Date: Fri, 21 May 1999 15:34:58 -0500
To: sasha@math.uchicago.edu
From: maxim@ihes.fr

At the moment I have (I hope) a quite clear understanding
of triangulated categories of "algebraic" origin,
like derived categories of sheaves of modules over algebras,
(and only a vague picture of topological examples, like spectra).
Modulo torsion, the algebraic picture should be sufficient.

First of all, we start with A-infinity categories.

The definition is clean, one define also easily the
deformation theory of
small A-infinity categories (as formal dg manifold with a base point).
By the way, everything works for Z/2Z-graded case as well.
The first question is what to ask about identity morphisms.
Either one can have strict units (as in my notes), or units
only in the homotopy category. There is no serious difference
because 1) deformation theory of a category with strict units, and
of the same category forgetting about units, are equivalent,
 2) if there are homotopy units, then there is a contarctible canonical space
 of ways to introduce units.

What is nice about A-infinity categories is that
for two categories A and B, A small,
there is a new A-ifinity category whose objects are A-infinity
functors from A to B.

Using functors one can define first what does it mean that two
objects X,Y of A are equivalent: an equivalence
is given by an
A-infinity functor from the category with two isomorphic objects
(and no higher Massey operations) to A.

Anagously to what is going on in topology, one defines
when a functor is an equivalence.

The second step is to introduce an idempotent (up to an equivalence)
 construction
on A-infty categories. If A is an A-infinity category, one can define twisted
 complexes in A (morally, results of finitely many
 subsequent extensions of shifts of objects in A.
 Apriori I do not assume that in A there are shifts, or even sums).
 Twisted complexes form a larger A-infinity category,
 Ahat, containg A as a full subcategory.
 Ahathat is naturally equivalent to Ahat.
 A is called closed if A-->Ahat is an equivalence.

 Theorem: Taking H^0 of a complete Ainfinity category we get a triangulated
  category.

  Principle: all triangulated categories of algebaric origin
  are naturally enhanced to an Ainfinity structure.

There are following properties of the completion:
if B is complete then
for any A
Fun(A,B), Fun(Ahat,B), Fun(A,B)hat are all naturally equivalent.

To check , for example the existence of triangles in H^0(complete category A),
you do the following: Triangles are constructed
starting from morphisms. A morphism is a functor
from the quiver A2: *--->*
to A. Because A=Ahat, we have Fun(A2,A)=Fun(A2hat,A).
A2hat is essentially the bounded derived category of finite-dimensionla
representations of A2^opp. This category has a Z/3Z symmetry
(modulo shifts). This gives triangles.
Analogously, octahedrons are constructed form pairs of arrows.
D^b(Repr A3) has 6 indecomposable objects (up to shifts),
they form vertices of the octahedron.

That's basically all.

I thought about technology for finiteness conditions, which seems
to be relevant for the mirror symmetry. Morally, I would like to generalize
D^b(Coh on smooth proper varieties).
In this world all functors have adjoint functors, there is a nice
way to build more complicated things from less complicated.
It seems that everything can  be done using following modification
of the completion: we also add direct summands for projectors
(Karoubi closure). First of all, D^b(Coh smooth projetcive)
is generated (= it is equiv. to teh completion) of the full subactegory
consisting of just one sufficiently large object.
Def: complete (in extended sense) category A is saturated iff
1) all Homs in A are finite dimensional (in total),
2) for any complete B the functor
(A^opp \otimes B)hat--->Fun(A,B) is an equivalence.

Property 2) is enough to check for the universal example
of the identity functor A-->A.
If A is saturated then A is generated just by one object
(i.e. by one finite-dimensional Ainfinity alegbra M).
The property 2) is constructive: morally, it means
that M is a perfect complex in the category of bi-modules.

>From all this it is easy to deduce the following result:
Let X and Y be two smopoth projective varieties.
Then Fun(the Ainfintiy category responsible for D^b(Coh(X),the same for Y)
= this category for X times Y.

--------------------

Also, from this point of view the category of (holonomic)
D-modules is ill. Not all functors have adjoint, there is no Serre
functor, etc.
It is clear why this happens: D-modules are "coherent sheaves"
over quantized cotangent bundle T^* X. It is not compact.
AFter the compactification by the projectivization of T^*X
one get a category with the same finiteness properties as usual coherent
sheaves.