Title: | Not precisely a seminar, followed by not precisely coffee |
This is the last activity of the special year |
Speaker: | Vincent Franjou (Nantes) |
Title: | Finite generation of invariants |
Abstract: | A classic problem in invariant theory, often referred to as Hilbert's 14th problem, asks, when a group acts on a finitely generated algebra by algebra automorphism, whether the ring of invariants is still finitely generated. I shall discuss joint work with W. van der Kallen treating the problem for a Chevalley group over an arbitrary base. If time allows, I shall also present progress on the corresponding conjecture for rational cohomology, recently proven over fields by W. van der Kallen and A. Touzé. |
References: | V. Franjou and W. van der Kallen, Power reductivity over an arbitrary base. Preprint, ArXiv:0806.0787. |
Speaker: | Clemens Berger (Nice) |
Title: | Generalised Reedy categories and crossed groups |
Abstract: | (This is joint work with Ieke Moerdijk). We extend the classical notion of a Reedy category so as to allow non-trivial automorphisms. Our extension includes many important examples occuring in topology such as Segal's category $\Gamma$, or the total category of a crossed simplicial group such as Connes' cyclic category $\Lambda$. We show that for any generalized Reedy category $\mathbb{R}$ and any (cofibrantly generated) Quillen model category $\mathcal {E}$, the functor category $\mathcal{E}^\mathbb{R}$ carries a canonical model structure of Reedy type. We then give sufficient conditions on $\mathcal{E}$ and $\mathbb{R}$ in order to get a monoidal model structure on the functor category. |
References: |
Speaker: | Ross Street (Macquarie) |
Title: | Categorical aspects of Mackey functors |
Abstract: | Mackey and Green functors were introduced into group representation theory in the 1970s. The purpose of the talk is to emphasise categorical aspects of the theory of Mackey functors and to use enriched categories to guide the development of that theory. Every representation of a group gives rise to a Mackey functor. We show that the category of finite-dimensional Mackey functors is star-autonomous. |
References: | E. Panchadcharam and R. Street, Mackey functors on compact closed categories. J. Homotopy Rel. Struct. 2 (2007) 261-293; ArXiv:0706.2922 |
Speaker: | Bruno Vallette (Nice) |
Title: | What can we do with cofibrant resolutions? |
Abstract: | Last time, I explained how to make cofibrant resolutions of operads and props explicit. This time, I will describe what we can do with this. First, I will make the deformation theory of morphisms of props explicit. It is represented by a cotangent complex which has nice properties. Then, I will prove the transfer of structure (homological perturbation lemma) in full generality. |
References: | B. Vallette, A Koszul duality for props,
Trans. Amer. Math. Soc. 359 (2007), 4865-4943. (Available online.)
S. Merkulov and B. Vallette, Deformation theory of representation of prop(erad)s, (to appear in Crelle). (Available online.) |
Speaker: | Paul Arne Østvær (Oslo) |
Title: | Homotopy theory of C*-algebras |
Abstract: | We construct from ground up a homotopy theory of C*-algebras.
This is achieved in parallel with the development of classical homotopy
theory by
first introducing an unstable model structure and second a stable model
structure.
The theory makes use of a full fledged import of homotopy theoretic
techniques
into the subject of C*-algebras.
