The Infinite Seminar

In the arXiv mailing from today, February 2nd 2008, there was a large chunk of $A_{\mathrm{\infty }}$-related preprints from Lyubashenko and Manzyuk. Most of these preprints are getting published in journals, or in Max-Planck-Institute report series.

For reference, the arXiv posts I’ll be talking about here are:
http://arxiv.org/abs/0802.2885
http://arxiv.org/abs/math/0211037
http://arxiv.org/abs/math/0306018
http://arxiv.org/abs/math/0312339
http://arxiv.org/abs/math/0701165

In general sweeps - acquired from scanning the abstracts - Lyubashenko and Manzyuk are talking about $A_{\mathrm{\infty }}$-categories as defined by Fukaya, and doing various basic constructions with them - demonstrating that various definitions of unitality coincide, showing a relation to Serre k-linear functors, constructing quotient categories with the same kind of structure et.c.

Now, one question that occurs to me quickly is the following:
Could an $A_{\mathrm{\infty }}$-algebra be considered to be a one-object $A_{\mathrm{\infty }}$-category? And if this is so, can the quotient constructions Lyubashenko and Manzyuk are using be extended (possibly significantly) to form a transferral of $A_{\mathrm{\infty }}$-(co)algebra structure across surjections in general?

And the way off target question: Does this somehow give us a way to transfer $A_{\mathrm{\infty }}$-(co)algebra structures along things like the restriction map in group (co)homology?

arXiv:0711.4499
Steffen Sagave - DG-algebras and derived A-infinity algebras

In this paper, the theory of $A_{\mathrm{\infty }}$-algebras as sketched in Kellers survey papers is developed to give bigraded (homological and “original”) resolutions of dg-algebras, and construct minimal models for dg-algebras using an extended notion of $A_{\mathrm{\infty }}$-algebras called derived $A_{\mathrm{\infty }}$-algebras. These take a bigraded structure into account.

Sagave proves a derived analogue of the minimality theorem applicable for any dg-algebra over any commutative ring.

arXiv:0712.0996

Valery A. Lunts: Formality of DG algebras (after Kaledin)

This paper, which states in the arXiv summary that it’s an early draft, and welcomes additional input, sets out to develop the background in order to prove a result stated in a paper by Dima Kaledin. Kaledin picks out a special cohomology class in a Hochschild cohomology ring, which acts as an obstruction to formality of the algebra studied. The core results deals with how this obstruction is realized in the Kaledin class and how formality of $A_{\mathrm{\infty }}$-algebras gets inherited in various constructions.

Some of the recent preprints seen in the arXiv mailings include:

Mikael Johansson: A partial $A_{\mathrm{\infty }}$-structure on the cohomology of $C_n\times C_m$

Suppose k is a field of characteristic 2, and n,m≥2 powers of 2. Then the $A_{\mathrm{\infty }}$-structure of the group cohomology algebras $H^{*}(C_n,k)$ and $H^{*}(C_m,k)$ are well known. We give results characterizing an $A_{\mathrm{\infty }}$-structure on $H^{*}(C_n\times C_m,k)$ including limits on non-vanishing low-arity operations and an infinite family of non-vanishing higher operations.

Alastair Hamilton & Andrey Lazarev: Cohomology theories for homotopy algebras and noncommutative geometry

This paper builds a general framework in which to study cohomology theories of strongly homotopy algebras, namely $A_{\mathrm{\infty }},C_{\mathrm{\infty }}$ and $L_{\mathrm{\infty }}$-algebras. This framework is based on noncommutative geometry as expounded by Connes and Kontsevich. The developed machinery is then used to establish a general form of Hodge decomposition of Hochschild and cyclic cohomology of $C_{\mathrm{\infty }}$-algebras. This generalizes and puts in a conceptual framework previous work by Loday and Gerstenhaber-Schack.

Alastair Hamilton & Andrey Lazarev: Symplectic $A_{\mathrm{\infty }}$-algebras and string topology operations

In this paper we establish the existence of certain structures on the ordinary and equivariant homology of the free loop space on a manifold or, more generally, a formal Poincar\’e duality space. These structures; namely the loop product, the loop bracket and the string bracket, were introduced and studied by Chas and Sullivan under the general heading `string topology’. Our method is based on obstruction theory for $C_{\mathrm{\infty }}$-algebras and rational homotopy theory. The resulting string topology operations are manifestly homotopy invariant.

Yes, this post includes a bit of shameless self-promotion.