"Cross-pollinations of descriptive set theory and homological algebra"Bergfalk, JeffreyThe core of this talk will be a series of works, joint with Martino Lupini and Aristotelis Panagiotopoulos, lying at the interface of descriptive set theory and homological algebra --- works taking their place within a broader contemporary impulse to systematically ``do algebra with topology''. Animating these works is a simple but far-reaching recognition, which is this: many of the classical functors $F:\mathcal{C}\to\mathcal{D}$ of homological algebra and algebraic topology factor through a ``definable version'' of the category $\mathcal{D}$; examples include Cech cohomology, $\mathrm{lim}^1$, and $\mathrm{Ext}$, each viewed as a functor to the category $\mathtt{Ab}$ of abelian groups. The lifts of these functors to a more rigid, ``definable'' version of $\mathtt{Ab}$, namely the category $\mathtt{GPC}$ of groups with a Polish cover, record significantly more information about their sources than their classical counterparts. Moreover, as Lupini has recently shown, the category $\mathtt{GPC}$ is of some algebraic significance in its own right. We will conclude by reviewing the derived category framework in which this significance manifests, and in which these works connect with the subjects of this author's previous Mary Ellen Rudin lecture, ``Cross-pollinations of set theory and algebraic topology''. |
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