CRM Research Programme for the academic year 2007-2008

Homotopy Theory and Higher Categories

HOCAT 2008. Homotopy Structures in Geometry and Algebra; Derived Categories, Higher Categories

Abstracts

• John Baez (University of California at Riverside):
  Groupoidification
  Abstract: There is a systematic process that turns groupoids into vector spaces and spans of groupoids into linear operators. "Groupoidification" is the attempt to reverse this process, taking familiar structures from linear algebra and enhancing them to obtain structures involving groupoids. Like quantization, groupoidification is not entirely systematic. However, examples show that it is a good thing to try! For example, groupoidifying the quantum harmonic oscillator yields combinatorial structures associated to the groupoid of finite sets. Groupoidifying the q-deformed oscillator yields combinatorial structures associated to finite-dimensional vector spaces over the field with q elements. We can also groupoidify some mathematics related to quantum groups and representations of finite groups. We first describe the basic ideas, and then as many examples as time permits.

• Yves Lafont (Université de la Meditérranée and Institut de Mathématiques de Luminy):
  A folk model structure on ω-Cat
  Abstract: Generalizing Lack's work for 2-categories, we build a model structure on the category of (strict) ω-categories. In fact, we define the generating cofibrations and the weak equivalences. The rest is given by Smith's theorem. This is joint work with François Métayer et Krzysztof Worytkiewicz.

• Clemens Berger (Université de Nice Sophia-Antipolis):
  The lattice path operad
  Abstract: (Joint work with M. Batanin.) We present a coloured operad in sets which permits a unified construction of E_n-operads in monoidal model categories with suitable cosimplicial object. The basic categorical tool consists in a separation of the unary operations of any coloured operad. In particular, our method sheds new light on previous constructions of E_n-operads by Barratt-Eccles, McClure-Smith and Berger-Fresse in the simplicial, topological, and dg-setting. We also recover Tamarkin's construction of a 2-operad action on the category of dg-categories, yielding thereby a "global" proof of the Deligne conjecture on Hochschild cochains.

• Richard Steiner (University of Glasgow):
  Chain complexes and opetopes
  Abstract: Inside the category of strict ω-categories there is a full subcategory with a simple algebraic description in terms of chain complexes. This subcategory includes the ω-categories associated to simplexes (the orientals), the ω-categories associated to cubes, and the simple ω-categories associated to Joyal's category of finite discs; it is also closed under tensor products. This talk gives a slightly more complicated but generally similar algebraic description of opetopes in terms of chain complexes, based on the combinatorial work of Kock, Joyal, Batanin and Mascari. This work should facilitate comparisons between different approaches to higher categories.

• Tom Fiore (Universitat Autònoma de Barcelona):
  The homotopy theory of n-fold categories
  Abstract: When are two categories the same? One possible notion of weak equivalence is an equivalence of categories, another is a functor whose nerve is a weak homotopy equivalence of simplicial sets. As is well known, these distinct notions of weak equivalence between categories have been encoded in model structures by Joyal-Tierney and Thomason. One can ask the same question for Ehresmann's internal categories in Cat: when are two double categories the same? There are several reasonable notions of weak equivalence. Together with Simona Paoli and Dorette Pronk, we have incorporated them into model structures. One intriguing aspect of the Thomason structure on Cat is that it is Quillen equivalent to SSet and hence also Top. In this talk I will also report on recent progress on a model structure for n-fold categories which extends the Thomason structure on Cat. This is joint work with Simona Paoli.

• Charles Rezk (University of Illinois at Urbana):
  Homotopy theory and (∞, 1)-categories
  Abstract: An "infinity topos" is a infinity category with “descent”. We describe how such objects are precisely the infinity categories which admit an internal model.
  

• Paul Balmer (University of California at Los Angeles):
  From spectrum to spectrum
  Abstract: We give a brief introduction and review of the ideas of (tensor) triangular geometry, that is, of the geometric study of tensor triangulated categories, as initiated by the notion of "spectrum" of such an object. We will motivate the computation of the spectrum by applications and then explain how this spectrum relates to the more classical (Zariski) spectrum of some commutative rings naturally appearing in this situation.

