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Title: Polycategories
Author: Garner, R. H. G.
Awarding Body: University of Cambridge
Current Institution: University of Cambridge
Date of Award: 2006
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We develop a categorically motivated description of Szabo’s notion of polycategory. Starting from the bicategory Mod of profunctors, we exhibit a pseudomonad Ŝ (encapsulating ‘input arities’) and a dual pseudocomonad T̂ (encapsulating ‘output arities’) on Mod, together with a pseudo-distributive law δ̂ of T̂ over Ŝ (encapsulating how composition of outputs with inputs should proceed). Polycategories are recovered as monads in the two-sided Kleisli category of this pseudo-distributive law on Mod, extending a well-known description of Lambek’s multicategories in similar terms. The above is the key idea at the heart of this thesis. However, producing the aforementioned pseudo-distributive law involves checking a prodigious amount of coherence data; some of this data is trivial, and some non-trivial. We tackle the ‘non-trivial’ data in detail. However, rather than gloss over the ‘trivial’ data, we develop a framework that will handle it for us. Indeed, our pseudo-distributive law δ̂ should be determined by its value at the terminal category 1, and be derivable at a general category C by simply ‘labelling with objects and arrows of C’. This property is highly reminiscent of Kelly’s theory of clubs. However, the theory of clubs only exists at a categorical level, rather than the bicategorical level necessary for application in Mod. A naïve generalisation of the theory of clubs to ‘bicategorical clubs’ would not help either, since Mod is poorly endowed with the (bicategorical) limits that such a theory would require. Instead, we develop ‘double clubs’, a fusion of Kelly’s clubs with the pseudo double categories of Grandis and Paré. These latter structures are an asymmetric generalisation of Ehresmann’s double categories: they have two directions, a vertical, ‘category-like’ direction and a horizontal, ‘bicategory-like’ direction. In particular, there is a pseudo double category Cat which vertically ‘looks like Cat’ and horizontally ‘looks like Mod’. Applying the machinery of double clubs in Cat, allows us to see that the ‘trivial’ coherence data for our pseudo-distributive law δ̂ is, as Freyd has it, ‘trivial for trivial reasons’.
Supervisor: Not available Sponsor: Not available
Qualification Name: Thesis (Ph.D.) Qualification Level: Doctoral
EThOS ID:  DOI: Not available