Pyknotic objects, I. Basic notions with Peter Haine
[ PDF ] Pyknotic objects are sheaves on the site of compacta.
These provide a convenient way to do algebra and homotopy theory with additional topological information present.
This appears, for example, when trying to contemplate the derived category of a local field.
In this article, we present the basic theory of pyknotic objects,
with a view to describing a simple set of everyday examples.
Exodromy for stacks with Peter Haine
[ PDF ] In this short note we extend the Exodromy Theorem
to a large class of stacks and higher stacks.
We accomplish this by extending the Galois category construction to simplicial schemes.
We also deduce that the nerve of the Galois category of a simplicial scheme is equivalent
to its étale topological type in the sense of Friedlander.
A comment on the vanishing of rational motivic Borel-Moore homology with Denis Nardin
[ PDF ] This note concerns a weak form of Parshin’s conjecture,
which states that the rational motivic Borel-Moore homology of a quasiprojective variety of dimension m
over a finite field in bidegree (s,t) vanishes for s > m+t.
It is shown that this conjecture holds if and only if
the cyclic action on the motivic cohomology of an Artin-Schreier field extension
in bidegree (i,j) is trivial if i < j.
On Galois categories & perfectly reduced schemes
[ PDF ] It turns out that one can read off facts about schemes
up to universal homeomorphism from their Galois categories.
Here we propose a first modest slate of entries in a dictionary between
the geometric features of a perfectly reduced scheme (or morphism of such)
and the categorical properties of its Galois category (or functor of such).
The main thing that makes this possible is Schröer’s total separable closure.
Exodromy with Saul Glasman and Peter Haine
[ PDF ] Let X be a quasicompact quasiseparated scheme.
Write Gal(X) for the category whose objects are geometric points of X
and whose morphisms are specializations in the étale topology.
We define a natural profinite topology on the category Gal(X) that
globalizes the topologies of the absolute Galois groups of the residue fields of the points of X.
One of the main results of this book is that
Gal(X) is a variant of MacPherson's exit-path category suitable for the étale topology:
we construct an equivalence between representations of Gal(X) and constructible sheaves on X.
We show that this exodromy equivalence holds with nonabelian coefficients and with finite abelian coefficients.
More generally, by using the pyknotic/condensed formalism,
we extend this equivalence to coefficients in the category of modules over profinite rings and
algebraic extensions of Ql.
As an exit-path category, the topological category Gal(X) also gives rise
to a new, concrete description of the étale homotopy type of X.
We also prove a higher categorical form of Hochster Duality,
which reconstructs the entire étale topos of a quasicompact and quasiseparated scheme
from the topological category Gal(X).
Appealing to Voevodsky's proof of a conjecture of Grothendieck,
we prove the following reconstruction theorem
for normal varieties over a finitely generated field k of characteristic 0:
the functor that carries X to Gal(X) from normal k-varieties to topological categories over BGk
and equivariant functors that preserve minimal objects is fully faithful.
Categorifying rationalization with Saul Glasman, Marc Hoyois, Denis Nardin, and Jay Shah
Forum of Mathematics, Sigma
[ PDF ] We construct, for any set of primes S and any exact ∞-category E,
an exact ∞-category S-1E of equivariant sheaves on the Cantor space
with respect to an action of a dense subgroup of the circle.
We show that this ∞-category is precisely the result of categorifying division by the primes in S.
In particular, Kn(S-1E) = S-1Kn(E).
Fibrations in ∞-category theory with Jay Shah
In 2016 MATRIX Annals, pp. 17-42
[ PDF ] In this short expository note, we discuss,
with plenty of examples, the bestiary of fibrations in quasicategory theory.
We underscore the simplicity and clarity of the constructions these fibrations make available
to end-users of higher category theory.
On the fibrewise effective Burnside ∞-category with Saul Glasman
[ PDF ] Effective Burnside ∞-categories are
the centerpiece of the ∞-categorical approach to equivariant stable homotopy theory.
In this étude, we recall the construction of the twisted arrow ∞-category, and
we give a new proof that it is an ∞-category,
using an extremely helpful modification of an argument due to Joyal-Tierney.
The twisted arrow ∞-category is in turn used to construct the effective Burnside ∞-category.
We employ a variation on this theme to construct a fibrewise effective Burnside ∞-category.
To show that this constuction works fibrewise,
we introduce a fragment of a theory of what we call marbled simplicial sets, and
we use a yet further modified form of the Joyal-Tierney argument.
A note on stable recollements with Saul Glasman
[ PDF ] In this short étude,
we observe that the full structure of a recollement on a stable ∞category
can be reconstructed from minimal data: that of a reflective and coreflective full subcategory.
The situation has more symmetry than one would expect at a glance.
We end with a practical lemma on gluing equivalences along a recollement.
Cyclonic spectra, cyclotomic spectra, and a conjecture of Kaledin with Saul Glasman
[ PDF ] With an explicit, algebraic indexing (2,1)-category,
we develop an efficient homotopy theory of cyclonic objects:
circle-equivariant objects relative to the family of finite subgroups.
