Operadic categories and Duoidal Deligne's conjecture

The purpose of this paper is two-fold. In Part 1 we introduce a new theory of operadic categories and their operads. This theory is, in our opinion, of an independent value. In Part 2 we use this new theory together with our previous results to prove that multiplicative 1-operads in duoidal categories admit, under some mild conditions on the underlying monoidal category, natural actions of contractible 2-operads. The result of D. Tamarkin on the structure of dg-categories, as well as the classical Deligne conjecture for the Hochschild cohomology, is a particular case of this statement.Comment: 54 pages, to appear in Advances in Mathematic

Bar constructions for topological operads and the Goodwillie derivatives of the identity

We describe a cooperad structure on the simplicial bar construction on a reduced operad of based spaces or spectra and, dually, an operad structure on the cobar construction on a cooperad. We also show that if the homology of the original operad (respectively, cooperad) is Koszul, then the homology of the bar (respectively, cobar) construction is the Koszul dual. We use our results to construct an operad structure on the partition poset models for the Goodwillie derivatives of the identity functor on based spaces and show that this induces the `Lie' operad structure on the homology groups of these derivatives. We also extend the bar construction to modules over operads (and, dually, to comodules over cooperads) and show that a based space naturally gives rise to a left module over the operad formed by the derivatives of the identity.Comment: Published by Geometry and Topology at http://www.maths.warwick.ac.uk/gt/GTVol9/paper20.abs.html Version 3: Reference to Salvatore added (see footnote 3, page 834

Koszul duality in operadic categories

Our aim is to set up the cornerstones of Koszul duality in general operadic categories. In particular, we will prove that operads (in our generalized sense) governing the most important operad- and/or PROP-like structures as classical operads, their variants as cyclic, modular or wheeled operads, and also diverse versions of PROPs such as properads, dioperads, 1/2-PROPs, and still more exotic stuff as permutads and pre-permutads are quadratic, and describe their Koszul duals. To this end we single out some additional properties of operadic categories ensuring that free operads admit a nice explicit description, and investigate how these properties interact with discrete operadic (op)fibrations which we use as a mighty tool to construct new operadic categories from the old ones. Particular attention is payed to the operadic category of graphs and to its clones, but several other examples are given as well. Our present work provides an answer to the questions "What does encode a type of operads?" and "How to construct Koszul duals to these objects?" formulated in the last Loday's 2012 talk in Lille.Comment: 108 pages; a section about the +-construction adde

Exploratory applications of the Fr\"ohlich-Morchio-Strocchi mechanism in quantum gravity

A manifestly diffeomorphism-invariant approach to canonical quantum gravity requires to use composite operators. These can be considered to be bound states of matter and/or gravitons, intrinsically non-perturbative objects. An analytical approach to determine the properties of such bound states could be the Fr\"ohlich-Morchio-Strocchi mechanism. We explore the necessary technology by applying it to various $n$-point functions, including geon propagators and black-hole-particle vertices.Comment: 27 page

A Process Algebra for Link Layer Protocols

We propose a process algebra for link layer protocols, featuring a unique mechanism for modelling frame collisions. We also formalise suitable liveness properties for link layer protocols specified in this framework. To show applicability we model and analyse two versions of the Carrier-Sense Multiple Access with Collision Avoidance (CSMA/CA) protocol. Our analysis confirms the hidden station problem for the version without virtual carrier sensing. However, we show that the version with virtual carrier sensing not only overcomes this problem, but also the exposed station problem with probability 1. Yet the protocol cannot guarantee packet delivery, not even with probability 1