Workshop on sheaves and duality


7 & 8 April 2014 - IMAPP - RU Nijmegen


Program

Monday 7 April 2014
Location: Huygens gebouw (map), room HG00.086

10:00-10:30 Coffee

10:30-11:15 Mai Gehrke: Are idempotents red herrings?
11:15-12:00 Sebastiaan Terwijn: Genericity, randomness, and differentiable functions

12:00-13:30 Lunch

13:30-14:15 Vincenzo Marra: Baker-Beynon Duality with coefficients in a subring of the reals: A glimpse of arithmetic PL geometry. (abstract)
14:15-15:00 Bart Jacobs: Kadison duality

15:00-15:15 Coffee

15:15-16:00 Paul-Andr� Melli�s: Lawvere theories with arities: a bridge between higher dimensional algebra and programming language semantics (abstract)

Tuesday 8 April 2014
Location: Aula RU Nijmegen (map)

12:30 PhD defense Sam van Gool: On sheaves and duality

Registration

There is no registration fee; however, it is necessary to register in advance.
Please send an e-mail to by 15 March to do so.

Support

This workshop is made possible through the generous support of the Wiskundecluster DIAMANT, the Foundation Compositio Mathematica, the Arend Heyting foundation, and the Algebra & Topology group of IMAPP, RU Nijmegen.

Abstracts


Vincenzo Marra: Baker-Beynon Duality with coefficients in a subring of the reals: A glimpse of arithmetic PL geometry.

I will sketch recent unpublished results that amount to an extension of the affine version of the classical Baker-Beynon Duality to the following generalised setting. Let C be a subring of the real numbers, and let Q be its field of fractions. Let P(C) be the category of compact polyhedra that are unions of finitely many (convex) polytopes whose vertices have coordinates in Q, with morphisms that are the continuous piecewise (affine) linear maps whose affine pieces are affine functions with coefficients in C. (The morphisms compose precisely because the coefficients range in a subring C of the reals, as opposed to merely ranging, say, in an additive subgroup.) Theorem: P(C) is dually equivalent to the category of finitely presented algebras in a (finitary) variety. As one extreme case, for C=R the entire field of real numbers, one obtains that P(R) is the classical compact polyhedral category of PL-topology, and the dual algebraic structures may be identified with the finitely presented vector lattices equipped with a (strong order) unit. In other words, this is affine Baker-Beynon Duality for unital vector lattices. As the other extreme case, for R=Z the ring of integers one obtains that that P(Z) is the category of rational polyhedra and PL-maps with integer coefficients that is familiar to MV-algebraists: the dual algebraic structures are precisely the finitely presented MV-algebras. In this case the polyhedral topology interacts most strongly with arithmetic phenomena in ways that are fascinating, but as yet ill-understood. As we move from Z to R across subrings C, the interactions between arithmetic and topological phenomena weaken, and they vanish for the classical purely geometric C=R case. Open Problem: To work out the relationship between the setting sketched above and the recent sheaf-theoretic representations of MV-algebras via Stone-Priestley Duality developed in Sam van Gool's Doctoral Thesis.


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Paul-Andr� Melli�s: Lawvere theories with arities: a bridge between higher dimensional algebra and programming language semantics

There is a beautiful and well-known correspondence between finitary monads and Lawvere theories on the category of sets. In this talk, I will explain how to generalize this correspondence to every category C equipped with a notion of arities Theta defined as a full and dense subcategory of C. I will illustrate the resulting correspondence between monads with arities and Lawvere theories with arities with examples coming from higher dimensional algebra and from programming language semantics.


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