We construct an ∞-category of cyclotomic spectra as the homotopy fixed points
of an action of the multiplicative monoid of the natural numbers on the category of cyclonic spectra.
Finally, we elucidate and prove a conjecture of Kaledin on cyclotomic complexes.
Spectral Mackey functors and equivariant algebraic K-theory (II) with Saul Glasman and Jay Shah
Tunisian journal of mathematics, vol. 2 (2020), no. 1, pp. 97-146
[ PDF ] We study the higher algebra of spectral Mackey functors,
which the first named author introduced in Part I of this paper.
In particular, armed with our new theory of symmetric promonoidal ∞-categories and
a suitable generalization of the second named author’s Day convolution,
we endow the ∞-category of Mackey functors with a well-behaved symmetric monoidal structure.
This makes it possible to speak of spectral Green functors for any operad O.
We also answer a question of A. Mathew, proving that the algebraic K-theory of group actions
is lax symmetric monoidal.
We also show that the algebraic K-theory of derived stacks provides an example.
Finally, we give a very short, new proof of the equivariant Barratt-Priddy-Quillen theorem, which states that
the algebraic K-theory of the category of finite G-sets is simply the G-equivariant sphere spectrum.
Dualizing cartesian and cocartesian fibrations with Saul Glasman and Denis Nardin
Theory and Applications of Categories, vol. 33 (2018), no. 4, pp. 67-94
[ PDF ] In this technical note,
we proffer a very explicit construction of the dual cocartesian fibration of a cartesian fibration, and
we show they are classified by the same functor to the category of ∞-categories.
Spectral Mackey functors and equivariant algebraic K-theory (I)
Advances in mathematics, vol. 304 (2017), no. 2, pp. 646-727
[ PDF ] Spectral Mackey functors are homotopy-coherent versions of
ordinary Mackey functors as defined by Dress.
We show that they can be described as excisive functors on a suitable ∞-category,
and we use this to show that universal examples of these objects are given by algebraic K-theory.
More importantly, we introduce the unfurling of certain families of Waldhausen ∞-categories bound together
with suitable adjoint pairs of functors;
this construction completely solves the homotopy coherence problem that
arises when one wishes to study the algebraic K-theory of such objects as spectral Mackey functors.
Finally, we employ this technology to lay the foundations of equivariant stable homotopy theory
for profinite groups and to study fully functorial versions of
A-theory, upside-down A-theory, and the algebraic K-theory of derived stacks.
Regularity of structured ring spectra and localization in K-theory with Tyler Lawson
[ PDF ] We identify a regularity property for structured ring spectra, and
with it we prove a natural analogue of Quillen’s localization theorem for algebraic K-theory
in this setting.
Multiplicative structures on algebraic K-theory
Documenta Mathematica 20 (2015) pp. 859-878
[ PDF ] Algebraic K-theory is the stable homotopy theory of homotopy theories,
and it interacts with algebraic structures accordingly.
In particular, we prove the Deligne Conjecture for algebraic K-theory.
From operator categories to higher operads
Geometry and Topology 22 (2018) pp. 1893-1959
[ PDF ] In this paper we introduce the notion of an operator category
and two different models for homotopy theory of ∞-operads over an operator category --
one of which extends Lurie’s theory of ∞-operads, the other of which is completely new,
even in the commutative setting. We define perfect operator categories, and
we describe a category attached to a perfect operator category that provides Segal maps.
We define a wreath product of operator categories and a form of the Boardman-Vogt tensor product that lies over it.
We then give examples of operator categories that provide universal properties for the operads
An and En (1 ≤ n ≤ +∞), as well as a collection of new examples.
On the Q construction for exact ∞-categories with John Rognes
[ PDF ] We prove that the K-theory of an exact quasicategory
can be computed via a higher categorical variant of the Q construction.
This construction yields a quasicategory whose weak homotopy type is a delooping of the K-theory space.
We show that the direct sum endows this homotopy type with the structure of a infinite loop space,
which agrees with the canonical one.
Finally, we prove a proto-devissage result, which gives a necessary and sufficient condition for
a nilimmersion of stable quasicategories to be a K-theory equivalence.
In particular, we prove that a well-known conjecture of Ausoni and Rognes is equivalent to
the weak contractibility of a particular quasicategory.
On exact ∞-categories and the Theorem of the Heart
Compositio Mathematica, vol. 151, no. 11 (Nov. 2015), pp. 2160-2186
[ PDF ] We introduce a notion of exact quasicategory,
and we prove an analogue of Amnon Neeman’s Theorem of the Heart for Waldhausen K-theory.
Quillen theorems Bn for homotopy pullbacks of (∞,k)-categories
with Dan Kan
[ PDF ] We extend the Quillen Theorem Bn
for homotopy fibers of Dwyer, et al. to similar results for homotopy pullbacks
and note that these results imply similar results for zigzags in the categories of relative categories and
k-relative categories, not only with respect to their Reedy structures but also their Rezk structure,
which turns them into models for the theories of (∞, 1)- and (∞, k)-categories, respectively.
