Notes

Corrects an error in a proof in [arXiv:1807.03281] and [arXiv:1904.09966] about covering pro-objects. The correction relies on a non-obvious closure property of classes of maps in an ∞-category that are closed under composition and pullback.

Handwritten notes for my minicourse at the Étale cohomology and étale homotopy conference in Frankfurt.

For an overview talk at the Banff workshop Enumerative Geometry Beyond Spaces.

Handwritten notes for my lectures at the Copenhagen Masterclass on Exit Paths and Stratified Homotopy Types.

For a talk that I gave at the Lloyd Roeling conference, UC Davis Algebraic Geometry Seminar, and Syzygies and Mirror Symmetry Virtual Seminar. To simplify the presentation, in the talk I ignored the difference between sheaves and hypersheaves. In these slides, the word sheaf should be interpreted as hypersheaf.

[arXiv:2210.00186]

Notes explaining why the functor sending a compact Hausdorff space K to the ∞-category of Postnikov complete sheaves on K is a sheaf on the site of compact Hausdorff spaces and finite jointly surjective families. We use this to show that condensed cohomology and sheaf cohomology agree for locally compact Hausdorff spaces.

For a talk for UC Berkeley's Mathematics Undergraduate Student Association (MUSA).

Extended notes from my talk at the 202One Talbot Workshop on Ambidexterity. These notes explain how to use Lurie's work on elliptic cohomology to recover character isomorphisms in chromatic homotopy theory due to Hopkins–Kuhn–Ravenel and Stapleton.

Notes from Hiro Tanaka's talk on Fukaya categories at the Viva Talbot! Workshop.

The purpose of this note is to generalize the following observation: given an (n+1)-connected map of spaces f: X → Y, the induced morphism Lf: LX → LY on free loop spaces is n-connected. We show that in an ∞-topos, cotensoring with a finite space with cells in dimensions ≤m decreases connectedness of morphisms by m.

Handwritten notes for a talk for the Spring 2021 Thursday Seminar on Redshift in Algebraic K-theory.

The purpose of this note is to prove a very general splitting result for free loop objects. For example, we show that in an ∞-category C with finite limits, given an A₂-algebra G in C that has inverses in an appropriate sense, the loop object LG splits as LG ≃ G × ΩG. We also sketch Aguadé & Ziller's proof that the free loop space LSⁿ splits if and only if n = 0, 1, 3, or 7.

A point of confusion for a lot of people learning about ∞-categories is why the different approaches to ∞-categorical enhancements of derived categories (via model categories, dg categories, and ∞-categorical localizations) produce the same answer. The explanation is scattered throughout §1.3 of Lurie's book Higher Algebra. Since so many people have asked me about this, I've collected the relevant references in one place.

Note explaining Bégueri's resolution of a commutative, finite locally free group scheme by smooth affine group schemes. This resolution is a technical tool used in Česnavičius and Scholze's paper Purity for flat cohomology to transfer questions about fppf cohomology to questions about étale cohomology.

There used to be notes for some talks at the Fall 2019 Juvitop Seminar on the foundations of differential cohomology theories as sheaves on the category of manifolds. They're now in an improved form in Part I of the book Differential Cohomology: Categories, Characteristic Classes, and Connections!

There used to be notes for some talks at the Fall 2019 Juvitop Seminar on the Segal–Sugawara Construction. They're now in an improved form in Chapter 23 of the book Differential Cohomology: Categories, Characteristic Classes, and Connections!

Notes for a talk at the Fall 2019 Thursday Seminar introducing the basic material needed to understand Elmanto, Hoyois, Khan, Sosnilo, & Yakerson's work on motivic infinite loop spaces.

Some informal notes sketching Ayala and Francis' proof of Nonabelian Poincaré Duality.

Notes on §4 of Hopkins and Lurie's paper Ambidexterity in K(n)-local stable homotopy theory phrased for those not familiar with the fibrational perspective.

Notes on the fundamental theorems of étale cohomology following Chapter VI of Milne's Étale Cohomology.

Notes on the basic setup of Goodwillie calculus in the ∞-categorical setting.

Notes for a talk at the MIT graduate student lunch seminar giving a very broad overview of motivic cohomology.

Notes for a talk given at the 2016 Kan Seminar.

arXiv:1604.06812

A note on a technical result about defining enhanced factorization systems on functor 2-categories.