Groupe de Travail: Categorification
Motivation:
Categorification consists in the search for higher structures (i.e. categories) having known objects as "shadows".
This additional layer of structure gives very often new insights about the objects we start with.
One common example is given by the semiring of natural numbers which can be seen as the shadow of the category of
(finitedimensional) vector spaces.
Indeed, forgetting the higher structure, that is, looking only at vector spaces up to isomorphism,
keeps only the dimensions, which are natural numbers.
In this context, the operations of addition and multiplication can be interpreted
as the shadow of the product and coproduct in the abovementioned category.
Another example is given by the homology theories in algebraic topology which can be interpreted as
categorifications of the Euler characteristic (this includes the case of the celebrated Khovanovlike homology theories for
knots and links).
Categorification can be achieved in many ways, depending on the way we want to look at the shadow of our higher structure.
For our purposes, we consider Grothendieck groups of additive, abelian or triangulated categories.
Most examples we know from topology and higher representation theory are produced using additive or abelian categories, but
in more and more cases we have the need to use triangulated structures, which are infamous for having a "bad behavior".
In most cases we know these triangulated categories are constructed as homotopy of derived DGcategories, and therefore come equipped with a
free DGenhancement which encodes the higher homotopy structure needed.
Yet, if we go a bit further and try to study functors between DGcategories, it seems to us that this notion is still
not enough for our purposes.
It has become apparent to us that it should be interesting to view DGcategories as A_{∞}structures,
where we have more space to construct interesting A_{∞}functors.
The purpose of this working group is to study A_{∞}categories and how they relate to other known structures.
We plan to do so in a completely downtoearth way, by studying "easy" examples (e.g. those obtained through DGcategories)
and their Grothendieck groups and homspaces of A_{∞}functors.
Calendar:
Date/Locale  Speaker  Topic  References 

20/09/2017 14h30 CYCL08 
Grégoire Naisse  Introduction & organization  
04/10/2017 14h30 CYCL07 
Elia Rizzo  Triangulated categories

Lectures on derived and triangulated categories (B. Noohi) Triangulated categories (A. Neeman) 
11/10/2017 14h30 CYCL08 
Grégoire Naisse  DGcategories I

On differential graded categories (B. Keller) Deriving dgcategories (B. Keller) 
25/10/2017 14h30 CYCL08 
Grégoire Naisse  DGcategories II

On differential graded categories (B. Keller) Lectures on dgcategories (B. Toën) 
31/10/2017 14h00 CYCL02 
Elia Rizzo  A_{∞}categories I

Introduction to A_{∞}algebras and modules (B. Keller) A_{∞}algebras, modules and functor categories (B. Keller) Fukaya Categories and PicardLefschetz Theory (P. Seidel) 
7/11/2017 14h00 CYCL02 
Elia Rizzo Grégoire Naisse  A_{∞}categories II

Pretriangulated A_{∞}categories (Bespalov et al) 
4/12/2017 16h15 CYCL09b 
Elia Rizzo Grégoire Naisse  (∞,1)categories

What is an A_{∞}category ? (J. Lurie) Higher Algebra (J. Lurie) 