On the algebraic K-theory of higher categories
Journal of Topology, vol. 9 (2016), pp. 245-347
[ PDF ] We prove that Waldhausen K-theory,
when extended to a very general class of quasicategories, can be described as a Goodwillie differential.
In particular, K-theory spaces admit canonical (connective) deloopings, and
the K-theory functor enjoys a universal property.
Using this, we give new, higher categorical proofs of both the additivity and fibration theorems of Waldhausen.
As applications of this technology, we study the algebraic K-theory of associative ring spectra
and spectral Deligne-Mumford stacks.
On the unicity of the theory of higher categories with Chris Schommer-Pries
[ PDF ] We axiomatise the theory of (∞, n)-categories.
We prove that the space of theories of (∞, n)-categories is a B(Z/2)n.
We prove that virtually all known purported models of (∞, n)-categories satisfy our axioms, whence
they are all equivalent, in a manner that is unique up to this action of (Z/2)n.
From partial model categories to ∞-categories with Dan Kan
[ PDF ] In this note we consider partial model categories,
by which we mean relative categories that satisfy a weakened version of the model category axioms
involving only the weak equivalences. More precisely, a partial model category will be
a relative category that has the two out of six property and admits a 3-arrow calculus.
We then show that Charles Rezk’s result that the simplicial space obtained from a simplicial model category
by taking a Reedy fibrant replacement of its simplicial nerve is a complete Segal space
also holds for these partial model categories.
We also note that conversely every complete Segal space is Reedy equivalent to
the simplicial nerve of a partial model category and in fact of a homotopically full subcategory of
a category of diagrams of simplicial sets.
n-relative categories: A model for the homotopy theory of n-fold homotopy theories
with Dan Kan
Homotopy, Homology, and Applications, vol. 15 (2013), no. 2, pp. 281-300
[ PDF ] We introduce, for every positive integer n,
the notion of an n-relative category and show that the category of the small n-relative categories is a model for
the homotopy theory of n-fold homotopy theories, i.e., homotopy theories of ... of homotopy theories.
A characterization of simplicial localization functors and a discussion of DK equivalences with Dan Kan
Indagationes Mathematicae 23 (2012), pp. 69-79
[ PDF ] In a previous paper we lifted Charles Rezk’s
complete Segal model structure on the category of simplicial spaces to a Quillen equivalent one on the category of
«relative categories». Here, we characterize simplicial localization functors among relative functors
from relative categories to simplicial categories as any choice of homotopy inverse
to the delocalization functor of Dwyer and the second author.
We employ this characterization to obtain a more explicit description of the weak equivalences in the model category
of relative categories mentioned above by showing that these weak equivalences are exactly the DK-equivalences,
i.e., those maps between relative categories which induce a weak equivalence between their simplicial localizations.
Relative categories: Another model for the homotopy theory of homotopy theories
with Dan Kan
Indagationes Mathematicae 23 (2012), pp. 42-68
[ PDF ] We lift Charles Rezk’s complete Segal space model structure
on the category of simplicial spaces to a Quillen equivalent one on the category of relative categories.
On left and right model categories and left and right Bousfield localizations
Homology, Homotopy and Applications, vol. 12 (2010), no. 2, pp. 245-320
[ PDF ] We verify the existence of left Bousfield localizations and
of enriched left Bousfield localizations, and we prove a collection of useful technical results
characterizing certain fibrations of (enriched) left Bousfield localizations.
We also use such Bousfield localizations to construct a number of new model categories,
including models for the homotopy limit of right Quillen presheaves, for Postnikov towers in model categories, and
for presheaves valued in a symmetric monoidal model category satisfying a homotopy-coherent descent condition.
We then verify the existence of right Bousfield localizations of right model categories, and
we apply this to construct a model of the homotopy limit of a left Quillen presheaf as a right model category.
Notes
CAVEAT LECTOR
Euler’s Gamma function and the field with one element
[ PDF ] A topics course at MIT on a web of ideas surrounding
the Gamma function and the field with one element.
The Bass-Quillen conjecture
[ PDF ] Notes for a talk at Hopkins’s Thursday Seminar at Harvard giving
Lindel’s proof of the geometric case of the Bass-Quillen conjecture.
The fundamental groupoid and the Postnikov tower
[ PDF ] Notes for a Moore-technique undergraduate course at MIT.
(In particular, there are no proofs in this document.)
Deligne cohomology
[ PDF ] Notes for a talk at Hopkins’s Thursday Seminar at Harvard
introducing Deligne cohomology.
Borel’s computation of the cohomology of SLn(OF)
[ PDF ] An old attempt to understand Borel’s computation.
The Atiyah-Hirzebruch spectral sequence for algebraic K-theory
[ PDF ] Notes for a talk at Hopkins’s Thursday Seminar at Harvard
describing Grayson’s approach to the weight filtration on algebraic K-theory.