A generalized map between Lie groupoids is obtained by inverting essential equivalences using the bicalculus of fractions approach. We develop a notion of homotopy between generalized maps which is suitable for applications to orbifolds. This notion of homotopy is invariant under Morita equivalence and generalizes the notions of natural transformation and strict homotopy for functors. Pretalk at 3:30 in same room.
Abstract: I will discuss how power operations in elliptic cohomology are related to the notion of replicability in Moonshine. I will describe equivariant power operations in Devoto's equivariant versions of K-Tate, and discuss the corresponding loop-space picture.
Let E_* be a good homology theory and A a commutative algebra in E_*E comodules. Then we have the following realization problem: is there a commutative S-algebra (aka E_\infty ring spectrum) so that E_*X = A? More generally, what is the homotopy type of the space of all such realizations? We describe a decomposition of this realization space as a tower of fibrations where the successive fibers depend only on algebraic data. This leads to an obstruction theory for deciding if the space is non-empty or connected. These spaces appear in both the Hopkins-Miller theorem for the Lubin-Tate spectra E_n and for the original proof of the existence of topological modular forms. This was joint work with Mike Hopkins, building on work with Bill Dwyer, David Blanc, and others.
[Please note date and time change! This is the ordinary Proseminar time.] We give a framework for comparing on the one hand theories of n-categories that are weakly enriched operadically, and on the other hand theories given using a globular operad. Examples of the former are the definition by Trimble and variants (May/Cheng/Gurski) and examples of the latter are the definition by Batanin and variants (Leinster). We will show how to take a theory of n-categories of the former kind and produce a globular operad whose algebras are the n-categories we started with. Our main aim is to show how the globular operad associated with Trimble's topological definition in this way, is related to the globular operad used by Batanin to define fundamental n-groupoids of spaces. This has the potential to provide an easier method for proving Batanin's Homotopy Hypothesis than the partial proofs currently in the literature; it also has the potential to provide a useful class of semi-strict n-categories.
Until recently, there was no construction of any categorical structure with objects the tricategories of Gordon, Power, and Street. On the other hand, bicategories give rise to a rich variety of categories, bicategories, and tricategories. After reviewing the cases of bicategories, I will discuss what sorts of structures we might higher categories to form, how that story plays out for tricategories, and some possible applications. This is joint work with Richard Garner.
A Lie 2-algebra is a categorified version of a Lie algebra where we have replaced the Jacobi identity by a natural isomorphism called the "Jacobiator," which in turn must satisfy a certain law of its own. This law is closely related to the Zamolodchikov tetrahedron equation, which is the higher-dimensional analogue of the third Reidemeister move. Lie 2-algebras can be classified in terms of third cohomology classes in Lie algebra cohomology. This classification allows us to construct for any finite-dimensional Lie algebra g a 1-parameter family of Lie 2-algebras, g_k, each having the simple Lie algebra g as its Lie algebra of objects, but with a Jacobiator proportional to a real number k and built from the Killing form. In this talk, we will explore current work defining a representation theory of Lie 2-algebras.
I will describe how to construct the Morava-Hecke algebra from deformations of isogenies of certain formal groups. This algebra acts naturally on the coefficients of K(n)-local algebras over the Morava E-theories and is therefore useful for E-theory calculations. This is joint work in progress with Neil Strickland.
I'll sketch a recent construction of operations on the homology of the free loop space of a smooth manifold. These operations are parameterized by the homology of the moduli space of Riemann surfaces with parameterized boundary components. This generalizes the BV structure introduced by Chas and Sullivan. It also is compatible with well-known operations on the homology of the manifold itself. If time permits, I'll also discuss how these structure might lead to proving that these are only homotopy invariants.
University of Chicago Algebraic Topology Seminar
University of Chicago Algebraic Topology Seminar