Topology, Geometry and Higher Structures — A Symposium in Honor of Ieke Moerdijk

For roughly fifteen years, beginning in the early 1990s, Ieke Moerdijk worked extensively on Lie groupoids and their applications, making several fundamental contributions to the field. This lecture will survey some of the work of Ieke, his students, and his collaborators in this area, as well as more recent developments in my own research that were influenced by these ideas.

Rui Loja Fernandes - University of Illinois

The convolution $C^*$-algebra of a Lie groupoid $G$ is often thought of as the noncommutative geometry of the differentiable stack presented by $G$. However, $C^*$-convolution is not functorial and forgets the smooth structure of $G$. We have proved that the bornologically completed convolution algebras of Lie groupoids and convolution bimodules of groupoid bibundles define a symmetric monoidal functor from the 2-category of differentiable stacks to the Morita 2-category of complete bornological algebras. I will explain the main result, give examples and immediate consequences, and discuss an application to the geometric quantization of Poisson manifolds. This is joint work with David Aretz.

Christian Blohmann - Max Planck Institute for Mathematics in Bonn

Segal’s category Γ (the dual of the category of finite sets and partial maps) comes equipped with an active/inert factorisation system, much exploited by Lurie in his formalism of ∞-operads.
Many categories of interest in algebraic topology are endowed with an active/inert factorisation system, together with an augmentation over Γ preserving this structure. Let us just mention the simplex category ∆, the cell categories Θn of Joyal, the dendrix category Ω of Moerdijk-Weiss, the cyclo-dendrix category Ξ of Hackney-Robertson-Yau, and its graphical extension Φ.
In all these cases, there is a precise analog of Segal’s Γ-spaces, as well as a concomitant homotopy theory, pertinent for various incarnations of operads. The purpose of my talk is to outline common features of these “Segalic” homotopy theories. If time permits I will discuss the following three topics:
(1) the equivalence between Lurie’s and dendroidal ∞-operads, based on the inclusion of the plus-construction Γ+ into Ω.
(2) the additivity theorem for En-operads, based on the wreath product Θn o Θm = Θn+m and on Batanin’s notion of n-operad.
(3) the derived modular envelope of the cyclic associative operad and its relationship with moduli spaces, based on the full embedding of Ξ into Φ.

Clemens Berger - Côte d'Azur University

Classical Tannakian duality over a field k provides a relation between certain symmetric monoidal k-linear categories and stacks over k. In this talk, I will discuss an extension of this correspondence to more general base rings, in the setting of infinity categories. It turns out that symmetric monoidal infinity categories are not very well-adapted to this kind of algebro-geometric problem. I will present a proposal for an enhancement of the notion of symmetric monoidal infinity category, that includes not only tensor products, but also derived symmetric towers. All of this is based on joint work with Bertrand Toën.

Joost Nuiten - University of Toulouse

Let g be a Lie algebra, with Lie group G. The space of connection 1-forms A¹(Δ¹, g) on the 1-simplex is naturally isomorphic to the space P_*G of based paths in G. The one-parameter semigroups in G correspond to the constant connection 1-forms, and G is the leaf space of a natural foliation on A¹(Δ¹, g), whose leaves are the gauge equivalence classes of connections. In this language, holonomy is the retraction from the space of connections to the gauge slice of constant connection 1-forms. (This map is well-defined in a small neighborhood of the constant small connections, and globally for nilpotent Lie algebras.)

Once couched in the language of homological perturbation theory for L_∞-algebras, this picture extends to all dimensions. Let g be an L_∞-algebra, and consider the L_∞-algebra A^*(Δⁿ, g). The condition of being a constant connection 1-form is replaced by lying in the kernel of the Dupont homotopy s_n: A^*(Δⁿ, g) → A^*−1(Δⁿ, g); this map is a contracting chain homotopy from the complex of differential forms to the complex W^*(Δⁿ, g) of Whitney elementary forms. Homological perturbation theory yields an L_∞-structure on W^*(Δⁿ, g), an L_∞-morphism f_n from A^*(Δⁿ, g) to W^*(Δⁿ, g), and taking the sets of Maurer-Cartan elements, a function MC(f): A^*(Δⁿ, g) → W^*(Δⁿ, g). This is the higher holonomy in dimension n.

In this talk, I will discuss a new approach to homological perturbation theory which is reminiscent of the proof of the Poincaré lemma: this makes the structure of the construction more transparent than in the usual treatments, and allows for an extension to cyclic L_∞-algebras (L_∞-algebras with an invariant inner product).

Ezra Getzler - Northwestern University