 
Summary: Abstracts
2Motives
Joseph Ayoub
The goal of this talk is to give a reasonable candidate for a category of mixed
2motives over a field k. By "reasonable" we mean a category M2(k) that shares
some of the mirific properties that the conjectural category of mixed 2motives
is expected to enjoy. The plan is as follows. First, we give the definition of
M2(k). Then we explain the ideas behind the verification that M2(k) is an abelian
category.
0.1. Definition. Let k be a perfect field. An object M DMeff (k) is called a
mixed 2motive, or simply a 2motive, if it satisfies the following conditions:
(a) Hi(M) = 0 for i {0, 1, 2};
(b) H0(M) is a 0motivic sheaf;
(c) H1(M) is a 1motivic sheaf;
(d) H2(M) is a 2motivic sheaf which is 1connected;
(e) if L is a nonzero 0motivic sheaf, then L[1] is not a direct factor of M
and Ext1
(H2(M), L) = 0.
The category of mixed 2motives is denoted by M2(k).
Some explanations are needed. Here, DMeff (k) is Voevodsky's category of ef
