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Fibred pseudo double categories for game semantics

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Clovis Eberhart, Tom Hirschowitz

We unify previous constructions from our work on concurrent game
semantics into a single categorical framework. From an operational
description of positions and moves in some game, called a
\emph{signature}, we produce a pseudo double category, in which
objects are positions and vertical morphisms are plays. The
considered games are multi-player, so it makes sense to consider
embeddings of positions: these are the horizontal
morphisms. Finally, cells may be thought of as embeddings of plays
preserving initial and final positions. In order to be suitable for
game semantics, the obtained pseudo double category should enjoy a
certain fibredness property. Under suitable hypotheses, we show that
our construction actually produces such a \emph{fibred} pseudo
double category, from which we can define relevant categories of
plays, and thus of strategies. We give a first necessary and
sufficient criterion for this to hold and then a sufficient
criterion that can be checked more easily.

Keywords:
concurrent game semantics, pseudo double categories, factorisation systems

2010 MSC:
18C50, 03B70, 68Q55

*Theory and Applications of Categories,*
Vol. 34, 2019,
No. 19, pp 514-572.

Published 2019-06-14.

http://www.tac.mta.ca/tac/volumes/34/19/34-19.pdf

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