Approaching higher limits from homotopy theory

Abstract: Higher limits are the higher derived functors of the inverse limit construction for functors taking values in abelian groups. Classically, they are computed using tools from homological algebra.

In algebraic topology, the cohomology of a homotopy colimit of a diagram spaces can be approached via a spectral sequence whose $E_2$ page consists precisely of the higher limits of the functor obtained from applying cohomology to the diagram of spaces. In particular, if the higher limits vanish, then the cohomology of the homotopy colimit is just the inverse limit of the cohomologies. There are many vanishing results, for example the Mittag-Leffler property [2, Section 3.5] or the pseudo-projectivity property [1].

We study the case where the category $P$ is a poset with an order preserving map $d: P\to\mathbb N$. If we consider the injective model category on the functor category Fun($P$, Ch(Ab)), a functor is pseudo-projective if it is cofibrant.

In this talk we will show how we can use the techniques from model categories, inspired by homotopy theory, to describe higher limits in this situation when the indexing category is a poset. We will give explicit bounds for the vanishing of higher limits in terms of properties of the functor improving previous results.

References:

[1] Diaz Ramos, A., A family of acyclic functors, Journal of Pure and Applied Algebra, 213.5 (2009), 783–794.

[2] Weibel, C., An Introduction to Homological Algebra, Cambridge University Press, 1994.

Mathematics

Audience: researchers in the topic

Opening Workshop (IRP Higher Homotopy Structures 2021, CRM-Bellaterra)