The spaces in C*-homotopy theory are certain hybrids of functors represented by C*-algebras and spaces studied in classical homotopy theory. In particular, we employ both the topological circle and the C*-algebra circle of complex-valued continuous functions on the real numbers which vanish at infinity. By using the inner workings of the theory we may stabilize the spaces by forming spectra and bispectra with respect to either one of these circles or their tensor product. These stabilized spaces or spectra are the objects of study in stable C*-homotopy theory. The stable homotopy category of C*-algebras gives rise to invariants such as stable homotopy groups and bigraded homology and cohomology theories. We work out examples related to the emerging field of noncommutative motives and introduce a new type of K-theory of C*-algebras which is not periodic. |
References: | No references exist. |
Speaker: | Carles Casacuberta (UB) |
Title: | Definable categories and supercompact cardinals |
Abstract: | In joint work with Bagaria, Mathias, and Rosický [1, 2], we prove the following result: Suppose that there is a proper class of supercompact cardinals. If K is a locally presentable category and L is an absolute orthogonality class in K, then L is reflective (i.e., the image of a localization). Recall that an orthogonality class is the class of objects X such that K(f, X) is bijective on hom-sets for all f in some given class of morphisms. The notion of absoluteness is crucial here. A class L is absolute if membership of L is defined by a formula of the language of set-theory without unbounded quantifiers. This result can be extended in at least two directions. On one hand, a variant of the same result should be true in homotopy categories of combinatorial Quillen model categories, thus ensuring the existence of localizations in very general situations, as in [3]. On the other hand, the assumption that the class L be absolute can be weakened. In fact there is a hierarchy of large-cardinal principles VP(n), converging to Vopenka's Principle, whose validity implies the reflectivity of orthogonality classes definable with up to n unbounded quantifiers in locally presentable categories. |
References: | [1] J. Bagaria, C. Casacuberta, A. R. D. Mathias,
Epireflections and supercompact cardinals.
CRM
Preprint No.740 (March 2007).
[2] J. Bagaria, C. Casacuberta, A. R. D. Mathias, J. Rosický, Definable orthogonality classes are small, in preparation. [3] C. Casacuberta, D. Scevenels, J. H. Smith, Implications of large-cardinal principles in homotopical localization. Adv. Math. 197 (2005), 120-139. Available online. |
Speaker: | Frank Neumann (Leicester) |
Title: | Moduli stacks of vector bundles over algebraic curves and Frobenii |
Abstract: | After giving a brief introduction into moduli problems and moduli stacks, I will outline how to determine the l-adic cohomology ring of the moduli stack of vector bundles on a given algebraic curve in positive characteristic and will explicitely describe the action of the various geometric and arithmetic Frobenius morphisms on the cohomology ring. It turns out that in using the language of algebraic stacks this becomes surprisingly easy. If time permits, I will indicate how to prove the analogues of the Weil conjectures for the moduli stack using Behrend´s Lefschetz trace formula for Artin stacks and how all these calculations might be governed by a conjectural etale homotopy type of the moduli stack. This is work in progress (in part with U. Stuhler (Goettingen)). |
References: |
G. Laumon, L. Moret-Bailly, Champs algébriques.
Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. Vol. 39.
Springer-Verlag, Berlin, 2000.
K. A. Behrend, The Lefschetz trace formula for algebraic stacks. Invent. Math. 112 (1993), 127-149. K. A. Behrend, Derived l-adic categories for algebraic stacks. Mem. Amer. Math. Soc. 163 (2003), no. 774. F. Neumann, U. Stuhler, Moduli stacks of vector bundles and Frobenius morphisms. Algebra and Number Theory, Proceedings of the International Conference held on the occasion of the silver jubilee of the School of Mathematics Hyderabad, HBA/AMS Delhi (2005) 126-146. |
Follow-up: | in the Working Seminar, Tuesday 10/06/2008. |
Speaker: | Bernard Keller (Paris 7) |
Title: | The periodicity conjecture via 2-Calabi-Yau categories |
Abstract: | The periodicity conjecture was formulated in mathematical physics at the beginning of the 1990s, in the work of Zamolodchikov, Kuniba-Nakanishi and Ravanini-Valleriani-Tateo. It asserts that a certain discrete dynamical system associated with a pair of Dynkin diagrams is periodic and that its period divides the double of the sum of the Coxeter numbers of the two diagrams. The conjecture was proved by Frenkel-Szenes and Gliozzi-Tateo for the pairs (A_n, A_1), by Fomin-Zelevinsky in the case where one of the diagrams is A_1 and by Volkov when both diagrams are of type A. We will sketch a proof of the general case which is based on Fomin-Zelevinsky's work on cluster algebras and on the theory relating cluster algebras to triangulated 2-Calabi-Yau categories. An important role is played by Amiot's construction of such categories from finite-dimensional algebras of global dimension two. |
References: |
C. Amiot, Cluster categories for algebras of global dimension
two and quivers with potential.
