Hochschild homology of the based loop space, for the case that base space is simply connected.

Posted by: Urs Schreiber on August 2, 2011 10:49 AM | Permalink | Reply to this

Re: Structured Ring Spectra 2011

I can’t open the link given above. It would be great if one could read more about Jack Morava’s and Stefan Schwede’s talks!

Posted by: Thomas on August 2, 2011 11:40 AM | Permalink | Reply to this

Re: Structured Ring Spectra 2011

The server which hosts the nLab is undergoing an upgrade at the moment. Will be back shortly (a few hours, hopefully).

Posted by: David Roberts on August 2, 2011 12:59 PM | Permalink | Reply to this

Re: Structured Ring Spectra 2011

It would be great if one could read more about Jack Morava’s and Stefan Schwede’s talks!

Yes. I have missed Schwede’s talk, unfortunately. Over lunch I tried to quiz people who did attend, but so far I have just very vague information. I’ll try to get ahold of more details.

I am hoping to be able to attend Morava’s talk on Friday. If I do, I’ll write about it here.

Posted by: Urs Schreiber on August 2, 2011 1:20 PM | Permalink | Reply to this

Re: Structured Ring Spectra 2011

Now Brooke Shipley is talking about her joint work with Kathryn Hess titled

The homotopy theory of coalgebras over a comonad

on model category structures on categories of coalgebras.

This is work in progress based on Kathryn Hess’s

  • A general framework for homotopy descent and codescent (pdf)

  • Homotopic Hopf-Galois extensions (slides)

but there does not seem to be any preview document online on this new work yet. (?)

So far we are being shown the co-analog of the usual theorem about transfer of model structures along right adjoints. But the meat here is all in the subtle technical details, which I think I won’t try to reproduce here.

Posted by: Urs Schreiber on August 2, 2011 1:32 PM | Permalink | Reply to this

Re: Structured Ring Spectra 2011

Now Kathryn Hess herself is speaking about the topic of the above two documents:

higher monadic descent.

(See the discussion at Sweedler coring for a pedestrian discussion of monadic descent.)

Posted by: Urs Schreiber on August 4, 2011 1:49 PM | Permalink | Reply to this

Re: Structured Ring Spectra 2011

Reference list is empty
so why Sweedler?

Posted by: jim stasheff on August 5, 2011 12:33 PM | Permalink | Reply to this

Re: Structured Ring Spectra 2011

why Sweedler?

Zoran Škoda kindly gave some indications on Sweedler over at the nnForum here:

Sweedler is a major author on Hopf algebras, authoring an old book in the subject, and a professor emeritus at Cornell. His work on noncommutative extensions dealing with Sweedler coring and related work of his colleague from Cornell Chase on noncommutative Galois, predating Hopf-Galois theory is from 1970s.

Posted by: Urs Schreiber on August 6, 2011 10:28 AM | Permalink | Reply to this

Re: Structured Ring Spectra 2011

Of course, Moss Sweedler will be forever known as having introduced Sweedler notation (for doing coalgebra computations).

Posted by: Todd Trimble on August 6, 2011 11:41 AM | Permalink | Reply to this

Re: Structured Ring Spectra 2011

I think Heyneman was in on the early development
jointly or independently?? of what is now known only as Sweedler notation

Posted by: jim stasheff on August 6, 2011 1:40 PM | Permalink | Reply to this

Re: Structured Ring Spectra 2011

so Sweedler was the first to consider corings?
or is this an instance of the Arnol’d principal?

Posted by: jim stasheff on August 6, 2011 1:44 PM | Permalink | Reply to this

coring, Sweedler coring

Though not much interested in the name-related terminology, I could say that the descent for (the fibered category of modules over) noncommutative algebras and the conceptTh of corings/cocategories were certainly known as examples of general machineries in category theory in 1960s, but specific study of Sweedler coring and of corings got specific attention and tangible publications were probably starting with him.

The descent for noncommutative algebras is a special case of comonadic descent which was studied much in 1960s (Beck etc.) and it was known at the time that one of the basic examples is the extension of algebras/rings and that commutativity is not critical. Still a very much cited reference is by Cippola from 1970s which describes the descent for noncommutative rings in elementary terms (without corings and comonads) parallel to the treatment in SGA I. The comonad for descent of noncommutative algebras is the tensoring with Sweedler coring, of course. The descent for modules over noncommutative algebras for covers by Gabriel localizations was considered in 1988 book of A. L. Rosenberg in Leites’ Stockholm seminars volume and then again independently by van Oystaeyen and his school in early 1990s and in both references unfortunately systematics viewpoint of comonadic descent has been obscured by elementary proofs using specifics of localization theory. 1988 Rosenberg’s Compositio paper *Noncommutative schemes* systematically takes the point of view that the relative generalization of category of quscicoherent sheaves over a general scheme (with relative functor to quasicoherent sheaves over the base scheme) should be axiomatized and studied with emphasis on exactness, as well as monadicity and comonadicity properties with the central role played by applications of Beck’s monadicity theorem. This fundamental application of monadicity and category theory to noncommutative algebraic geometry has not been much noticed by the Hopf algebraic community which needed the more basic understanding of Hopf-Galois descent and coalgebra Galois descent and generalizations. In Spring 2002 I have talked to Valery Lunts who proposed to me that the analogue of modules over Borel construction for the comodule algebras should replace the definition of relative Hopf modules, making the descent theorems like Schneider’s obvious (the basic ideas of this are explained in much later paper of mine “Some equivariant constructions in noncommutative geometry”). Also in 2002, Pjotr Hajac proposed that Sweedler’s corings should be used for the same purpose – in fact the modules over “coborel constuction” and modules over the associated coring are just two viewpoints on the same thing. Then the whole subject of corings in Hopf context got developed much further by Tomasz Brzeziński, Gabi Böhm etc. and influenced by the study of mixed distributive laws which was much studied in this context few years before. Cf. also [descent in noncommutative algebraic geometry](http://ncatlab.org/nlab/show/descent+in+noncommutative+algebraic+geometry).

Posted by: Zoran Skoda on August 8, 2011 11:29 AM | Permalink | Reply to this

Re: Structured Ring Spectra 2011

Compare

www.math.uic.edu/~radford/SlidesSYSU%20(2).pdf

will see if Radford can elab orate the history

Posted by: jim stasheff on August 6, 2011 1:47 PM | Permalink | Reply to this

Re: Structured Ring Spectra 2011

In case anyone has further information on Moss Sweedler to share, please add it to the entry Moss Sweedler on the nnLab.

Posted by: Urs Schreiber on August 6, 2011 1:29 PM | Permalink | Reply to this

Re: Structured Ring Spectra 2011

I had written:

but there does not seem to be any preview document online on this new work yet. (?)

Mike kindly points out that now it has appeared:

Kathryn Hess, Brooke Shipley, The homotopy theory of coalgebras over a comonad (arXiv:1205.3979)

Posted by: Urs Schreiber on May 18, 2012 5:53 PM | Permalink | Reply to this

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