• Giordano Favi (Universität Basel):
  Triangular geometry, Rickard idempotents and the telescope conjecture
  Abstract: Triangular geometry is the study of an essentially small tensor triangulated category K via its spectrum Spc(K) as introduced by Balmer. We will explain some features of this topological space and describe a canonical presheaf of triangulated categories on it. We will then move on and assume that K is the category of compact objects of some compactly generated category T, itself equipped with a tensor structure. Standard Bousfield techniques applied to T allow us to define idempotent objects e(V) in T attached to any subset V of Spc(K), generalizing a construction of Rickard. Finally we use this construction and the above presheaf to explain the local behaviour of the so-called telescope conjecture.

• Thomas Hüttemann (Queens University, Belfast):
  Colocalisation and derived categories of toric varieties
  Abstract: A quasi-coherent sheaf on a scheme is determined by a collection of modules, the sections of the sheaf over open affines, together with patching data: the given modules must be compatible wherever the corresponding affines overlap. After passing to chain complexes of sheaves, there are two different interpretations of "compatibility" one recovering the usual notion of chain-complexes of sheaves, the other giving a seemingly weaker notion of "homotopy sheaves" with chain complexes agreeing up to quasi-isomorphism on intersections of affine open sets. By applying techniques from model category theory to certain generalised diagram categories, I will prove two fundamental results:
  1. The "usual" derived category of a (quasi-compact, semi-separated) scheme can be described as the homotopy category of homotopy sheaves. The main ingredient is a strictification construction that allows to replace a homotopy sheaf by an equivalent quasi-coherent sheaf.
  2. In case the variety in question is a smooth complete toric variety, the homotopy sheaves can in turn be characterised using the machinery of Bousfield colocalisation. This gives a new description of the derived category, and provides a purely combinatorial construction for a set of line bundles generating the derived category.

• Mihai Halic (Dhahran):
  Strong exceptional sequences of vector bundles on certain Fano varieties
  Abstract: Exceptional sequences of vector bundles/sheaves over a variety X are special generators of the triangulated category D^b(Coh X). Kapranov proved the existence of tilting bundles over homogeneous varieties. King conjectured the existence of tilting sequences of vector bundles on projective varieties which are obtained as quotients of Zariski open subsets of affine spaces. Although the conjecture does not hold in general, it remains the problem of constructing examples of varieties admitting tilting bundles. For toric varieties, examples of exceptional bundles have been given by Altmann and Hille, and by Costa and Miró-Roig. The goal of this paper is to give further examples of projective varieties carrying exceptional sequences of vector bundles. The varieties are obtained as geometric invariant quotients of affine spaces by linear actions of reductive groups, as in King's conjecture.

• So Okada (Australian National University, Canberra):
  Joyce invariants for K3 surfaces and mock theta functions
  Abstract: For Bridgeland stability conditions of K3 surfaces, we study moduli stacks of semistable objects in terms of Donaldson-Thomas type invariants, which were introduced by Joyce, and mock theta functions, which were introduced by Ramanujan.

• Tom Bridgeland (University of Sheffield):
  Spaces of stability conditions
  Abstract: Spaces of stability conditions are complex manifolds naturally associated to certain triangulated categories. The definition was motivated by ideas in string theory. I will explain the definition and give some examples. If there is time I will also talk about some recent work on wall-crossing behaviour on these spaces.

• Ezra Getlzer (Northwestern University):
  Open-closed topological field theories in two dimensions
  Abstract: A close examination of Harer's triangulation of Teichmüller space establishes a filtration F_k, k = 0, 1 ,..., 2g - 2 + n, of the Harvey bordification, and hence of the modular operad governing topological field theories in two dimensions, with the following property: the pair (F_k, F_k-1) is k-connected. This generalizes a series of theorems, including Turaev's characterization of G-equivariant topological fields theories in two dimensions (this follows from the case k = 1), the Moore-Seiberg theorem (k = 2), and open-closed generalizations of these results, due to Moore and Segal and others.

• Gonçalo Tabuada (Universidade de Nova Lisboa):
  Higher K-theory via universal invariants
  Abstract: Using the formalism of dg categories, we give a conceptual characterization of Quillen-Waldhausen's K-theory.

• Bernhard Keller (Université de Paris VII):
  Quiver mutation and derived equivalence
  Abstract: Quiver mutation is the central ingredient of Fomin-Zelevinsky's theory of cluster algebras. In this talk, we will interpret quiver mutation as the shadow of a derived equivalence between suitable differential graded algebras. We will rely on Derksen-Weyman-Zelevinsky's recent work on mutations of quivers with potentials and on an important construction due to Ginzburg. This is joint work with Dong Yang.