ArXiv:0805.1035.
S. Fomin and A. Zelevinsky, Cluster algebras IV: Coefficients. Compositio Mathematica 143 (2007), 112-164. ArXiv:math/0602259 S. Fomin and A. Zelevinsky, Y-systems and generalized associahedra. Ann. of Math. 158 (2003), 977-1018. Available online. |
Follow-up: | in the Working Seminar, Thursday 05/06/2008. |
Speaker: | Tom Fiore (UAB) |
Title: | Selecting coherence diagrams for pseudo algebras over theories |
Abstract: | Generalizing algebras to pseudo algebras is a fundamental idea which has recently become important in axiomatization of conformal field theory. The question of which coherence diagrams to select is not as straightforward as one might think, as evidenced already by Laplaza's coherence diagrams for pseudo commutative semi-rings in 1972. In this talk I will describe how to use algebraic theories and operads to select coherence diagrams. The main example is the algebraic structure on worldsheets, which is used in the definition of conformal field theory. This is joint work with Po Hu and Igor Kriz. |
References: |
P. Hu and I. Kriz, Conformal field theory and elliptic
cohomology.
Adv. Math. 189 (2004), 325-412.
Available
online.
T. Fiore, Pseudo limits, biadjoints, and pseudo algebras: categorical foundations of conformal field theory. Mem. Amer. Math. Soc. 182 (2006), no. 860, x+171 pp. ArXiv:math/0408298. T. Fiore, P. Hu and I. Kriz, Laplaza sets, or how to select coherence diagrams for pseudo algebras. To appear in Advances in Mathematics. ArXiv:0803.1408. |
Speaker: | Christian Ausoni (Bonn) |
Title: | Two-vector bundles and algebraic K-theory of ku |
Abstract: | |
References: |
Speaker: | Jiri Rosický (Masaryk Univ.) |
Title: | Homotopy accessible categories |
Abstract: | Finitely accesible categories are based on filtered colimits (i.e., those which commute with finite limits in sets). Analogously, algebraic categories are based on sifted colimits (commuting with finite products). Their enriched versions use weighted colimits. If the base category is a model category, there is a homotopy theory of enriched categories. I did it for simplicial categories - it is interesting that homotopy sifted categories coincide with totally coaspherical ones. The results are parallel to those obtained by J. Lurie in the language of quasicategories. But one can introduce homotopy weighted colimits over any model category base and define homotopy accessible enriched categories there (joint work with S. Lack). The talk will overview these concepts and results. |
References: |
A. Joyal, Quasicategories and Kan complexes.
J. Pure Appl. Alg. 175 (2002), 207-222.
J. Lurie, Derived Algebraic Geometry I. ArXiv:math/0608228. G. Maltsiniotis, La Theorie de l'Homotopie de Grothendieck. Asterisque 301, 2005. J. Rosický, On homotopy varieties. Adv. Math. 214 (2007), 525-550. |
Speaker: | Georg Biedermann (MPIM & CRM) |
Title: | Introduction to Goodwillie's calculus of homotopy functors |
Abstract: | Calculus of homotopy functors attempts to study homotopy
functors as analogues of differentiable functions by
considering a logarithmic Taylor series. It interpolates
between stable and unstable homotopy theory. We will describe
the basic definitions and give an overview over some fundamental
results.
This is an overview talk aimed at a general (topological) audience; it will be followed by more detailed and technical talks the following week (12-16 May). |
References: |
T. G. Goodwillie, Calculus III:
Taylor series. Geom. Top. 7 (2003), 645-711 (electronic).
