"subtraction" in derived categories

From: Urs Schreiber (Urs.Schreiber_at_uni-essen.de)
Date: 03/19/05 

Date: 19 Mar 2005 16:15:01 -0500

Last time I asked about notions of multiplication in derived
categories, this time I would like to inquire about notions of
"subtraction" in the following sense:

Consider any derived category D(X) which is triangulated. From the
point of view of applications in physics, the distinguished triangles
in D(X) represent Poincare duals of "Feynman diagrams" depicting
possible reaction paths of objects in D(X).

So the existence of a distinguished triangle

 A \to B \to C \to A[1]

translates in physics to the statement that

1) C may decay into A and B

or

2) A and B may bind to form C

(think of a chemical rection diagram if you like).

Of course from this point of view either of 1) or 2) must be preferred
over the other in that one of these "reactions" occurs "spontaneously"
while the other does not.

Correspondingly, in physics these triangulated derived categories come
equipped with a certain label that tells you which of 1) or 2) is the
"spontaneous" process.

My first question is if such an assignment is known under any name in
the mathematical literature.

Given such an assignment, it is natural to hope that there is what I
shall call a "B-functor" here, namely a functor which sends any object
E in D(X) to an object

  B(E) = \osum_i (A^E)_i

where all the (A^E)_i are objects in D(X) which are "stable" under the
above described "reactions" given by distinguished triangles, i.e.
which are such that there are no labelled distinguished triangles
which describe the "spontaneous decay" of any of the (A^E)_i.

Given such a B-functor it is in turn natural to hope that it is
compatible with direct sums in the sense that

   B( E1 \oplus B( E2 \oplus E3 ) )
= B( B( E1 \oplus E2 ) \oplus E3 )
= B( E1 \oplus E2 \oplus E3 ) .

Has anything like that ever been considered in the mathematical
literature?

One of my motivations to be interested in such a B-functor is that it
would equip D(X) with additive inverses.

For instance, by definition, for any object A in D(X) there is the
distinguished triangle

 A \to A \to 0 \to A[1]

which describes the reaction of A with an "anti-A"=A[1] to produce 0.
If we had a B-functor as above we could "subtract A from A to get 0"
in D(X) by writing

 B( A \oplus A[1] ) \sim 0 .

Has anything like this ever been considered in the literature?