Workshop // 12-14 July 2017 // Osnabrück

Infinity-Operads and Applications

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Gijs Heuts, Javier J. Gutiérrez and Ieke Moerdijk

Lectures on dendroidal topology

Dendroidal sets and dendroidal spaces form a natural extension of simplicial sets and spaces, and relate to topological operads in the same way as simplicial sets do to topological categories. In the dendroidal theory, the category of finite linear orders parametrizing simplicial objects is replaced by a category or finite rooted trees. In a series of nine lectures, together we will present some basic aspects of the theory, which parallels the simplicial theory to some extent but is technically more involved due to the presence of automorphism groups of trees. Special attention will be paid to the relation with infinite loop spaces, Goodwillie calculus, and self-equivalences of model categories.

In more detail, the plan is as follows.

• Lectures 1 and 2 (JG). The category of dendroidal sets will be introduced, together with its tensor product, and the operadic Quillen model structure modelling infinity operads will be discussed.
• Lecture 3 (GH). We will start the preparations for the relation to Goodwillie calculus by giving an overview of the classical theory. Following work of Ching and Arone, we will discuss the operad structure on the derivatives of the identity functor and the role of bar-cobar duality.
• Lecture 4 (IM). We will present the Quillen equivalent theory of complete Segal dendroidal spaces and its equivalence to topological (or simplicial) operads.
• Lecture 5 (IM). We will describe the covariant model structure on dendroidal sets and spaces, which serves to describe the homotopy theory of algebras over operads.
• Lecture 6 (GH). We will discuss stabilization of infinity categories and enter into the details of constructing a stable infinity operad out of a sufficiently nice such category. We will see that such stable infinity operads are closely related to cooperads in the category of spectra.
• Lecture 7 (JG). We will use the model of complete segal dendroidal spaces to identify the (∞,1)-category of auto-equivalences of the (∞,1)-category of infinity-operads: it is a contractible ∞-groupoid, which informally tells us that if there is essentially only one way to compare two models for the theory of ∞-operads.
• Lecture 8 (IM). We will discuss the relation between the covariant model structure mentioned earlier to infinite loop spaces and Segal's category of Gamma spaces.
• Lecture 9 (GH). We will return to the question introduced in his first lecture, rephrased as: given a stable infinity-operad O, can one classify the infinity-categories whose stabilization agrees with O? The approach to this problem will be inductive, by considering the truncations of this operad O. Examples include Quillen's rational homotopy theory, as well as various other localizations of unstable homotopy theory.

Selected references

• D. Ara, M. Groth and J.J. Gutiérrez, On autoequivalences of the (infinity,1)-category of infinity-operads, Math. Z. 281 (2015).
• P. Boavida and I. Moerdijk, Dendroidal spaces, Gamma spaces and the special Barratt-Priddy-Quillen theorem, ArXiv 2017.
• D.-C. Cisinski and I. Moerdijk, Dendroidal sets and simplicial operads, J. Topol. 6 (2013).
• T. Goodwillie, Calculus III, Taylor series, Geom. Top 7 (2003).
• G. Heuts, Goodwillie approximations to hiogher categories, ArXiv 2015.
• J. Lurie, Higher Topos Theory, Princeton University Press, Princeton (2009).
• J. Lurie, Higher Algebra (2016).
• I. Moerdijk, Lectures on dendroidal sets (notes by J. Gutiérrez), CRM Barcelona 2007.
• B. Toën, Vers une axiomatisation de la théorie des catégories supérieures. K Theory 34(3) (2005).

Research Talks

Pedro Boavida

On En

I'll survey some recent results concerning little disks operads and their mapping spaces. These mapping spaces turn out to be relevant both in topology (via a relation to embedding spaces) and in algebra (via graph complexes and deformation theory). Infinity operads play an important role in making some of such connections.

Philip Hackney

The homotopy theory of Segal cyclic operads

Cyclic operads were introduced by Getzler and Kapranov as a suitable general setting for defining cyclic cohomology. Roughly, a cyclic operad is an operad-like structure where we relax the distinction between 'inputs' and 'outputs.' Many familiar operads admit a cyclic structure, for instance the associative, Lie and commutative operads, the A-infinity operad, and the framed little disks operads. In support of a project of Boavida, Horel, and Robertson on profinite completions of the framed little disks operad, we lay the foundations for homotopy-coherent versions of cyclic operads.

In pursuit of this goal, we take as inspiration the theory of dendroidal objects, which is used to model homotopy-coherent operads. There is a category of unrooted trees which is closely related to the Moerdijk-Weiss category of rooted trees used in the dendroidal picture. Cyclic operads can be regarded as those presheaves on the category of unrooted trees which satisfy a strict Segal condition. Segal cyclic operads are precisely those (reduced) presheaves satisfying a weak Segal condition. We show that there is a Quillen model category structure on this category of presheaves whose fibrant objects are precisely the Segal cyclic operads.

This work is joint with Marcy Robertson and Donald Yau.

Rune Haugseng

Infinity-operads as polynomial monads

I will describe a new model for infinity-operads as polynomial monads. This extends to the infinity-categorical setting the description of (one-coloured) operads as associative monoids in symmetric sequences, as well as the natural generalization of this to coloured operads. This is joint work with David Gepner and Joachim Kock.

Brice Le Grignou

Linear operads up to homotopy

Operads are tools to encode algebraic structures. But they are themselves an algebraic structure encoded by an (colored) operad. In the framework of chain complexes, the Koszul duality allows us to compute a resolution of this operad of operads and then to describe a notion of linear operads up to homotopy. All of this is related to an adjunction called bar-cobar which links operads to conilpotent cooperads. We show that conilpotent cooperads provide a convenient framework to describe the homotopy theory of operads up to homotopy. Finally, we build a bridge relating these ones to dendroidal sets.

Marcy Robertson

Operads of genus zero curves and the Grothendieck-Teichmüller group

We show that the group of homotopy automorphisms of the profinite completion of the genus zero surface operad is isomorphic to the (profinite) Grothendieck-Teichmüller group. Using a result of Drummond-Cole, we deduce that the Grothendieck-Teichmüller group acts nontrivially on the operad of stable curves of genus zero.
This result uses, in a fundamental way, the fact that the profinite completion of an operad is an infinity operad. This is joint work with Pedro Boavida and Geoffroy Horel.

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