T. G. Goodwillie, Calculus. II. Analytic functors. K-Theory 5 (1991/92), 295-332. N. J. Kuhn, Goodwillie towers and chromatic homotopy: an overview. ArXiv:math/0410342. G. Biedermann, B. Chorny, O. Röndigs, Calculus of functors and model categories. Adv. Math. 214 (2007), 92-115. ArXiv:math/0601221. |
Follow-up: | in the Working Seminar, Tuesday 13/05/2008 and Friday 16/05/2008. |
Speaker: | Nicola Gambino (CRM) |
Title: | The identity type weak factorisation system |
Abstract: | Martin-Löf type theory is generally studied in mathematical logic as a foundation for constructive mathematics and in theoretical computer science as an idealized programming language. The seminar will take a different point of view: we investigate what may be referred to as the homotopy theory of Martin-Löf type theory. I will show how the syntax of Martin-Löf type theory gives rise to a category that admits a non-trivial weak factorisation system. I will then relate this weak factorisation system the natural Quillen model structure on the category of groupoids. No prior knowledge of Martin-Löf type theory will be assumed. This is joint work with Richard Garner. |
References: | S. Awodey and M. A. Warren, Homotopy-theoretic models
of identity types. ArXiv:0709.0248. N. Gambino and R. Garner, The identity type weak factorisation system. ArXiv:0803.4349. |
Speaker: | Julie Bergner (Kansas State University) |
Title: | Complete Segal spaces arising from simplicial categories |
Abstract: | While model categories or, more generally, simplicial categories, are natural models for homotopy theories, it can be more convenient to work instead in the setting of complete Segal spaces. After describing complete Segal spaces, we'll consider several different (but equivalent) functors taking simplicial categories to complete Segal spaces. The functor given by Rezk is particularly useful for characterizing the complete Segal space arising from a given simplicial category. One can then use this characterization to show that a definition of homotopy fiber product of model categories does in fact correspond to a homotopy pullback in the complete Segal space model structure. |
References: | C. Rezk, A model for the homotopy
theory of homotopy theory. Trans. Amer. Math. Soc.
353 (2001), 973-1007 (electronic).
J. Bergner, Three models for the homotopy theory of homotopy theories. Topology 46 (2007), 397-436. ArXiv:math/0504334. J. Bergner, Complete Segal spaces arising from simplicial categories. ArXiv:0704.1624. J. Bergner, A survey of (\infty,1)-categories. ArXiv:math/0610239. |
Follow-up: | in the Working Seminar, Thursday 24/04/2008. |
Speaker: | Julie Bergner (Kansas State University) |
Title: | Diagrams encoding algebraic structures on spaces |
Abstract: | A diagram given by an algebraic theory can be used to describe a corresponding algebraic structure on a space via a product-preserving functor from the theory to the category of spaces. We can consider any such structure either strictly or up to homotopy, and a theorem of Badzioch allows us to rigidify a homotopy structure to an equivalent strict one. Using his result, we can actually find simpler diagrams (other than the respective theories) encoding monoid and group structures on spaces. |
References: | B. Badzioch, Algebraic theories in homotopy theory.
Ann. of Math. (2) 155 (2002), 895-913.
J. Bergner, Simplicial monoids and Segal categories. IN: Street Festschrift: Categories in algebra, geometry and mathematical physics, 59-83, Contemp. Math., 431, Amer. Math. Soc., Providence, RI, 2007. J. Bergner, Adding inverses to diagrams encoding algebraic structures. ArXiv:math/0610291. |
Speaker: | Georges Maltsiniotis (Paris 7) |
Title: | Quillen model structures on presheaf categories (d'après Cisinski) II |
Abstract: | The aim of these two lectures is to report on some results of Denis-Charles Cisinski concerning Quillen model structures on categories of presheaves. See below. |
References: | D.-C. Cisinski, Les préfaisceaux comme modèles des types
d'homotopie. Astérisque 308. electronic |
Speaker: | Georges Maltsiniotis (Paris 7) |
Title: | Quillen model structures on presheaf categories (d'après Cisinski) I |
Abstract: | The aim of these two lectures is to report on some results of Denis-Charles
Cisinski concerning Quillen model structures on categories of presheaves.
