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Glynn Winskel presented a paper with Hugo Paquet from Inria, Paris at the Mathematical Foundations of Program Semantics Conference (MFPS) in Ljubljana. The paper presents a very general model of concurrent games on event structures with symmetry, in which strategies satisfy the laws of composition up to a weak notion of equivalence. It brings together several existing approaches to symmetry and gives a neat characterisation of concurrent strategies with symmetry as Street fibrations.

The conference also included a tribute session for Alex Simpson on his reaching 60. The immensely talented John Longley composed, played and sang a wonderful Gilbert-and-Sullivan-inspired song on mathematicians cum computer scientists.

Abstract
Recently the theory of event structures has been used as a foundation for an expressive model of concurrent games and strategies. In this paper we revisit the problem of adding symmetry to concurrent games and strategies. Symmetry is essential for many applications but a source of mathematical complexity, in part because strategies with symmetry can be considered up to several notions of equivalence.

Our first contribution is a complete characterization of two classes of strategies for which composition admits an identity. For these two classes the identity laws hold, respectively, up to a 'weak' and a 'strong' notion of equivalence. We show that the weak class of strategies corresponds precisely to those inducing Street fibrations over the game. We show how the established 'thin' approach to symmetry also fits in this general framework.

A second contribution is a formal connection between games and higher-categorical profunctors. This follows a long tradition of relating games and relational models, but here additionally includes a very general model of symmetries. Our characterization of strategies as Street fibrations makes the connection to profunctors significantly easier to establish.

The final version of the paper will be published in September.