Clark Barwick

  • Categorifying rationalization with Saul Glasman, Marc Hoyois, Denis Nardin, and Jay Shah
    • [ PDF ]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, \(K_n(S^{-1}E)\cong S^{-1}K_n(E)\).
  • Parametrised higher category theory and higher algebra with Emanuele Dotto, Saul Glasman, Denis Nardin, and Jay Shah
    • A general introduction by Barwick, Dotto, Glasman, Nardin, and Shah
      • [ PDF ]We introduce the study of parametrised higher category theory and parametrised higher algebra, and we describe the main theorems of the series of Exposés that make up the monograph.
    • Exposé I – Elements of parametrised higher category theory by Barwick, Dotto, Glasman, Nardin, and Shah
      • [ PDF ]We introduce the basic elements of the theory of parametrised ∞-categories and functors between them. These notions are defined as suitable fibrations of ∞-categories and functors between them. We give as many examples as we are able at this stage. Simple operations, such as the formation of opposites and the formation of functor ∞-categories, become slightly more involved in the parametrised setting, but we explain precisely how to perform these constructions. All of these constructions can be performed explicitly, without resorting to such acts of desperation as straightening. The key results of this Exposé are: (1) a universal characterization of the \(T\)-∞-category of \(T\)-objects in any ∞-category, (2) the existence of an internal Hom for \(T\)-∞-categories, and (3) a parametrised Yoneda lemma.
    • Exposé II – Parametrised limits and colimits by Shah
      • [ PDF ]We develop foundations for the category theory of ∞-categories parametrised by a base ∞-category. Our main contribution is a theory of parametrised homotopy limits and colimits, which recovers and extends the Dotto–Moi theory of \(G\)-colimits for \(G\) a finite group when the base is chosen to be the orbit category of \(G\). We apply this theory to show that the \(G\)-∞-category of \(G\)-spaces is freely generated under \(G\)-colimits by the contractible \(G\)-space, thereby affirming a conjecture of Mike Hill.
    • Exposé III – The stratified geometry of parametrised homotopy theory by Barwick and Glasman
      • [ PDF ]In this exposé we introduce the theory of orbital \(\infty\)-categories and
    • Exposé IV – Stability with respect to an orbital ∞-category by Nardin
      • [ PDF ]In this exposé we develop a theory of stability for \(T\)-categories (presheaf of categories on the orbit category of \(T\)), where \(T\) is a finite group. We give a description of Mackey functors as \(T\)-commutative monoids exploit it to characterise \(T\)-spectra as the \(T\)-stabilization of \(T\)-spaces. As an application of this we provide an alternative proof of a theorem by Guillou and May.
  • Fibrations in \(\infty\)-category theory with Jay Shah
    • [ 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 \(\infty\)-category with Saul Glasman
    • [ PDF ]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
    • [ PDF ]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
    • [ 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 \(\infty\)-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
    • [ 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 \(\infty\)-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
    • [ PDF ]In this technical note, we proffer a very explicit construction of the dual cocartesian fibration \(p^{\vee}\) of a cartesian fibration \(p\), and we show they are classified by the same functor to \(Cat_{\infty}\).
  • 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 \(\infty\)-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.
  • 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) 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
      To appear in Geometry and Topology
    • [ PDF ]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
    • [ 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 \(\infty\)-categories and the Theorem of the Heart
  • Quillen theorems \(B_n\) for homotopy pullbacks of \((\infty,k)\)-categories with Dan Kan
      To appear in Homology, Homotopy, and Applications
    • [ PDF ]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
    • [ 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 homotopy theory of higher categories with Chris Schommer-Pries
    • [ PDF ]We propose four axioms that a quasicategory should satisfy to be considered a reasonable homotopy theory of \((\infty,n)\)-categories. This axiomatization requires that a homotopy theory of \((\infty,n)\)-categories, when equipped with a small amount of extra structure, satisfies a simple, yet surprising, universal property. We further prove that the space of such quasicategories is homotopy equivalent to \((RP^{\infty})^n\). In particular, any two such quasicategories are equivalent. This generalizes a theorem of Toën when \(n = 1\), and it verifies two conjectures of Simpson. We also provide a large class of examples of models satisfying our axioms, including those of Joyal, Kan, Lurie, Rezk, and Simpson.
  • From partial model categories to \(\infty\)-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
  • 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
  • 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.
  • Hunters and farmers
    • [ PDF ]A short talk for the University of Edinburgh undergraduate mathematics society, MathSOC.
  • 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 \(\mathbf{F}_1\).
  • The Bass-Quillen conjecture
    • [ PDF ]Notes for a talk at Hopkins's Thursday Seminar at Harvard giving a 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 \(SL_n(O_F)\)
    • [ PDF ]An old attempt to relate Borel's astonishing 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.
  • 121 Exercises on locally compact abelian groups
    • [ PDF ]A problem set that got way out of hand.
  • Descent problems for algebraic K-theory
    • [ PDF ]Notes for an old talk motivating some aspects of the Carlsson conjecture.
  • Applications of derived algebraic geometry to homotopy theory
    • [ PDF ]Notes for a minicourse on derived algebraic geometry in Salamanca.
  • Operator categories, multicategories, and homotopy-coherent algebra
    • [ PDF ]My very first attempt (ca. 2007) to set right the foundations of homotopy coherent algebra. Now superceded by the Lurie juggernaut and (to a lesser extent) my text on operator categories.
  • Topological rigidification of schemes
    • [ PDF ]Notes on how to invert universal homeomorphisms within the category of schemes.
  • \(D\)-crystals
    • [ PDF ]Notes from a seminar I gave long ago (2006?) on \(D\)-crystals. Now completely superceded by Gaitsgory, of course.
  • Notes on mathematical writing
    • [ PDF ]A short letter on mathematical style, targeted toward Ph.D students and other mathematicians at the beginning of their careers.
  • The future of homotopy theory
    • [ PDF ]A short letter on the health of homotopy theory as a discipline, targeted toward the new young leaders in homotopy theory today.