In the first lecture we present a complete classification of cofibrantly generated closed model structures, on such categories, having as cofibrations exactly the monomorphisms, in terms of "accessible localizers". In the second lecture, we describe such a structure associated to a "homotopy structure", and we introduce the notions of "anodyne extensions" and "absolute weak equivalences". A crucial step for this study is to consider the model structures on the categories of presheaves over a given presheaf. |
References: | D.-C. Cisinski, Les préfaisceaux comme modèles des types
d'homotopie. Astérisque 308. electronic |
Speaker: | Mathieu Anel (CRM) |
Title: | Fundamental infinity groupoids of higher stacks, and applications (Joint work with J. Heller (UWO)) |
Abstract: | Using results of Dugger and Isaksen, we define an
extension of the functor of singular chains from the category
Top of locally contractible topological spaces to the model
category of topological stacks (simplicial presheaves on Top
with the projective local model structure) and establish its
link with the cohomology with constant coefficients. As an application, given a topological group G, we compare three notions of classifying object for G: the classifying space, the classifying stack and the classifying homotopy type. To compare them we embbed them in the category of topological stacks, and prove that although there are non equivalent as stacks, they have the same fundamental infinity-groupoid and thus the same cohomology with constant coefficients. |
References: | M. Anel, J. Heller, Fundamental infinity-groupoid of
stacks. In preparation. D. Dugger, Universal homotopy theories. Adv. Math. 164 (2001), 144-176. ArXiv:math/0007070. D. Dugger, D. Isaksen, Topological hypercovers and A1-realizations. Math. Z. 246 (2004), 667-689. |
Speaker: | Tom Leinster (Glasgow) |
Title: | The cardinality of a metric space, II |
Abstract: | In this second talk I will define the cardinality of a metric space
and discuss its properties.
This is very recent work, done entirely in the five weeks that I have been at the CRM, and contains some highly speculative aspects. Conjecturally, the most important invariants of geometric measure theory (such as volume, Euler characteristic and Hausdorff dimension) are special cases of general concepts for enriched categories. The connection between the two subjects is made by the observation that a metric space is a kind of enriched category. |
References: | F.W. Lawvere, Metric spaces, generalized logic and closed
categories.
Reprints in Theory and Applications of Categories, No. 1 (2002),
1-37. electronic.
Tom Leinster, Metric spaces, post at The n-Category Café, February 2008. |
Speaker: | Tom Leinster (Glasgow) |
Title: | The cardinality of a metric space, I |
Abstract: | Recent work suggests that there might be substantial connections
between category theory and certain parts of geometry traditionally
associated with probability theory and combinatorics. This is part of
a programme to understand "cardinality" in a very general context.
In this first talk I will run through the necessary background material. On the one hand, I will say something about the subject variously known as integral geometry, geometric measure theory, and geometric probability (the most famous part of which is Buffon's needle problem). In particular, I will talk about measures on convex sets. On the other, I will explain how to define the cardinality or Euler characteristic of a category. These two things will be brought together in the second talk. |
References: | D. A. Klain, G.-C. Rota, Introduction to Geometric
Probability. Lezioni Lincee, Cambridge University Press, 1997.
