Wednesday, September 24, 14:00, 7.527

Simone Virili (Barcelona), A new construction of bounded realization functors.

Abstract:
Let T be a triangulated category, and t=(U,V) a t-structure in T with heart H:=U ∩ V. A (bounded) realization functor is a triangulated functor real_t:D^b(H) → T that extends the obvious inclusion H → T, and there is a classical construction of such functors due to Beilinson. Indeed, whenever there is a so-called f-category structure (FT, f:FT → T), the t-structure t=(,V) lifts to a new t-structure in FT whose heart is equivalent to Ch^b(H). At this point, it is enough to observe that the inclusion Ch^b(H) → FT, followed by the "forgetting the filtration" functor f:FT →T, sends q.isomorphisms to isomorphisms, and so it factors through the projection Ch^b(H) → D^b(H).

In this talk we will show that, whenever T=D(1) is the base of a strong and stable derivator D, all these results can be obtained almost trivially as a consequence of a "coherent" analog of the Dold-Kan correspondence, that can itself be easily deduced from the derivator axioms combined with the stability condition. Moreover, we will show that the realization functors obtained this way are compatible with monidal structures, and that similar results hold for (bounded) weight complex functors T → K^b(H) (where H is the co-heart of a bounded co-t-structure in T).