• Julie Bergner (Kansas State University):
  Algebraic applications of the homotopy theory of homotopy theories
  Abstract: Given a model category or, more generally, a simplicial category, one can associate to it a complete Segal space, or simplicial space modeling a homotopy theory. In particular, one can characterize the complete Segal space arising from a model category, up to homotopy. One area of interest from this perspective is considering stable complete Segal spaces associated to derived categories, with the viewpoint that they retain much higher-order homotopy-theoretic information.

• Dmitry Tamarkin (Northwestern University):
  On Swiss cheese 2-operads
  Abstract: The notion of Swiss cheese 2-operad is due to M. Batanin. I will explain how this notion can be used in order to establish an action of Voronov's Swiss cheese operad on the pair (C(A, A); A), where A is a dg-associative algebra and C(A, A) is its Hochschild cochain complex.

• Michael Batanin (Macquarie University, Sydney):
  Crossed interval groups and operations on the Hochschild cohomology
  Abstract: This is joint work with Martin Markl. We introduce crossed interval groups and construct a crossed interval analog IS of the Fiedorowicz-Loday symmetric category ΔS. We prove that the functor F_S(-) of the free IS-extension of a cosimplicial object does not change homotopy type. We then observe that the operad B of all natural operations on the Hochschild cohomology equals F_S(T), where T is an operad whose homotopy type is known. We conclude from these facts that B has the homotopy type of the operad of singular chains on the little disks operad. This shows that up to homotopy the Deligne's conjecture gives the ultimate answer about the natural operations on the Hochschild cohomology.

• Mark Weber (Université de Paris 7):
  Higher operads and multitensors
  Abstract: (Joint work with Michael Batanin and Denis-Charles Cisinski) In the combinatorial approach to defining and working with higher categorical structures, one uses globular operads to say what the structures are in one go. However in the simplicial approaches to higher category theory, one proceeds inductively following the idea that a weak (n + 1)-category is something like a category enriched in weak n-categories. In this talk the inductive content hidden within the globular operadic approach will be revealed. This content will be expressed formally as an equivalence between normalised higher operads in the sense of Batanin, and certain lax monoidal categories. Moreover the algebras of a given operad coincide with categories enriched in the corresponding lax monoidal category. An application of this theory is that one can capture the Gray tensor product formally from the operad for Gray categories, and the possibility exists to extend this example to shed some light on the question of what higher dimensional semi-strict categories may be in general.

• Javier Gutiérrez (CRM):
  Algebras over coloured operads and localization functors
  Abstract: We give sufficient conditions for homotopical localization functors to preserve algebras over coloured operads in monoidal model categories. Our approach encompasses a number of previous results about preservation of structures under localizations, such as loop spaces or infinite loop spaces, and provides new results of the same kind. For instance, under suitable assumptions, homotopical localizations preserve ring spectra (in the strict sense, not only up to homotopy), modules over ring spectra, and algebras over commutative ring spectra, as well as ring maps, module maps, and algebra maps. It is principally the treatment of module spectra and their maps that led us to the use of coloured operads (also called enriched multi-categories) in this context. This is a joint work with C. Casacuberta, I. Moerdijk and R. M. Vogt.

• David Benson (University of Aberdeen):
  Classifying localising subcategories of the stable module category of a finite group
  Abstract: Let G be a finite group and k be a field of characteristic p. Let StMod(kG) denote the stable category of not necessarily finitely generated kG-modules. I shall describe joint work with Krause and Iyengar in which we classify the localising subcategories of StMod(kG).

• Søren Galatius (Stanford University):
  Spaces of graphs
  Abstract: I will discuss the homotopy type of various spaces of graphs in Euclidean space. This is part of a calculation (math/0610216) of the homology of Aut(F_n) in the "stable range". Here F_n is a free group on n generators and Aut(F_n) is its automorphism group. Hatcher and Vogtmann established a stable range: H_k(Aut(F_n)) is independent of n as long as n > 2k + 1.