S. H. Schanuel, What is the length of a potato? An introduction to geometric measure theory. In Categories in Continuum Physics, Lecture Notes in Mathematics 1174, Springer, 1986. |
Speaker: | Amnon Neeman (CRM and ANU Canberra) |
Title: | Invariant theory, statistics, stable splittings |
Abstract: | |
References: | P. Hall, A. Neeman, R. Pakyari, R. Elmore, Nonparametric inference in multivariate mixtures. Biometrika 92 (2005), 667-678. |
Speaker: | Luchezar Avramov (Nebraska) |
Title: | Reduction formulas for Hochschild cohomology |
Abstract: | |
References: | L. Avramov and S. Iyengar, Gorenstein algebras and Hochschild cohomology. Preprint 2007. |
Speaker: | Michel Van den Bergh (Hasselt) |
Title: | Degenerations of some Calabi-Yau categories |
Abstract: | |
References: |
Speaker: | Wolfgang Pitsch (UAB) |
Title: | Relative homological algebra via model approximations |
Abstract: | We show how to use model approximations to do relative homological algebra in the category of unbounded chain complexes over an abelian category. Here "relative" means that one fixes a-priori a class of injectives or equivalently a class of exact sequences. This approach avoids the problems that arise when one tries to put a model structure directely on the category of unbounded chain complexes as pointed out in the work of Christensen and Hovey. |
References: | S. Eilenberg and J. C. Moore,
Foundations of relative homological algebra.
Mem. Amer. Math. Soc. 55 (1965).
D. Christensen and M. Hovey, Quillen model structures for relative homological algebra. Math. Proc. Cambridge Philos. Soc. 133 (2002), 261-293. |
Speaker: | Henning Krause (Paderborn) |
Title: | Support varieties and the classification of localizing subcategories |
Abstract: | This talk provides an introduction to support varieties for triangulated categories, and I explain how they are used for classifying localizing subcategories in various context, for instance in commutative algebra and representation theory. |
References: | D. Benson, S. Iyengar, and H. Krause,
Local cohomology and support for
triangulated categories.
ArXiv:math/0702610.
A. Neeman, The chromatic tower of D(R). Topology 31 (1992), 519-532. |
Follow-up: | in the Working Seminar, Monday 22/10/2007 (Henning Krause), Wednesday 24/10/2007 (Henning Krause), and Monday 29/10/2007 (Joachim Kock). |
Speaker: | Denis-Charles Cisinski (Paris 13) |
Title: | A general approach to model categories and homotopy colimits |
Abstract: | The theory of abstract Quillen model categories
suffers some kind of gaps. Mainly the theory of homotopy
colimits does not fit in this setting in the sense
that we cannot say that colimit functors are left
Quillen functors. This is due to the fact that we don't
even know in general if the categories of functors from
a given small category to a model category is
itself a model category. Thomason introduced in an unpublished manuscript a notion of model category which has better properties and proved that we could see homotopy colimits as left Quillen functors. The problem is that any proper Quillen model category is a Thomason model category, but the non proper model categories still remain out of this picture. I will introduce a notion of model category with the property that any model category in the sense of Quillen or of Thomason is a model category, and such that homotopy colimits are actually left Quillen functors. |
References: | D.-C. Cisinski, Catégories dérivables,
preprint.
W. G. Dwyer, P. S. Hirschhorn, D. M. Kan, J. H. Smith, Homotopy Limit Functors on Model Categories and Homotopical Categories. Math surveys and Monographs, vol. 113, AMS, 2004. D. Quillen, Homotopical algebra. LNM 43, Springer-Verlag, 1967. A. Radulescu-Banu, Cofibrations in homotopy theory. ArXiv:math/0610009. C. Weibel, Homotopy ends and Thomason model categories. Selecta Math 7 (2001), 533-564. |
Follow-up: | in the Working Seminar, Monday 15/10/2007 and Wednesday 17/10/2007. |
Speaker: | Vincent Franjou (Nantes) |
Title: | Functor cohomology |
Abstract: | |
References: | see separate page with detailed, commented references. |
Follow-up: | in the Working Seminar, Wednesday 10/10/2007. |
Speaker: | Tom Fiore (UAB) |
Title: | Homotopy theory of double categories |
Abstract: | When are two categories the same? One possible notion of sameness
is equivalence of categories, another is weak homotopy equivalence
of their nerves. 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? We collate the various notions of weak equivalence into model structures on the category of double categories. This complements recent work on model structures on BiCat and 2-Cat in Lack and Hess et al as well as the work of Kieboom and collaborators on model structures for internal categories. This is joint work with Simona Paoli and Dorette Pronk. |
References: | T.M. Fiore, S. Paoli, D.A. Pronk, Model structures on the
category of small double categories. ArXiv:0711.0473.
