A note about some problems with ordinals in "model-independent" homotopy theory: Ordinals
A somewhat roundabout, but "model-independent" proof that filtered colimits of anima are left exact: Filtered colimits are left exact
A brief introduction to adic spaces, written for Babytop at MIT, Spring 2024 : Adic spaces
Notes about equivariant stable homotopy theory from the point of view of spectral Mackey functors (à la Barwick) - I have not proofread them as much as I would like to, so they will most likely change in the (hopefully near) future - any form of feedback is appreciated ! : Equivariant stable homotopy theory
A short incomplete note about full dualizability of presentable (stable) categories (I'd like to know a lot more than what's in there, if you do know more, let me know !): Full dualizability
A note about a proof that n-excisive functors "preserve" finite colimits - more precisely send finite colimits to other types of finite colimits, giving a sort of "formula" for f(colim), when f is only n-excisive : Finite colimits and excisiveness
A short (and incomplete in terms of references) note about a Hochschild homology obstruction to some endomorphism algebra being an F_p-algebra : HH as an obstruction to algebraic structure
A short note about finiteness obstructions in equivariant stable homotopy theory - a short explanation about the difference between finite and compact for G-spectra, based on a computation of the K-theory of compact G-spectra; also mentions quickly Wall's finiteness obstruction: Finiteness obstructions for G-spectra
Notes written for "Topics in Algebraic Topology" at Copenhagen University (some stuff on Picard spaces, on multiplicative infinite loop space machines, and on chromatic homotopy theory)
A note about square zero extensions in the associative setting - in this case, ordinary algebra behaves as well as homotopical algebra and we do have a classification in terms of derivations: Associative square-zero extensions
A note about the universal coefficient theorem with twisted coefficients - I wrote this after a discussion with David Roberts about whether such a thing existed, and a disappointment at the current state of affairs (it appears to only be in the literature in the form of an exercise of Spanier's), even though it requires no extra work with the appropriate point of view : A twisted universal coefficient theorem
Notes I wrote for myself, in an effort to understand some p-adic stuff in homotopy theory; the notes go back to the basics of p-adic stuff as I needed to understand some of the usual story first. The level varies along the document: the beginning is elementary (some knowledge of algebra and topology is required), then categories start being used, and from some point onwards, infinity-categories are used to make everything easier - the document is not fully finished, I intend to modify it (updated 18/08/21) : p-adic stuff
A note about the higher algebraic phenomenon that modding out by n need not make n null, but it does make it nilpotent (updated 21/05 - added a section on how one must be careful in formulating that statement) f^2 = 0
An introduction to algebraic K-theory, written for my "ENS diploma" validation. For simplicity, I focused on direct sum K-theory, but I tried to give a broad overview of applications (in french) : Introduction to algebraic K-theory
An infinity-categorical interpretation of the Bousfield-Kan formula for homotopy colimits, from "first principles" in higher category theory : The Bousfield-Kan formula
A definition of the "center" of a monad, together with a proof that the center of a monad is a Morita invariant, generalizing the classical result for rings (if you know how to do this with oo-categories, let me know !) : Center of a monad
A short and elementary proof that a dualizable idempotent algebra has to be a retract of the unit (I kept forgetting it, so this is just to record it) : Small idempotent algebras
A proof that if you invert an element in a commutative algebra, the result does not depend on whether you inverted it as a commutative algebra, or as an associative algebra: Inverting an element in a commutative algebra
Documents written for the quarantined tea of the maths department of the ENS - these are very short introductions to some mathematical object, and as such they're very far from complete; the format is Introduction-Definition-Examples-A nice theorem, they were supposed to last less than 5 minutes, and so unsurprisingly they lack a lot of detail and precision (in french): Fundamental infinity-groupoid of a space , The infinity-category of spectra
A short note about the notion of a symmetric monoidal adjunction - I prove there the well-known fact that if two adjoint functors have lax symmetric monoidal structures such that the unit and co-unit are symmetric monoidal, then the left adjoint is strong monoidal : Symmetric monoidal adjunctions