The fundamental feature of operadic categories is that the objects under study are viewed as algebras over (generalized) operads in a specific operadic category. For instance, operads are algebras over the terminal operad in the operadic category of rooted trees, modular operads are algebras over the terminal operad in the operadic
category of genus-graded connected graphs, wheeled PROPs are algebras over directed graphs, &c. Moreover, operadic categories provide natural environments for Batanin's n-operads, tubings on a graph, decomposition spaces, decalage comonads, and other exotic structures. Operadic categories offer a concise framework for constructing infinity versions of operad-like objects. Operadic Grothendieck's construction is available as a powerful tool for obtaining new operadic categories from old ones. 

The (lax) Gray tensor product (or more accurately its associated closed structure) plays a crucial role in, among other things, Street's formal theory of monads. The goal of this talk is to homotopify the statement "the Gray tensor product forms part of a monoidal closed structure on 2-Cat" into a 2-quasi-categorical version, with an eye towards developing the formal theory of homotopy coherent monads.

The first half of the talk will be devoted to describing the main combinatorial tool I used to prove this result, namely Oury's inner horns. These horns provide a combinatorially tractable characterisation of Ara's model structure.

Bibliography:

Dimitri Ara. Higher quasi-categories vs higher Rezk spaces. Journal of K-Theory. K-Theory and its Applications in Algebra, Geometry, Analysis & Topology, 14(3):701, 2014.

John W. Gray. Formal category theory: adjointness for 2-categories. Lecture Notes in Mathematics, Vol. 391. Springer-Verlag, Berlin-New York, 1974.

David Oury. Duality for Joyal’s category Θ and homotopy concepts for Θ_2-sets. PhD thesis, Macquarie University, 2010.

Ross Street. The formal theory of monads. J. Pure Appl. Algebra, 2(2):149–168, 1972.

The classical Kan--Quillen model structure on the category of simplicial sets is a fundamental object in homotopy theory. Many proofs of its existence have been found, but (until recently) all of them relied on principles of classical logic: the law of excluded middle and the axiom of choice. For the purposes of interpretation of Homotopy Type Theory, such arguments were insufficient.

A fully constructive proof is possible, but it requires a careful adjustment of the definitions of cofibrations and weak homotopy equivalences of simplicial sets. In the talk, I will outline one such constructive argument, explaining how various standard techniques of simplicial homotopy theory need to be adapted to constructive logic.

This is joint work with Nicola Gambino and Christian Sattler, inspired by Simon Henry who was the first to obtain the result.

The problem of dealing with infinitely many coherence constraints in oo-category theory when trying to define oo-functors has lead to a fibrational approach, in which one represents diagrams of the form B-->Cat_oo as a suitable kinds of fibrations over B. While this is a theorem, due to Lurie, in the case of oo-categories (i.e. (oo,1)-categories), so far there has been no combinatorial definition of  cartesian fibrations of (oo,2)-categories.

In this talk, we will define cartesian fibrations in this context, prove some of their basic properties and show they are equivalent (under a suitable equivalence of (oo,2)-categories) to the counterpart in the context of categories enriched over marked simplicial sets (where the definition is given, mutatis mutandis, based on what happens with 2-categories). Furthermore, we will prove some statements made by Gaitsgory and Rozemblyum concerning locally cartesian fibrations and (oo,2)-categories fibred over (oo,1)-categories, with the intent of substantiating the validity of our definition.

I'll describe modules for disklike categories, and then how to use modules as boundary data for the blob complex. This leads to a natural gluing formula.

I will explain several new results about fusion rings and all flavors of fusion category using number theory so elementary that proper number theorists have audibly laughed when I called it that.  Most of these results are theorems about weakly integral fusion, braided, or modular tensor categories C, restated so that FPdim(C) can live freely in your favorite totally real algebraic number field.  Highlights include a dimensional grading for all fusion rings, bounds on the order of multiplicative central charge based on FPdim(C), and proof that all of the dimensions appearing in the fusion categories coming from quantum groups are kind of boring.  Please encourage attendance by graduate students as I will discuss many approachable open problems. (joint work with Terry Gannon)