Periodic Reporting for period 1 - CatT (Cohomology and the transfinite)
Reporting period: 2024-03-01 to 2026-02-28
This project's most overarching ambition is to bring two iconic but largely orthogonal approaches to mathematical objects — the set theoretic and the cohomological — into sustained and mathematically productive dialogue. Its four research units intertwine the study of cohomological structures and of infinitary combinatorics in highly original and mutually illuminating ways: in one representative direction, cohomological perspectives dramatically extend our conception of the higher-dimensional and ZFC combinatorics of small cardinals; in another direction, applications of these same combinatorics in derived and homotopical settings substantially extend our conceptions of their significance. We could also put it like this: relations between the small and large, the local and global, are thematic in both set theory and, say, homological algebra, albeit in, at least superficially, very different ways. A number of recent advances suggest deeper connections between fundamental questions in these two settings, and these form our project's defining concern.
Concretely, the project organizes into four interrelated research streams: (1) Cohomological localizations of categories of spaces; (2) The set theory of higher derived limits; (3) Higher walks and the cohomology of the ordinals; (4) Borel definable homological algebra. These streams' constituent problems function, in aggregate, as a laboratory for more methodical minglings of the approaches evoked above, i.e. for developing what have so far been persistent but ad hoc interactions between set theory, homological algebra, and category theory into a robust and programmatically interdisciplinary research field which is of decisive value to each of them. Some of these problems -- as in (1) -- are notorious fifty-year-old open ones; others -- as in portions of (2), (3), and (4) -- reflect such fresh contact between fields as to only really have been conceivable within the past few years.
Concretely, the project organizes into four interrelated research streams: (1) Cohomological localizations of categories of spaces; (2) The set theory of higher derived limits; (3) Higher walks and the cohomology of the ordinals; (4) Borel definable homological algebra. These streams' constituent problems function, in aggregate, as a laboratory for more methodical minglings of the approaches evoked above, i.e. for developing what have so far been persistent but ad hoc interactions between set theory, homological algebra, and category theory into a robust and programmatically interdisciplinary research field which is of decisive value to each of them. Some of these problems -- as in (1) -- are notorious fifty-year-old open ones; others -- as in portions of (2), (3), and (4) -- reflect such fresh contact between fields as to only really have been conceivable within the past few years.
We record the project’s main achievements under its four research headings:
In Research Unit I, regular meetings with Joan Bagaria and Carles Casacuberta resulted in substantial advances on the problem of the existence of cohomological localizations of the stable homotopy category; their further development is a work in progress at the time of writing.
The focus of Research Unit II is the infinitary combinatorics of derived limits, particularly in its interactions with strong homology and with condensed mathematics. Together with Matteo Casarosa, we have answered multiple open questions on the vanishing of the higher derived limits of wide inverse systems (A_kappa in the relevant literature). Two further works on interactions between set theory and condensed mathematics were completed with MSCA support, one, joint with Chris Lambie-Hanson and Jan Saroch, principally on the theme of forcing, and one, joint with Chris Lambie-Hanson, principally on the theme of infinitiary combinatorics and derived functors.
The focus of Research Unit III is on higher walks and the cohomology of the ordinals; the major existing work describing the former was completed during the fellowship. A book-length manuscript treating the latter, joint with Jing Zhang and Chris Lambie-Hanson and building on advances attained during their research visits, is in preparation.
Research Unit IV explored interactions between descriptive set theory and algebraic topology (broadly speaking) on a variety of fronts; in the course of the fellowship, (1) foundational works in Borel definable homological algebra were finalized for publication; (2) joint work with Martino Lupini applying this technology to the homological algebra of locally compact abelian Polish groups was drafted; and (3) joint work with Iian Smythe on the Borel complexity of manifold classification problems was drafted as well.
In Research Unit I, regular meetings with Joan Bagaria and Carles Casacuberta resulted in substantial advances on the problem of the existence of cohomological localizations of the stable homotopy category; their further development is a work in progress at the time of writing.
The focus of Research Unit II is the infinitary combinatorics of derived limits, particularly in its interactions with strong homology and with condensed mathematics. Together with Matteo Casarosa, we have answered multiple open questions on the vanishing of the higher derived limits of wide inverse systems (A_kappa in the relevant literature). Two further works on interactions between set theory and condensed mathematics were completed with MSCA support, one, joint with Chris Lambie-Hanson and Jan Saroch, principally on the theme of forcing, and one, joint with Chris Lambie-Hanson, principally on the theme of infinitiary combinatorics and derived functors.
The focus of Research Unit III is on higher walks and the cohomology of the ordinals; the major existing work describing the former was completed during the fellowship. A book-length manuscript treating the latter, joint with Jing Zhang and Chris Lambie-Hanson and building on advances attained during their research visits, is in preparation.
Research Unit IV explored interactions between descriptive set theory and algebraic topology (broadly speaking) on a variety of fronts; in the course of the fellowship, (1) foundational works in Borel definable homological algebra were finalized for publication; (2) joint work with Martino Lupini applying this technology to the homological algebra of locally compact abelian Polish groups was drafted; and (3) joint work with Iian Smythe on the Borel complexity of manifold classification problems was drafted as well.
The aforementioned manuscript substantially extends, both conceptually and computationally, our knowledge of the sheaf cohomology groups of the ordinals. These form what set theorists might term a graded family of incompactness principles exhibiting a mix of ZFC and ZFC-independent behaviors; as such, they form important invariants of the set theoretic universe itself. A range of new techniques for their evaluation were developed in the course of the fellowship.
The work on manifold classification consisted, first of all, in the provision of a sufficiently general, or modular, approach to parametrizing manifolds (of a variety of types) as standard Borel spaces. Within this framework, a number of complexity computations were recorded (for hyperbolic manifolds, for topological 2-manifolds, for finite-type hyperbolic 3-manifolds, etc.); at least as significant, though, are the problems which this framework renders newly accessible, i.e. the field which this work opens up.
Our work on condensed mathematics involved, on one front, an analysis of its constructions in terms of set theoretic forcing; we showed that the condensation of nice topological spaces X are, in essence, nothing other than “an organized presentation of all forcing names for elements of X”. On another front, we showed that infinitary combinatorics closely related to the cohomology computations referenced above carry implications for the structure of condensed categories; this was by way of an analysis of those combinatorics’ bearing on multiple derived functor computations.
In our work on the category theory of Polish spaces and groups, we analyzed the derived category of locally compact Polish abelian groups, identifying the injective and projective objects of its heart.
The work on manifold classification consisted, first of all, in the provision of a sufficiently general, or modular, approach to parametrizing manifolds (of a variety of types) as standard Borel spaces. Within this framework, a number of complexity computations were recorded (for hyperbolic manifolds, for topological 2-manifolds, for finite-type hyperbolic 3-manifolds, etc.); at least as significant, though, are the problems which this framework renders newly accessible, i.e. the field which this work opens up.
Our work on condensed mathematics involved, on one front, an analysis of its constructions in terms of set theoretic forcing; we showed that the condensation of nice topological spaces X are, in essence, nothing other than “an organized presentation of all forcing names for elements of X”. On another front, we showed that infinitary combinatorics closely related to the cohomology computations referenced above carry implications for the structure of condensed categories; this was by way of an analysis of those combinatorics’ bearing on multiple derived functor computations.
In our work on the category theory of Polish spaces and groups, we analyzed the derived category of locally compact Polish abelian groups, identifying the injective and projective objects of its heart.