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Geometric Topology Seminar
September 24, 2018 15:30–17:30, Moscow, Math Department of the HSE (Usachyova, 6), room 212

Corrected derived limits, or How to expel set theory from algebraic topology

S. A. Melikhov

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Abstract: We address the following problem
"Does direct limit ($colim$) commute with inverse limit ($\lim$) when higher derived limits ($\lim^p$) are taken into account?"
in a natural topological context: $\lim$ and $colim$ are applied to the homology (or cohomology) of finite simplicial complexes which approximate in two ways (as an inverse system of direct systems, or as a direct system of inverse systems) a given separable metrizable space $X$.
It is likely that this problem (which is stated precisely in terms of a Bousfield-Kan spectral sequence) cannot be solved in the usual set theory ZFC. At least, this is known to be so for some specific (very simple) spaces $X$. But we show that the problem has positive solution (for any $X$ in the case of cohomology, and for finite-dimensional $X$ in the case of homology), if the derived functors $\lim^p$ are "corrected" so as to take into account a natural topology on the indexing poset. The correction involves sheaves (particularly, Leray sheaves), as well as probability measures and measurable functions. Further details can be found in
The theory of the "corrected" $\lim^p$ is still in its infancy, and there are a lot of very basic problems about them (see the last page in the linked paper) which could well be the subject of a course/diploma/thesis work by any student who has some idea about sheaves and homology algebra.

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