The projection formula, extranatural transformations and surface diagrams

Abstract: This talk is motivated by trying to understand how closed monoidal categories fit into higher categorical frameworks, in particular to understand why, in higher categorical terms, the projection formula -- f_!(a x f^*b) = f_!(a) x b -- holds for an adjunction f_! -| f^* when f^* strong closed monoidal. The circle of ideas involves the notion of extranatural transformation, which is key to a formal definition of closed monoidal category. These can be represented graphically using 'surface diagrams' and it transpires that these actually have a natural interpretation in terms of the monoidal double category of categories, functors and profunctors. Along the way we will see the notion of conjugation for adjunctions of two variables which will help formalize the projection formula result.

Computer sciencecategory theorylogic

Audience: learners

( video )

Topos Institute Colloquium