posic

~~Will be~~ Has been made publicly accessible after some time ~~passes~~ passed.

Dear Tony and Sasha,

I am writing to you as people who pointed my attention to the question
of the proper definition of curved (weak) A_\infty-algebras and their
(exotic) derived categories. This is to let you know about the recent
developments, which haven't been quite definitely verified yet, but
look correct and potentially relevant.

A short summary: curved A_\infty-coalgebras are better than curved
A_\infty-algebras in a number of respects. (But applicability to
any particular problems outside of the abstract Koszul duality
theory have not been considered, as always.)

0. A preliminary remark: one has to keep in mind that in the bar-cobar
constructions and Koszul duality business curved structures on the left
hand side correspond to unital, nonaugmented structures on the right
hand side, while uncurved structures correspond to nonunital or
augmented structures. So curvature appears when one applies Koszul
duality to something with a (co)unit and without a (co)augmentation.

I. Curved A_\infty-algebras.

1. As is well-known, one can indeed define curved A_\infty-algebras,
but there is no notion of a curved A_\infty-morphism between them.
The reason is that the structure maps of an A_\infty-algebra are
the components of a differential on the bar-construction, which is
a tensor (cofree conilpotent) coalgebra. At the same time,
an A_\infty-morphism between A_\infty-algebras is essentially
the same that a DG-coalgebra morphism between the bar-constructions.
Now, conilpotent coalgebras have coderivations that do not preserve
coaugmentations, but no coalgebra morphisms between them not
preserving the coaugmentations exist.

1a. One conclusion is that a CDG-morphism (between CDG-algebras) with
a nonzero change-of-connection element does not induce any morphism
of the bar-constructions.

1b. One can also notice that the bar-construction of a curved
A_\infty-algebra with a nonzero curvature element is always totally
acyclic (even the counit on the cohomology coalgebras is zero).
One checks this by passing to the dual topological graded algebra
to the bar-construction and showing that its unit is a coboundary.

1c. The assertion 1b applies to nonunital or augmented curved
A_\infty-algebras. The bar construction of a strictly unital curved
A_\infty-algebra (with a nonzero unit) is a CDG-coalgebra, so
the question of its cohomology does not even arise to confuse us.
The assertions 1 and 1a apply equally to nonunital or unital curved
A_\infty-algebras.

2. Let us consider a strictly unital, nonaugmented curved
A_\infty-algebra A (as the more general case). The derived category
of strictly unital curved A_\infty-modules over A can be simply
defined as the homotopy category of CDG-comodules over
the CDG-coalgebra Bar(A) that are cofree as graded comodules.
This is equivalent to the homotopy category of CDG-contramodules
over Bar(A) that are free as graded contramodules. This is also
the same that the coderived category of CDG-comodules over Bar(A)
or the contraderived category of CDG-contramodules over Bar(A).
This is the definition that I learned from Sasha.

2a. It appears that the exotic derived category of curved
A_\infty-modules defined in 2. should be indeed called the *derived*
category and thought of as a derived category of the first kind.
The reason is that in the particular case when the curved
A_\infty-algebra has zero curvature, this definition reduces to
the familiar definition of the derived category of an A_\infty-algebra.
On the other hand, in the particular case when the curved
A_\infty-algebra reduces to a CDG-algebra, the derived category of
A_\infty-modules defined in 2. is not at all equivalent to
the coderived or the contraderived category of CDG-modules. In fact,
it is a certain common quotient category of these coderived and
contraderived categories. Neither is this derived category invariant
with respect to CDG-isomorphisms of CDG-algebras.

II. Curved A_\infty-coalgebras.

3. A curved A_\infty-coalgebra C without counit is defined as
a graded vector space together with an odd derivation with zero square
on the free graded algebra generated by C[-1]. This graded algebra
with this derivation is the cobar-construction Cobar(C). This is
equivalent to saying that a sequence of higher comultiplications
\mu_n is defined on C, where n=0,1,2,... These maps satisfy
a convergence condition: for any element c\in C all \mu_n with large
enough n vanish on c. The notion of a curved A_\infty-morphism between
curved A_\infty-coalgebras makes perfect sense, it is just a DG-algebra
morphism between the cobar-constructions. Essentially, it is due to
the natural convergence conditions that one can define curved
A_\infty-morphisms for curved A_\infty-coalgebras.

