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Towards a Calculus of Substitution for Dinatural Transformations

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In 1972, with the papers “Many-Variable Functorial Calculus, I” and “An Abstract Approach of Coherence”, Kelly started a long-term project on achieving an abstract theory of coherence. To do so, he argues that a “tidy calculus of substitution” of functors in many variables and appropriately general natural transformations is in order; such a calculus ought to generalise the usual Godement calculus of functors in one variable and ordinary natural transformations. Inspired by the work he did with Eilenberg on extranatural transformations in 1966, he developed this calculus for (many-variable) covariant functors and natural transformations. Upon trying the mixed-variance case, he ran into problems linked to the fact that extranaturals do not compose. In this talk, I will show how we realised that the full mixed-variance case wanted by Kelly involves a simple generalisation of dinatural transformations; a sufficient and essentially necessary condition for two consecutive dinatural transformations to compose will be mentioned, and I will present a new definition of horizontal composition of dinaturals. Armed with these results, I will show how to achieve he first steps made by Kelly (in the mixed-variance case, this time) towards a full substitution calculus. There are still some conceptual difficulties about the remaining steps which are yet to be overcome.

This talk is part of the Theoretical computer science seminar series.

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