Title: ***Paranatural Category Theory*** 

What is the appropriate notion of transformation between "difunctors", that is, functors C^op × C --> Set? In this talk, we'll examine several applications in the semantics/metatheory of functional programming and type theory where we find ourselves in a "Goldilocks"-type situation: the usual notion of natural transformation is too narrow (excluding the examples we're interested in) and standard notions of 'diagonal natural transforms' are too broad (so broad, indeed, that they fail to be closed under composition). I'll talk about the notion which I think is 'just right': strong dinatural transformations. Strong dinatural transformations (which I advocate for rechristening as 'paranatural transformations') are a concept neglected in the present literature, I claim, and one which can serve as the centerpiece of a striking and widely-useful mathematical theory (which I dub 'paranatural category theory'). I hope to bring attention to some of the basic results in this fascinating branch of category theory, and instigate further research into it.

A preprint covering this material is available on the arXiv at arxiv.org/abs/2307.09289.