In the paper [Som90] p.105, Somekawa conjectures that his Milnor Kgroup K(k, G1, . . . , Gr) attached to semi-abelian varieties G1,. . . ,Gr over a field k is isomorphic to ExtrMk (Z, G1[−1] ⊗ . . . ⊗ Gr[−1]) where Mk is a certain category of motives over k. The purpose of this note is to give remarks on this conjecture, when we take Mk as Voevodsky’s category of motives DM (k) .

Today, Nicky spoke on a few approaches to higher K-theory. Let C be a pointed ∞-category with finite colimits (as in Lurie’s approach) or a category with cofibrations and weak equivalences satisfying certain axioms (as in Waldhausen’s approach). Recall that K0(C) was defined to be the free abelian group on isomorphism classes of objects of C modulo [X] = [X′] + [X′′] whenever we have a pushout ...

The category of crossed complexes gives an algebraic model of CW-complexes and cellular maps. Free crossed resolutions of groups contain information on a presentation of the group as well as higher homological information. We relate this to the problem of calculating non-abelian extensions. We show how the strong properties of this category allow for the computation of free crossed resolutions ...

For any abelian category C satsifying (AB5) over a separated, quasicompact scheme S , we construct a stack of 2-groups GL(C) over the flat site of S. We will give a concrete description of GL(C) when C is the category of quasi-coherent sheaves on a separated, quasi-compact scheme X over S. We will show that the tangent space gl(C) of GL(C) at the origin has a structure as a Lie 2-algebra.

In Chapter 1 we associate with every Cartan matrix of finite type and a non-zero complex number ζ an abelian artinian category FS. We call its objects finite factorizable sheaves. They are certain infinite collections of perverse sheaves on configuration spaces, subject to a compatibility (”factorization”) and finiteness conditions. In Chapter 2 the tensor structure on FS is defined using funct...

The category of crossed complexes gives an algebraic model of CW -complexes and cellular maps. Free crossed resolutions of groups contain information on a presentation of the group as well as higher homological information. We relate this to the problem of calculating non-abelian extensions. We show how the strong properties of this category allow for the computation of free crossed resolutions...

Let G be a reductive, connected algebraic group over an algebraic closure of a finite field. We define a tensor structure on the category of perverse sheaves on G which are direct sums of unipotent character sheaves in a fixed two-sided cell; we show that this is equivalent to the centre with a known monoidal abelian category (a categorification of the J-ring associated to the same two-sided ce...

We prove a conjecture of Rognes that establishes a localization cofiber sequence of spectra K(Z) → K(ku) → K(KU) → ΣK(Z) for the algebraic K-theory of topological K-theory. We deduce the existence of this sequence as a consequence of a devissage theorem identifying the K-theory of the Waldhausen category of Postnikov towers of modules over a connective A∞ ring spectrum R with the Quillen K-theo...

for any smooth projective variety X over a field k. Here CH(X)Q is the Chow group of codimension p algebraic cycles modulo rational equivalence on X with Q-coefficients, and DMM(X) is the bounded derived category of the (conjectural) abelian category of mixed motivic sheaves onX . By the adjoint relation for the structure morphism aX : X → Spec k, the conjecture would be equivalent to the bijec...

Let G denote the full subcategory of topological abelian groups consisting of the groups that can be embedded algebraically and topologically into a product of locally compact abelian groups. We show that there is a full coreflective subcategory S of G that contains all locally compact groups and is *-autonomous. This means that for all G,H in S there is an “internal hom” G −◦H whose underlying...