• Jérôme Scherer (Universitat Autònoma de Barcelona):
  Beyond p-compact groups
  Abstract: This is joint work with Natàlia Castellana and Juan A. Crespo. The best homotopy analogue of a compact Lie group is Dwyer and Wilkerson's notion of p-compact group, i.e. a loop space with mod p finite cohomology. They proved that the mod p cohomology of the classifying space is Noetherian. These objects have now been classified (at all primes). Apart from p-completed classifying spaces of compact Lie groups there are a few exotic p-compact groups. We relax the finiteness assumption and study loop spaces with Noetherian mod p cohomology. We analyze the structure of these objects and prove that they are extensions of p-compact groups by an Eilenberg-Mac Lane space. The cohomology of their classifying spaces is finitely generated as an algebra over the Steenrod algebra and is as small as possible in a certain sense.

• Georg Biedermann (Max-Planck-Institut für Mathematik, Bonn):
  Homotopy n-nilpotent groups
  Abstract: (joint with B. Dwyer) We study the connection between the Goodwillie tower of the identity and the lower central series of the loop group on connected spaces. We define the simplicial theory of homotopy n-nilpotent groups. This notion interpolates between infinite loop spaces and loop spaces. We prove that the set-valued algebraic theory obtained by applying π₀ is the theory of ordinary n-nilpotent groups and that the Goodwillie tower of a connected space is determined by a certain homotopy left Kan extension. We prove that n-excisive functors of the form ΩF have values in homotopy n-nilpotent groups.

• Andy Tonks (London Metropolitan University):
  n-categories and diagonal approximation
  Abstract: This talk explores the categorical expression of higher analogues of a basic map in algebraic topology: the Alexander-Whitney diagonal approximation for the chain complex on a simplicial set X,

  f : C_*X → C_*X ⊗ C_*X.

  Ideally we would like to generalise this to sequences of n-categories C_m,n and maps

  f_m,n : C_m,n → C_m,n ⊗ C_m,n

  where ⊗ is the tensor product of ∞-categories of Steiner, Crans, etc., C_0,n are the simplicial orientals of Street, and C_m+r,n-r is an r-fold 'looping' (à la Baez-Dolan) of C_m,n. For m = 0, the map f_m,n will be analogous to the original Alexander-Whitney map, and for m = 1 to the cubical version used in by Baues. For m = 2 the notions of (weak) n-categories will give insight into the associahedron and permutohedron diagonal approximations of Saneblidze-Umble and, conjecturally, into the problem of iterating the cobar construction.

• Andrew Blumberg (Stanford University):
  Localization theorems in topological Hochschild homology
  Abstract: I will discuss the construction of localization cofiber sequences for the topological Hochschild homology (THH) and topological cyclic homology (TC) of spectral categories. Using a "global" construction of the THH and TC of a scheme in terms of the perfect complexes in a spectrally enriched version of the category of unbounded complexes, these sequences provide cofiber sequences associated to the inclusion of an open subscheme. This is joint work with Michael Mandell.

• Ralph Meyer (Georg-August-Universität Göttingen):
  Homological algebra in triangulated categories
  Abstract: I will present some general machinery for doing homological algebra in triangulated categories, which is mainly due to Beligiannis and Christensen, and which I have found well-adapted to problems in non-commutative topology. The additional input is a given homological functor on the category or, equivalently, the ideal it defines: its kernel on morphisms. This ideal generates a notion of exactness for chain complexes in the category and leads to an Abelian category that approximates the given category. This Abelian category is related to the original category by a spectral sequence, which generalises the Adams spectral sequence in stable homotopy theory.

• Michel Van den Bergh (Universiteit Hasselt):
  Hochschild cohomology and Atiyah classes
  Abstract: We discuss our proof of the fact that the Gerstenhaber algebra structures on the Hochschild cohomology and the tangent cohomology of an algebraic variety are compatible with the HKR map up to twisting with the square root of the Todd genus. This is joint work with Damien Calaque.

• Behrang Noohi (Florida State University):
  Explicit HRS-tilting
  Abstract: For an abelian category A equipped with a torsion pair, we give an explicit description for the abelian category B introduced by Happel-Reiten-Smalø, and also for the categories Ch(B) and D(B). We also describe the dg structure on Ch(B). As a consequence, we find new proofs of certain results of Happel-Reiten-Smalø. The main ingredient is the category of decorated complexes.

• Bertrand Toën (Université Paul Sabatier, Toulouse):
  Chern character, loop spaces and derived algebraic geometry
  Abstract: The purpose of this talk is to present a general framework for the Chern character map, based on techniques from derived algebraic geometry and higher category theory. I will explain in particular how it can be useful in order to define a Chern character map for sheaves of categories rather than sheaves of modules.