From Koszul duality to Poincar\'e duality

We discuss the notion of Poincar\'e duality for graded algebras and its connections with the Koszul duality for quadratic Koszul algebras. The relevance of the Poincar\'e duality is pointed out for the existence of twisted potentials associated to Koszul algebras as well as for the extraction of a good generalization of Lie algebras among the quadratic-linear algebras.Comment: Dedicated to Raymond Stora. 27 page

Koszul duality for locally constant factorization algebras

Generalising Jacob Lurie's idea on the relation between the Verdier duality and the iterated loop space theory, we study the Koszul duality for locally constant factorisation algebras. We formulate an analogue of Lurie's "nonabelian Poincare duality" theorem (which is closely related to earlier results of Graeme Segal, of Dusa McDuff, and of Paolo Salvatore) in a symmetric monoidal stable infinity category carefully, using John Francis' notion of excision. Its proof depends on our earlier study of the Koszul duality for E_n-algebras. As a consequence, we obtain a Verdier type equivalence for factorisation algebras by a Koszul duality construction.Comment: 32 pages. Section 2.0 slightly simplified, References updated. Comments welcome

Harmonic analysis on the SU(2) dynamical quantum group

Dynamical quantum groups were recently introduced by Etingof and Varchenko as an algebraic framework for studying the dynamical Yang-Baxter equation, which is precisely the Yang-Baxter equation satisfied by 6j-symbols. We investigate one of the simplest examples, generalizing the standard SU(2) quantum group. The matrix elements for its corepresentations are identified with Askey-Wilson polynomials, and the Haar measure with the Askey-Wilson measure. The discrete orthogonality of the matrix elements yield the orthogonality of q-Racah polynomials (or quantum 6j-symbols). The Clebsch-Gordan coefficients for representations and corepresentations are also identified with q-Racah polynomials. This results in new algebraic proofs of the Biedenharn-Elliott identity satisfied by quantum 6j-symbols.Comment: 51 pages; minor correction

Realizability Toposes from Specifications

We investigate a framework of Krivine realizability with I/O effects, and present a method of associating realizability models to specifications on the I/O behavior of processes, by using adequate interpretations of the central concepts of `pole' and `proof-like term'. This method does in particular allow to associate realizability models to computable functions. Following recent work of Streicher and others we show how these models give rise to triposes and toposes

Classical realizability in the CPS target language

AbstractMotivated by considerations about Krivine's classical realizability, we introduce a term calculus for an intuitionistic logic with record types, which we call the CPS target language. We give a reformulation of the constructions of classical realizability in this language, using the categorical techniques of realizability triposes and toposes.We argue that the presentation of classical realizability in the CPS target language simplifies calculations in realizability toposes, in particular it admits a nice presentation of conjunction as intersection type which is inspired by Girard's ludics

Descent properties of topological chiral homology

We study descent properties of Jacob Lurie’s topological chiral homology. We prove that this homology theory satisfies descent for a factorizing cover, as defined by Kevin Costello and Owen Gwilliam. We also obtain a generalization of Lurie’s approach to this homology theory, which leads to a product formula for the infinity 1-category of factorization algebras, and its twisted generalization