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 (finite-dimensional) 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 above-mentioned 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 Khovanov-like 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 DG-categories, and therefore come equipped with a free DG-enhancement which encodes the higher homotopy structure needed. Yet, if we go a bit further and try to study functors between DG-categories, 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 DG-categories 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 down-to-earth way, by studying "easy" examples (e.g. those obtained through DG-categories) and their Grothendieck groups and hom-spaces of A_∞-functors.