**Fridays at 14:15**

MA 10

### Program

Date | Title | Speaker |
---|---|---|

28.09.12 |
Cochains as an E_n-algebra | Justin Young EPFL |

05.10.12 | Brace bar-cobar duality and applications | Justin Young EPFL |

12.10.12 | Duality, descent and extensions | Kathryn Hess EPFL |

19.10.12 | A Boardman-Vogt tensor product for operadic bimodules | Kathryn Hess EPFL |

02.11.12 | Moduli of rational homotopy types | Michael Schlessinger UNC-Chapel Hill |

09.11.12 | Spaces of long embeddings and iterated loop spaces | Kathryn Hess EPFL |

16.11.12 | An introduction to the theory of derivators | Moritz Groth Nijmegen |

07.12.12 at 13h15 |
On topological triangulated orbit categories | Marcy Robertson Lille |

14.12.12 | Approximating Taylor towers with mapping spaces | Greg Arone Virginia |

01.02.1310-12CM 106 |
Masters project oral defenses | Marielle Corthésy Dimitri Zaganidis EPFL |

08.02.13MA 30 |
Cellularization of D(R) and point-free reconstruction of affine schemes | Wolfgang Pitsch UA Barcelona |

11-15.02.13 9h15-12h20 MA 10 |
Working group: Introduction to triangulated categories |
Various |

22.02.13 | Weak bialgebras of fractions | Steve Bennoun UBC |

01.03.13 | Cellular covers of groups with free kernel | José Luis Rodríguez Almería |

08.03.13 | Realization of conjugation spaces | Jérôme Scherer EPFL |

15.03.13 | What is a semi-abelian category? | Marino Gran Louvain-la-Neuve |

22.03.13 | Quillen model structures on combinatorial categories | Jean-Marie Droz Bremen |

19.04.13 | Quillen models for Kasparov categories | Ivo Dell’Ambrogio Lille |

26.04.13 | Are complete intersections symmetric? | Lev Kiwi Fribourg |

03.05.13 | Determinants and deloopings of algebraic K-theory | Jesse Wolfson Northwestern |

07.05.13CM 013 |
Rigidity and algebraic models for rational equivariant stable homotopy theory |
Brooke Shipley UIC |

17.05.13 | Bicategorical homotopy fiber sequences | María Calvo Granada |

07.06.13MA 12 |
Metrics on diagrams and persistence of metric space valued functions | Jonathan Scott Cleveland State |

13.06.13MA 12 10h15-11h30 |
Bousfield localization and commutative monoids | David White Wesleyan |

14.06.13MA 12 10h15-11h |
A combinatorial model of the algebraic K-theory spectrum | Marc Stephan EPFL |

14.06.13MA 12 |
Stasheff’s associahedra and R.J. Thompson’s groups | Ariadna Fossas Tenas EPFL |

21.06.13MA 12 |
Algebraic K-theory of higher categories (after Barwick) | Michael Gröchenig EPFL |

02.07.13MA 12 9h30-15h |
Mini-course on derivators | Moritz Groth Nijmegen |

03.07.13MA 129h30-15h 04-05.07.13 GRA331 14h-15h |
Mini-course on infinity categories | Moritz Groth Nijmegen |

(See also the program of the topology seminar in 2011/12, 2010/11, 2009/10, 2008/09, 2007/08, 2006/07, and 2005/06.)

### Abstracts

**Young (first talk): **We will discuss E_n algebras in general, and why the cochain complex of a space is an example. Then, we’ll review Mandell’s theorem, which says that the E_infinity structure on cochains determines the homotopy type of a space and use that as a jumping off point to consider the “simpler” E_n structures. This leads to the main theorem of the talks that after inverting enough primes in the ring R, S^*(X, R) is equivalent as an E_2 algebra to a commutative algebra.

**Young (second talk)**: We will review the classical bar-cobar duality between algebras and coalgebras. Then, we will see how this can be enhanced to a slightly weaker duality between E_2 algebras and Hopf algebras. As a result, we will see that the work of Hess-Parent-Scott-Tonks leads to a canonical Adams-Hilton model for chains on a loop space as a Hopf algebra. Finally, via the bar construction, we will reformulate the main theorem of the last talk in terms of Hopf algebras, and show how it follows from a relatively straightforward rigidification of a theorem of Anick.

**Hess (first talk):** In recent work with Alexander Berglund, we studied the relationships among the notions of Koszul duality for dg algebras, Grothendieck descent for morphisms of dg algebras and Hopf-Galois extensions of dg algebras. We showed in particular if B is a multiplicative acyclic closure of a dg algebra A, and a dg Hopf algebra H coacts on B by algebra maps, then H is Koszul dual to A if and only if the inclusion map of A into B is an H-Hopf-Galois extension satisfying Grothendieck descent.

