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Title: A unified approach to the construction of categories of games
Author: Bowler, Nathan James
Awarding Body: University of Cambridge
Current Institution: University of Cambridge
Date of Award: 2011
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Our aim is to explain a way to construct categories of games by combining 2-dimensional structures called playpens. We use a running example of a category of games based on digraphs. We explain some basic properties of digraphs, including a construction of a tensor product on this category by means of a promonoidal structure on the category consisting of a parallel pair of maps. We describe a corefiective subcategory of the category of digraphs which is naturally thought of as a category of trees. To motivate the 2-dimensional structures we shall use, we introduce the concept of unwiring. This formalises the idea of a structure whose elements can be decomposed over the shapes provided by a monad. We show that this construction picks out a well-behaved class of exponentiable objects, and allows the lifting of exponentiation functors to categories of algebras. We explore the details of the special case of this construction for free multicategory monads. Examining what happens for the free fc-mulicategory monad leads to the introduction of playpens. We explore simple constructions and examples of playpens, and show how these may be used to produce an fc-multicategory of games based on digraphs. We outline the theory of represent ability for fc-multicategories, and we demonstrate that the fc-multicategory we've produced underlies a strict double category, whose horizontal part is a standard category of games. We explore a link between this construction and slicing in fc-multicategories. We explain how to modify the construction so far to produce categories with different conventions for play, with different notions of strategy, and with different combinatorial notions of game (including trees of the kind mentioned earlier). By means of these examples, we hope to indicate the variety of potential further extensions of our construction.
Supervisor: Not available Sponsor: Not available
Qualification Name: Thesis (Ph.D.) Qualification Level: Doctoral
EThOS ID:  DOI: Not available