A research seminar on algebraic topology and related subjects, intended for interested staff members and advanced students with some background in algebraic topology.
Organisers: Ieke Moerdijk, Urs Schreiber, Moritz Groth.
The plan is to begin with a series of lectures by Moritz Groth on derivators. After that, we plan to study (parts of) Dominic Joyce‘s paper on algebraic geometry over C-∞ rings, and presumably move a bit in the direction of derived differential geometry.
by Moritz Groth
Abstract The theory of derivators – going back to Grothendieck and Heller – is a purely (2-)categorical approach to an axiomatic homotopy theory. The usual passage from a model category (resp. an abelian category) to the underlying homotopy category (resp. derived category) results in a loss of information. The typical defects of triangulated categories (e.g. the non-functoriality of the cone construction) can be seen as a reminiscent of this fact. The basic idea of a derivator is that one should instead simultaneously form homotopy/derived categories of “all” diagram categories and also keep track of the restriction and homotopy Kan extension functors. The aim of these talks is to give an introduction to the theory of derivators and to (hopefully) advertise it as a convenient ‘weakly terminal’ approach to axiomatic homotopy theory. Along the way, we will see that there is a threefold hierarchy of such structures, namely derivators, pointed derivators, and stable derivators. A nice fact about this theory is that ‘stability’ is a property of a derivator as opposed to an additional structure. If time permits we will also cover some monoidal aspects of the theory.