Seminar: Simon Willerton - The projection formula, extranatural transformations and surface diagrams in a monoidal double category (Apr 14, 2026)

Speaker: Simon Willerton (University of Sheffield)
Title: The projection formula, extranatural transformations and surface diagrams in a monoidal double category
Time: April 14, 2026 at 19:00 Istanbul local time (17:00 Sheffield local time)
Place: Online
Meeting ID: 935 6390 4955
Passcode: 699568
Abstract: This talk is motivated by trying to understand how closed monoidal categories fit into higher categorical frameworks, in particular to understand why, in abstract 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 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.
[See also https://arxiv.org/abs/2501.01881]