654 research outputs found
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 -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
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