monoidal derived categories

From: Urs Schreiber (Urs.Schreiber_at_uni-essen.de)
Date: 03/14/05 

Date: Mon, 14 Mar 2005 21:04:57 +0100

I am in the process of trying to understand some aspects of derived
categories. In particular, I would like to understand monoidal structures on
derived categories

 D(C)

which come from monoidal C.

Certainly I have not looked at the right place, but what I have found is a
remark in math.KT/0209029 where the author says (in application 2.1)

"Supppose (C,\otimes,e) is a monoidal category such that the underlying
category C is an exact category, and such that \otimes is an exact functor.
Then \otimes can be extended naturally to the bounded derived category
D^b(C) and it is not difficult to see that D^b(C) becomes in this way a
suspended monoidal category, whose unit object is simply e considered as an
object of D^b(C) in the usual way [...]"

My problem is that right now I do find it difficult to see this. Probably I
am missing something.

I'd guess that if

  0 -> E1 -d1-> E2 -d2-> ...

is an object in D^b(C) (i.e. a complex in C) and

  0 -> F1 -d'1-> F2 -d'2-> ...

is another one, then their \otimes-product should be the complex

 0 -> E1 \otimes F1 - d1 \otimes d'1-> E2 \otimes F2 -> ...

Now with e the unit object in C it seems to me that by the above description
the unit object in D^b(C) would be

 0 -> e -> 0 -> 0 -> ....

But that does not seem to be right. How does it really work?

Actually, I have a specific example which is what I am really interested in:

Given a graph and its graph category G, the functors

 f : G -> Vect

form a monoidal category, Rep(G), with the product coming from the tensor
product in Vect.

I would like to understand any monoidal structure induced by that on the
derived category D(Rep(G)) of Rep(G).

Relevant Pages

  • Re: monoidal derived categories
    ... I would like to understand monoidal structures on ... > category C is an exact category, and such that \otimes is an exact functor. ... whose unit object is simply e considered as an ... this left-derived tensor product is associative up to a natural ...
    (sci.math.research)
  • Re: monoidal derived categories
    ... > category C is an exact category, and such that \otimes is an exact functor. ... where d is used for all three differentials. ... > the unit object in D^bwould be ... Dan ...
    (sci.math.research)