This talk covers most of section 4 in the Mathew-Naumann-Noel paper [MNN15]. We first discuss nilpotence in an arbitrary symmetric monoidal stable ∞-category. We then discuss the historical origins of nilpotence in the stable homotopy category, namely the Ravenel conjectures and the Nilpotence theorem proved by Devinatz-Hopkins-Smith.

This paper is about importing the stable homotopy theory of symmetric spectra [4] and more generally presheaves of symmetric spectra [8] into the Morel-Voevodsky stable category [9], [11], [12]. Loosely speaking, the latter is the result of formally inverting the functor X 7→ T∧X on the category of pointed simplicial presheaves on the smooth Nisnevich site of a field within the Morel-Voevodsky ...

There is an ongoing massive effort by many researchers to link category theory and geometry, especially homotopy coherence and categorical coherence. This constitutes just a part of the broad undertaking known as categorification as described by Baez and Dolan. This effort has as a partial goal that of understanding the categories and functors that correspond to loop spaces and their associated...

We give sufficient conditions for the existence of a model structure on operads in an arbitrary symmetric monoidal model category. General invariance properties for homotopy algebras over operads are deduced.

We give sufficient conditions for the existence of a model structure on operads in an arbitrary symmetric monoidal model category. General invariance properties for homotopy algebras over operads are deduced.

We take another look at the Chu construction and show how to simplify it by looking at it as a module category in a trivial Chu category This simpli es the construction substantially especially in the case of a non symmetric biclosed monoidal category We also show that if the original category is accessible then for any of a large class of polynomial like functors the category of coalgebras has...

We exhibit a complete set of identities for CNOT, the symmetric monoidal category generated by the controlled-not gate, the swap gate, and the computational ancillæ. We prove that CNOT is a discrete inverse category. Moreover, we prove that CNOT is equivalent to the category of partial isomorphisms of finitely-generated non-empty commutative torsors of characteristic 2. Equivalently this is the...

We define the phrase ‘category enriched in an fc-multicategory’ and explore some examples. An fc-multicategory is a very general kind of 2-dimensional structure, special cases of which are double categories, bicategories, monoidal categories and ordinary multicategories. Enrichment in an fc-multicategory extends the (more or less well-known) theories of enrichment in a monoidal category, in a b...

We show that the generic symmetric monoidal category with a commutative separable algebra which has a Σ-family of actions is the category of cospans of finite Σ-labelled graphs restricted to finite sets as objects, thus providing a syntax for automata on the alphabet Σ. We use this result to produce semantic functors for Σautomata.

The monoidal category of Soergel bimodules can be thought of as a categorification of the Hecke algebra of a finite Weyl group. We present this category, when the Weyl group is the symmetric group, in the language of planar diagrams with local generators and local defining relations. Date: February 27, 2009.