Towards an axiomatization of the theory of higher categories
We define a notion of "theory of (1,infty)-categories", and we prove that such a theory is unique up to equivalence.
math.CT↗
arXiv subjects
B. Toen
Publications and source records attributed to B. Toen.
Toward a Galoisian interpretation of homotopy theory
Given any pointed CW complex (X,x), it is well known that the fondamental group of X pointed at x is naturally isomorphic to the automorphism group of the functor which associates to a locally constant sheaf on X its fibre at x. The purpose of this work is to generalize this fact to higher homotopy. For this we introduce the (infinite) category of locally constant stacks on X, and we prove that the loop-space of endomorphisms of its fibre functor at x is naturally equivalent to the loop space of X based at x.
math.AT↗
On motives for Deligne-Mumford stacks
We define and compare two different definitions of Chow motives for Deligne-Mumford stacks, associated with two definitions of Chow rings. The main result we prove is that both categories of motives are equivalent to the usual category of motives of algebraic varieties, but the motives of a given stack associated with both theories are not isomorphic. We will also give some examples of motives associated with some alegbraic stacks.
math.AG↗
Notes on G-theory of Deligne-Mumford stacks
Based on the methods used by the author to prove the Riemann-Roch formula for algebraic stacks, this paper contains a description of the rationnal G-theory of Deligne-Mumford stacks over general bases. We will use these results to study equivariant K-theory, and also to define new filtrations on K-theory of algebraic stacks.
math.AG↗
Riemann-Roch Theorems for Deligne-Mumford Stacks
The goal of this paper is to prove Riemann-Roch type theorems for Deligne-Mumford algebraic stacks. To this end, we introduce a "cohomology with coefficients in representations" and a Chern character, and we prove a Grothendieck-Riemann-Roch theorem for the Riemann-Roch transformation it defines. As a corollary we obtain an Hirzebruch-Riemann-Roch formula for the Euler characteristic of a coherent sheaf, and some formulas for the different topological Euler characteristics of complex algebraic stacks.
math.AG↗