CRM Research Programme for the academic year 2007-2008

Homotopy Theory and Higher Categories

Thematic Week

Higher Operads Week

Aim

Organisers: André Joyal and Joachim Kock

Schedule

SCHEDULE	Monday 14/4	Tuesday 15/4	Wednesday 16/4	Thursday 17/4	Friday 18/4	Saturday 19/4
09:30—10:30	Leinster	Batanin	Batanin	Maltsiniotis	Maltsiniotis	Cisinski
10:50—11:50	Batanin	Berger	Berger	Cisinski	Cisinski	Casacuberta
12:00—13:00	Berger	Weber	Weber	Vallette	Vallette	Kock

Abstracts and references

Introduction to higher operads

References:
• M. Batanin, Monoidal globular categories as a natural environment for the theory of weak n-categories, Adv. Math. 136 (1998), 39--103. (Available online.)
• T. Leinster, Higher operads, higher categories, Cambridge University Press, 2003. ArXiv:math.CT/0305049.

Michael Batanin:

1: Higher operads and symmetric operads. Symmetrisation and dessymmetrisation.

Abstract: There is a classical adjunction between the categories of nonsymmetric operads (1-operads in my terminology) and symmetric operads which is usually considered as a triviality. I will show in my lecture that there is an analogous adjunctions between n-operads for all n and symmetric operads, which is much more interesting. I will develop a techniques of internal algebras of cartesian monads and as an application I will obtain an explicit formula for symmetrisation of n-operads.

2: The homotopy type of symmetrisation and compactification of real configuration spaces

Abstract: I will study the homotopy type of symmetrisation functor. I will show that under some natural restrictions the category of n-operads has a model structure and that symmetrisation and dessymetrisation form a Quillen pair. I will show that the (derived) symmetrisation of the terminal topological n-operad is an operad equivalent to the little n-cube operad, which gives a new recognition principle for n-fold loop spaces. To do this I will explain how to get the Fulton-Macpherson operad of compactified configuration spaces fmⁿ as a symmetrisation of a contractible n-operad. From this calculations we will see how the coherence laws for n-fold loop spaces can be understood.

3: Locally constant n-operads as higher braided operads

Abstract: I will show first that the category of braided operads sits as a full subcategory inside the category of 2-operads. I will introduce a category of locally constant n-operads which we should think of as the category of higher braided operads. I will show that the homotopy category of locally constant n-operads is equivalent to the homotopy category of nonsymmetric, braided and symmetric operads for n=1,2,∞ correspondingly.

References:
• M. Batanin, Symmetrisation of n-operads and compactification of real configuration spaces, Adv. Math. 211 (2007), 684--725. ArXiv:math/0606067.
• M. Batanin, The Eckmann-Hilton argument and higher operads, Adv. Math. 217 (2008), 334--385. ArXiv:math/0207281.
• M. Batanin, Locally constant n-operads as higher braided operads, MPIM preprint, April 2008. (Will be available in the ArXiv soon but already available by request.)
• D.-C. Cisinski Locally constant functors, ArXiv:0803.4342.

Clemens Berger:

1: Geometric Reedy categories and categorical wreath product

Abstract: The concept of a geometric Reedy category is an attempt to axiomatise those properties of the simplex category Δ which make simplicial sets so powerful in homotopy theory. Using Cisinski's work on local test-categories, the topos of presheaves on a flat geometric Reedy category is shown to carry a Quillen model structure for the homotopy category of topological spaces.
Joyal's category Θ_n is presented as an iterated wreath product of the simplex category Δ. This endows Θ_n with the structure of a flat geometric Reedy category, and induces a homotopically meaningful, fully faithful nerve functor for strict n-categories.

2: Homogeneous n-graphical theories and n-operads

Abstract: The epi-mono factorisation system of Δ refines to a triple factorisation system thanks to the distinction between inner and outer face operators. The iterated wreath product Θ_n carries an analogous triple factorisation system, which characterises Θ_n as the terminal homogeneous n-graphical theory.
There is a one-to-one correspondence beween homogeneous n-graphical theories and set-valued n-operads in Batanin's sense. This naturally gives rise to a Segal-type model for n-fold loop spaces, namely reduced Θ_n-spaces.

3: Θ-spectra and small CW-models for Eilenberg-Mac Lane spaces

Abstract: The Segal-spectrum functor for Γ-spaces factors through the category of Θ-spectra. Evaluating this functor at the Γ-set model for the Eilenberg-Mac Lane spectrum HA gives comparatively small Θ_n-set models for Eilenberg-Mac Lane spaces of type K(A,n). We study these models for A=Z/2Z and obtain explicit cocycle representatives for the generators of the cohomology H^*(K(Z/2Z,n);Z/2Z).

References:
• C. Berger, A cellular nerve for higher categories. Adv. Math. 169 (2002), 118--175. (Available online.)
• C. Berger, Iterated wreath product of the simplex category and iterated loop spaces. Adv. Math. 213 (2007), 230--270. (Available online.)

Mark Weber:

Monads with arities and parametric right adjoints

Abstract: The ultimate goal when trying to describe any higher categorical structure algebraically, is to give a complete description of a monad whose algebras are the structure in question. Experience has shown us that the monads that so arise satisfy some rather nice general conditions. In my two lectures I will describe two rather abstract conditions on a monad T — (1) "being parametrically representable" and (2) "having arities" — and how just these conditions enable one to consider nerves of T-algebras. In this way the homotopical point of view arises in a natural way from the algebraic one.