In this talk I will briefly recall the notions of Koszul duality, homotopic Grothendieck descent and homotopic Hopf-Galois extension, then describe the common categorical framework into which all of these notions fit and sketch the proof of the result stated above. I will also explain how to how to construct families of examples based on Hirsch algebras to which our framework can be applied.

**Hess (second talk): ** (Joint work with Bill Dwyer) The Boardman-Vogt tensor product is a symmetric monoidal structure on the category of simplicial operads. I’ll explain how to lift this tensor product to bimodules over operads. This lifting has deep geometric meaning, which I will illustrate with applications to modelling spaces of long links with n strands in R^{m} and to modelling manifolds as right modules over little balls operads.

**Schlessinger: ** The set of nilpotent rational homotopy spaces with given cohomology algebra H forms a “moduli space”. In fact, it has the form M = Aut H\ W /U, where W is the cone over a projective algebraic variety, and U is a unipotent algebraic groupoid. We outline the construction, which proceeds from the observation that W is the base of the “miniversal deformation” of the formal space F with cohomology H. We also relate the construction to the classifying space BAut(F) and give examples and problems.

**Hess (third talk):** (Joint work with Bill Dwyer) Let K(m,n) denote the space of long m-knots in R^{n}, i.e., the space of smooth embeddings of R^{m} into R^{n} that agree with a fixed linear embedding outside a compact subspace. Let B_{k} denote the little k-balls operad. Modulo a framing adjustment, the embedding calculus of Goodwillie, Klein and Weiss leads to an identification of K(m,n) with the derived mapping space of linear B_{m}-bimodule maps from B_{m} to B_{n}. Starting from this, we identify K(m,n) as the (m+1)-fold loop space on the derived mapping space of operad maps from B_{m} to B_{n}.

**Groth: ** The theory of derivators –going back to Grothendieck and Heller– is a purely (2-)categorical approach to axiomatic homotopy theory. It adresses the problem that the rather crude passage from model categories to homotopy categories results in a serious loss of information. In the stable context, the typical defects of triangulated categories (non-functoriality of cone construction, lack of homotopy colimits) can be seen as a reminiscent of this fact.

The basic idea behind a derivator is that it forms homotopy categories of ‘all’ diagram categories and also encodes the calculus of homotopy Kan extensions.

The aim of this talk is to give an introduction to derivators and to (hopefully) advertise them as a convenient, ‘weakly terminal’ approach to axiomatic homotopy theory. We will see that there is a threefold hierarchy of such structures, namely derivators, pointed derivators, and stable derivators. A nice fact about this theory is that ‘stability’ is a property of a derivator as opposed to being an additional structure.

**Robertson: ** In 2005, Keller showed that the orbit category associated to the bounded derived category of a hereditary category under an auto equivalence is triangulated. As an application he proved that the cluster category is triangulated. We show that this theorem generalizes to triangulated categories with topological origin (i.e. the homotopy category of a stable model category). As an application we construct a topological triangulated category which models the cluster category. This is joint work with Andrew Salch.

**Arone: **Let F be a topological functor. The sequence of derivatives of F forms a module over a certain operad (the operad depends on the domain and target of F). If one tries to recover F from the module structure on the derivatives, one obtains a certain “best possible” approximation to F in terms of mapping spaces between operad modules. In some cases, the approximation coincides with the Taylor tower of F. Even when it doesn’t, the approximation is interesting in its own right. The difference between our approximation and the Taylor tower is measured by the Tate homology of the derivatives of F. As one consequence, we obtain an amusing new perspective on classical rational homotopy theory. The talk is based on joint work with Michael Ching.

**Pitsch: **In this talk we will revisit the classification of compactly generated localizing subcategories in the derived category of a ring D(R) by Thomason, Neeman, Balmer. We will show how the description of cellularization in D(R) with respect to quotients R/I of the ring by a finitely generated ideal, due to Dwyer-Greenlees, leads to a natural bijection between the lattice of compactly generated localizing subcategories and the Hochster open sets in Spec R, avoiding almost all noetherian assumptions. We will show in particular that the algebraic version of the “nilpotence theorem” of Devinatz, Hopkins and Smith is in fact a consequence of the fact that Spec R with this topology is a Tychonoff space. If time permits we will show how to dualize this construction in D(R) to reconstruct the usual affine scheme Spec R.

This is joint work with Joachim Kock (Universidad Autonoma de Barcelona)

**Bennoun: ** The notions of weak bialgebra and weak Hopf algebra were introduced by Böhm, Nill and Szlachanyi as generalizations of the well-known notions of bialgebra and Hopf algebra. One important result about weak bialgebras is that any fusion category is equivalent to a category of modules over a weak Hopf algebra.