T. Everaert, R.W. Kieboom and T. Van der Linden, Model structures for homotopy of internal categories. Theory Appl. Categ. 15 (2005/2006), 66-94, electronic. |
Speaker: | Brooke Shipley (University of Illinois at Chicago) |
Title: | Topological equivalences of differential graded algebras |
Abstract: | I discussed derived equivalences of rings and differential graded algebras. The case of DGAs differs from rings in two ways: (1) derived equialences and Quillen equivalences of module categories do not agree for DGAs (they do agree for rings) and (2) one must consider tilting spectra rather than just tilting complexes to produce all Quillen equivalences. This led to considering topological equivalences of DGAs. |
References: | D. Dugger and B. Shipley, Topological equivalences
of differential graded algebras.
Adv. Math. 212 (2007), 37-61.
B. Shipley, Morita theory in stable homotopy theory. IN: Handbook of Tilting Theory (eds. Angeleri Huegel, Happel, Krause) London Math. Soc. Lecture Notes Series, 332 (2007), 393-409. D. Dugger and B. Shipley, K-theory and derived equivalences. Duke Math. J. 124 (2004), 587-617. S. Schwede, Morita theory in abelian, derived and stable model categories. IN: Structured Ring Spectra, pp. 33-86, London Mathematical Society Lecture Notes 315, Cambridge Univ. Press, Cambridge, 2004. S. Schwede, An untitled book project about symmetric spectra. Preliminary version, 154 pp., available at http://www.math.uni-bonn.de/people/schwede/. |
Follow-up: | in the Working Seminar, Wednesday 03/10/2007. |
Speaker: | Nicola Gambino (CRM) |
Title: | Homotopy limits for 2-categories |
Abstract: | We study homotopy limits for 2-categories using the theory of Quillen model categories. In order to do so, we establish the existence of projective and injective model structures on diagram 2-categories. Using this result, we describe the homotopical behaviour not only of conical limits but also of weighted limits for 2-categories. Finally, homotopy limits are related to pseudo-limits. |
References: | N. Gambino, Homotopy limits for 2-categories.
(To appear.)
electronic.
S. Lack, A 2-categories companion. ArXiv:math/0702535. |
Speaker: | Amnon Neeman (CRM and ANU Canberra) |
Title: | Brown representability follows from Rosický |
Abstract: | A triangulated category satisfies Brown representability if any
homological functor, which satisfies the obvious necessary condition
about respecting products, is representable. We reviewed the history of
Brown representability theorems, and explained that, for the best part of
a decade, we have all been stumped by the problem of trying to extend the
theorem to the dual of well generated triangulated categories.
Then we explained how, using a recent result of Rosicky, it is possible to give a very simple solution to the problem that had us all stuck. Rosickys theorem seems to be a very powerful tool, and needs to be understood better. |
References: | A. Neeman,
Brown representability follows from Rosický.
CRM Preprint
762 (2007).
Revised version.
J. Rosický, Generalized Brown representability in homotopy categories. Theory Appl. Categ. 14 (2005), 451-479, electronic. |
Follow-up: | in the Working Seminar, Monday 17/09/2007 (Amnon Neeman), Wednesday 19/09/2007 (Amnon Neeman), Wednesday 26/09/2007 (Henning Krause), and Monday 01/10/2007 (Amnon Neeman). |