References:
• M. Weber, Generic morphisms, parametric representations and weakly Cartesian monads. Theory Appl. Categ. 13 (2004), 191--234 (electronic).
• M. Weber, Familial 2-functors and parametric right adjoints. Theory Appl. Categ. 18 (2007), 665--732 (electronic).
• M. Batanin and M. Weber, Algebras of higher operads as enriched categories, ArXiv:0803.3594.

Georges Maltsiniotis:

1: Lax ∞-groupoids (d'après Grothendieck)

Abstract: We present Grothendieck's theory of lax ∞-groupoids as is developed in "Pursuing Stacks", and his construction of the corresponding fundamental ∞-groupoid of a topological space, a construction that he generalizes for any closed model category in which all objects are fibrant. We explain his conjecture asserting that lax ∞-groupoids classify homotopy types, and we suggest a strategy for proving it.

Reference: • G. Maltsiniotis, Infini groupoïdes non stricts, d'après Grothendieck. Preprint (2007).

2: Still another definition of lax ∞-categories

Abstract: Although Grothendieck introduced a notion of a lax ∞-groupoid he did not define a concept of lax ∞-category. We realized that a slight modification of his definition leads to a theory of ∞-categories very similar and perhaps equivalent to the theory of Batanin.

Reference: • G. Maltsiniotis, Infini catégories non strictes, une nouvelle définition. Preprint (2007).

Denis-Charles Cisinski:

Weakening higher structures

Abstract: Weak higher categories can be thought in different ways: either we can understand them using homotopical algebra, or we can try to define them as an algebraic structure. The homotopical point of view is already well developped for weak higher categories in which the i-cells are invertible for i>1: we have simplicial categories (Dwyer and Kan), Segal categories (Simpson et al.), complete Segal spaces (Rezk), and quasi-categories (Joyal). Moreover, we know that all these points of view are equivalent in a precise sense (Bergner, Joyal and Tierney). In this talk, I will present a way to extend this homotopical point of view to general higher categories. The starting point is a parametric right adjoint monad T (in the sense of Weber). To such a monad, we can associate different notions of weak T-algebras: complete Segal T-spaces (in the spirit of Rezk) and quasi-T-algebras (in the spirit of Joyal). These two points of view are always equivalent. This will produce in particular a reasonable theory of weak n-categories (with n finite or not). The notion of Segal n-category will also find its way into this picture (and will also be equivalent to the other points of view). More generally, under some mild assumptions on T, there is a theory of Segal categories enriched in weak T-algebras which is equivalent to the the theory of weak U-algebras, where U is the parametric right adjoint monad whose algebras are the categories enriched in T-algebras (this is closely related with Berger's wreath product construction). Another interesting example: the notion of quasi-operad (dendroidal inner Kan complex), developped by Moerdijk and Weiss, fits into this general picture as well. We will also study in which sense, for a general parametric right adjoint monad T, the theory of weak T-algebras can be related to homotopy types. This will lead to an understanding of the relationship between the notion of weak T-algebras and the notion of algebra on a weakening of T in the sense of Batanin. If time permits, I will give a precise program to prove that Batanin's point of view on weak infinity categories is also equivalent to the homotopical point of view (but this strategy requires some tools which are not completely available yet).

Bruno Vallette:

How to make explicit cofibrant models for operads?

Abstract: Categories of algebraic structures are modeled by different kinds of operads: non-symmetric operads, "symmetric" operads, colored operads, properads, props. Consider such a category and its associated operad P. With the data of P, one can define the suitable lax notion of P-algebra up to homotopy, the (co)homology theory for (homotopy) P-algebras and prove the homological perturbation lemma for (homotopy) P-algebras. But to make all these results explicit, one needs a cofibrant replacement for P. The purpose of these two talks is to explain the methods we have at hand to do this.
In the first talk, we will review the operadic homological constructions (twisting morphisms, bar and cobar constructions, twisted tensor product) and Koszul duality theory which is the first efficient method, when it works. We will explain it conceptually and we will show how to use it in practice.
In the second talk, we will show how to go beyond the Koszul case. We will give a few new methods to treat non-Koszul operads. Once again, we will apply it to new examples (associative bialgebras, Batalin-Vilkovisky algebras).

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.)

Carles Casacuberta:

Localization of algebras over coloured operads

Abstract:

Reference:
• C. Casacuberta, J. Gutiérrez, I. Moerdijk, R. Vogt, Localization of algebras over coloured operads, Preprint.

Joachim Kock:

Combinatorics of opetopes

Abstract: (This is joint work with André Joyal, Michael Batanin, and Jean-François Mascari.) Opetopes parametrise higher-dimensional many-in/one-out shapes. More precisely they are defined as operations for a certain sequence of higher operads (which are not globular). I will present an elementary, intuitive, and purely combinatorial definition of opetopes, and show how this description can be extracted from the classical definitions (due to Baez-Dolan and Leinster) using the theory of polynomial functors. Ingredients: trees. (May contain traces of nuts.)

Reference:
• J. Kock, A. Joyal, M. Batanin, J.-F. Mascari, Polynomial functors and opetopes, ArXiv:0706.1033.