Subject: Re: response and question
Date: Wed, 23 Oct 2002 11:12:14 +0200 (CEST)
From: Pepijn van der Laan
To: dmd1@lehigh.edu
> From: jim stasheff
>
> I have a vague recollection the following has been worked out already:
> the closed model category of L_\infinty modules over a fixxed L_\infty
> algebra
> oand/or the corresponding A_\infty vrsion
>
> references? thanks
>
One approach can be derived from Hinich's paper "Homological algebra of homotopy
algebras" (Comm. Algebra 25 (1997), no. 10, 3291--3323.): Interpret an
L_\infty algebra as an algebra over an operad. For a fixed algebra, modules in
the operadic sense correspond to L_\infty modules. Hinich provides a model
category. The shortest proof of this is probably by adapting the strategy of
Berger and Moerdijk's "Axiomatic homotopy theory for operads"
(arXiv:math.AT/0206094) to modules. This strategy can be followed for other
\infty algebras as well.
However, it might NOT be the model structure you look for since it is based on
"strict" maps. That is, instead of the usual L_\infty type morphisms we only
consider those morphisms that strictly commute with all the higher brackets. So
maybe you end up with too few arrows.
Best regards,
Pepijn.
--
Pepijn van der Laan - Mathematisch Instituut, UU
vdlaan@math.uu.nl - P.O.box 80.010, 3508 TA Utrecht, NL