3a. The cohomology of the cobar construction of a curved
A_\infty-coalgebra, in particular a CDG-coalgebra, with a nonzero
curvature element can be well nonzero, as it should be. There is
no acyclicity phenomenon like in 1b.

3b. Analogously, strictly counital (with a nonzero counit) curved
A_\infty-coalgebras C also have cobar-constructions Cobar(C), which are
exactly those CDG-algebras that are free as graded algebras. Strictly
counital curved A_\infty-morphisms between such coalgebras are
just CDG-algebra morphisms between the cobar-constructions.

4. Let us consider a strictly counital, noncoaugmented curved
A_\infty-coalgebra C (as the more general case). (Strictly counital)
curved A_\infty-comodules over C can be described as CDG-modules over
Cobar(C) that are free as graded modules. Analogously, curved
A_\infty-contramodules over C are CDG-modules over Cobar(C) that are
cofree as graded modules. The coderived category of curved
A_\infty-comodules over C is defined as the homotopy category of
CDG-modules of the former kind over Cobar(C), while the contraderived
category of A_\infty-contramodules over C is defined as the homotopy
category of CDG-modules of the latter kind over Cobar(C).

4a. These two homotopy categories of CDG-modules are equivalent,
as they both project equivalently into the coderived category of
CDG-modules over Cobar(C), which coincides with the contraderived
category of CDG-modules over Cobar(C). The assertions in the latter
phrase hold, essentially, because Cobar(C) as a graded algebra without
the CDG-structure has a finite homological dimension. The result that
we have obtained can be called the comodule-contramodule correspondence
for curved A_\infty-coalgebras.

4b. It appears that the exotic derived categories of curved
A_\infty-comodules and curved A_\infty-contramodules defined in 4.
should be indeed called the *coderived* and *contraderived* categories
and considered as derived categories of the second kind. The reason
is that in the particular case when the curved A_\infty-coalgebra
reduces to a CDG-coalgebra, these definitions are equivalent to
the familiar definitions of coderived and contraderived categories of
CDG-comodules and CDG-contramodules.

4c. It remains to say some words about the notion of an (uncurved)
A_\infty-coalgebra, but here it turns out that there are two versions
of it. An A_\infty-coalgebra in a narrow sense is a curved
A_\infty-coalgebra (as defined above) with a zero curvature map.
This means a sequence of higher comultiplications \mu_1, \mu_2, ...
satisfying the convergence condition. If one drops the convergence
condition, one obtains the notion of an A_\infty-coalgebra C in
the wide sense. Its cobar-construction can be only defined as
the completion of the free graded algebra generated by C[-1] with
respect to the ideal of augmentation (the differential does not act
on the noncompleted free graded algebra). Accordingly, there is
the narrow and the wide way to define homotopy categories of
A_\infty-co/contramodules and A_\infty-morphisms between them, not
to speak of the corresponding derived categories. Roughly speaking,
the wide theory must be a technically hardened, and likely worsened,
version of the familiar theory of A_\infty-algebras and modules,
while the narrow theory should be best viewed as a particular case
of the curved A_\infty-coalgebras theory. My further comment is
related to the narrow theory.

4d. Let C be a (strictly unital) curved A_\infty-coalgebra (as defined
above) with a zero curvature map. Then one can define the derived
category of A_\infty-comodules over C as the quotient category of
the homotopy category of A_\infty-comodules and A_\infty-morphisms
between them (i.e., the coderived category, as defined in 4.) by
the thick subcategory consisting of A_\infty-comodules with zero
cohomology. Analogously one can define the derived category of
A_\infty-contramodules. These are then in general two different
quotient derived categories of the first kind of one derived category
of the second kind (coderived category of A_\infty-comodules =
contraderived category of A_\infty-contramodules).

This is all I have to say now.

with best wishes,
Lenya.