In this presentation I will start by defining weak bialgebras and weak Hopf algebras. I will then briefly present some examples and basic properties. Next, generalizing results of Hayashi for bialgebras, I will explain under which hypotheses one can construct the weak bialgebra of fractions of a given weak bialgebra. I will moreover discuss the relationship between the weak bialgebra of fractions and the weak Hopf envelope.

**Rodríguez: **In this talk, we will review some results on cellular covers of groups, motivated by its counterpart in homotopy theory of spaces. Recall that an epimorphism π: G → H is called a cellular cover if it induces a bijection π*: End(G) ≅ Hom(G,H), where π*(ψ)=πψ. We pay attention to the case when H and G are cotorsion-free abelian groups (or more generally, R-modules over a cotorsion-free ring). We provide uncountably many new examples where the rank of H is 2, and the kernel of the cellular cover is free of countable rank. This extends results from Göbel-Rodríguez-Strüngmann, and Rodríguez-Strüngmann.

**Scherer: **I will introduce the beautiful subject of conjugation spaces and conjugation manifolds, as defined by Hausmann, Holm, and Puppe. Roughly speaking they are even dimensional spaces (or manifolds) equipped with an involution such that their mod two cohomology is isomorphic to that of the fixed points after dividing degrees by two. The main problem to which I will try to give some answers is that of realizing a given space as fixed points of a conjugation space. This is joint work with Wolfgang Pitsch.

**Gran: **The problem of finding an axiomatic context capturing some typical properties of the category of groups was already mentioned by S. Mac Lane in the article [1]. Only recently the introduction of the notion of semi-abelian category [2] made it possible to treat many fundamental properties the categories of groups, Lie algebras, crossed modules and compact groups have in common, in a similar way to the one the notion of abelian category allows for a unified treatment of module categories and of their categories of sheaves. The theory of semi-abelian categories provides a suitable categorical setting to treat some fundamental aspects of non-abelian homological algebra, radical theory and commutator theory. This introductory talk will focus on some basic aspects of the theory, and on a couple of more recent results obtained in collaboration with T. Everaert, T. Van der Linden and M. Duckerts [3,4].

*References*

[1] S. Mac Lane, *Duality of groups*, Bull. Am. Math. Soc. 56 (6), 486-516 (1950)

[2] G. Janelidze, L. Marki and W. Tholen, *Semi-abelian categories*, J. Pure Appl. Algebra 168, 367-386 (2002)

[3] T. Everaert, M. Gran and T. Van der Linden, *Higher Hopf formulae for homology via Galois Theory*, Adv. Math. 217, 2231-2267 (2008)

[4] M. Duckerts, T. Everaert and M. Gran, *A description of the fundamental group in terms of commutators and closure operators*, J. Pure Appl. Algebra 216, 1837-1851 (2012)

**Droz: ** Various Quillen model structures can be constructed on categories of combinatorial objects. We will show that classical graph theoretical notions, like the core of a graph and the set of connected components, are notions of homotopy types in specific Quillen model structures. However, we will also demonstrate that, often, what could be regarded as natural candidates for homotopy types in a category of graphs cannot arise from a model structure. Finally, we will consider the classification of all model structures on simple categories of combinatorial objects and count the model structures on a category of finite graphs.

**Dell’Ambrogio: ** We embed the equivariant Kasparov category of a locally compact group into the homotopy category of a stable symmetric monoidal model category, as a full tensor triangulated subcategory. Following Østvær, the model is constructed by the recipe of Dugger’s universal homotopy theories, by localizing simplicial copresheaves on C*-algebras, and the embedding is then simply induced by the Yoneda functor. As a consequence our construction has very nice functorial properties which extend the already rich functoriality of KK-theory. As an immediate application, we obtain a proof of the additivity of traces in KK-theory.

**Kiwi: ** I will first explain the terms in the title. Since it is my thesis question, I will convince you that it is an interesting question and tell you how we manage to answer it so far.

**Wolfson: ** When C is an idempotent complete exact category, we construct a universal determinantal theory and show that it realizes the K-theory of Tate spaces in C as a delooping of the K-theory of C. This is joint work in progress with Braunling and Groechenig.

**Shipley: ** A homotopy category is called rigid if the underlying Quillen model category is unique up to Quillen equivalence. I will discuss several examples of rigid homotopy categories including the rational G-equivariant stable homotopy category for G a finite group, a profinite group, and the circle group. I will also discuss Patchkoria’s recent related result for the two-local G-equivariant stable homotopy category for G a finite group.

**Calvo: **Small B