*Planned travel*- July–December 2018
- Newton Institute.
- Homotopy harnessing higher structures

- June 2019
- Universität Regensburg.
- SFB 1085/1 Higher Invariants

- August 2019
- Oberwolfach.
- Homotopy Theory

- January–May 2020
- Mathematical Sciences Research Institute.
- Higher Categories and Categorification

- July 2020
- Institut des Hautes Études Scientifiques.
- Summer school on motivic, equivariant, and non-commutative homotopy theory

- July–December 2018

*Pyknotic objects, I. Basic notions*with Peter Haine- 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- 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- 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*- 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- On the étale homotopy type of a scheme \(X\), we identify a natural stratification, and we prove what we call an
*exodromy equivalence*between representations of this*stratified étale homotopy type*– which can be regarded as a suitable topological category of*points*of \(X\) – and constructible sheaves on \(X\). We also prove a higher categorical form of Hochster's theorem, which ensures that the étale ∞-topos of a coherent scheme can be reconstructed from its stratified homotopy type. A conjecture of Grothendieck – proved by Voevodsky – implies our main*Stratified Anabelian Theorem*: the stratified étale homotopy type is fully faithful as a functor on reduced normal schemes of finite type over a finitely generated field of characteristic zero. That is, a morphism between such schemes can be reconstructed from a Galois-equivariant functor between their associated topological categories.

- On the étale homotopy type of a scheme \(X\), we identify a natural stratification, and we prove what we call an
*Categorifying rationalization*with Saul Glasman, Marc Hoyois, Denis Nardin, and Jay Shah- We construct, for any set of primes \(S\) and any exact ∞-category \(E\), an exact ∞-category \(S^{-1}E\) 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, \(K_n(S^{-1}E)\cong S^{-1}K_n(E)\).

*Fibrations in \(\infty\)-category theory*with Jay Shah*On the fibrewise effective Burnside \(\infty\)-category*with Saul Glasman- Effective Burnside \(\infty\)-categories are the centerpiece of the \(\infty\)-categorical approach to equivariant stable homotopy theory. In this étude, we recall the construction of the twisted arrow \(\infty\)-category, and we give a new proof that it is an \(\infty\)-category, using an extremely helpful modification of an argument due to Joyal-Tierney. The twisted arrow \(\infty\)-category is in turn used to construct the effective Burnside \(\infty\)-category. We employ a variation on this theme to construct a fibrewise effective Burnside \(\infty\)-category. To show that this constuctionworks 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- In this short étude, we observe that the full structure of a recollement on a stable \(\infty\)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- 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 \(\infty\)-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- 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 \(\infty\)-categories and a suitable generalization of the second named author's Day convolution, we endow the \(\infty\)-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*Spectral Mackey functors and equivariant algebraic K-theory (I)*- Advances in mathematics, vol. 304 (2017), no. 2, pp. 646-727
- 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 \(\infty\)-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 \(\infty\)-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.

*Multiplicative structures on algebraic K-theory*- Documenta Mathematica 20 (2015) pp. 859-878
- 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
- In this paper we introduce the notion of an operator category and two different models for homotopy theory of \(\infty\)-operads over an operator category -- one of which extends Lurie's theory of \(\infty\)-operads, the other of which is completely new, even in the commutative setting. We define perfect operator categories, and we describe a category \(\Lambda(\Phi)\) attached to a perfect operator category \(\Phi\) 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 \(A_n\) and \(E_n\) \((1\leq n\leq+\infty)\), as well as a collection of new examples.

*On the \(Q\) construction for exact \(\infty\)-categories*with John Rognes- 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 \(\infty\)-categories and the Theorem of the Heart*- Compositio Mathematica, vol. 151, no. 11 (Nov. 2015), pp. 2160-2186
- 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 \(B_n\) for homotopy pullbacks of \((\infty,k)\)-categories*with Dan Kan- To appear in Homology, Homotopy, and Applications
- We extend the Quillen Theorem \(B_n\) 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 \((\infty, 1)\)- and \((\infty, k)\)-categories, respectively.

*On the algebraic K-theory of higher categories*- Journal of Topology, vol. 9 (2016), pp. 245-347
- 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- We axiomatise the theory of (∞, n)-categories. We prove that the space of theories of (∞, n)-categories is a \(B(\mathbb{Z}/2)^n\). 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 \((\mathbb{Z}/2)^n\).

*From partial model categories to \(\infty\)-categories*with Dan Kan- 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.

- 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.
*\(n\)-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
- 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
- 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
- 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
- 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.

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