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Generalising Lawvere theories to an axiomatically defined base

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We generalise the correspondence between Lawvere theories and finitary monads on Set in two ways. First, we allow our theories to be enriched in a category V that is locally finitely presentable as a symmetric monoidal closed category: symmetry is convenient but not necessary. And second, we allow the arities of our theories to be finitely presentable objects of a locally finitely presentable V-category A. We extend the correspondence for ordinary Lawvere theories to one between such generalised Lawvere theories and finitary V-monads on A. A leading example of the utility of this generalisation is given by the generalised Lawvere theory on the ordinary category Cat for which the models are all small cartesian closed categories.

This talk is part of the Category Theory Seminar series.

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