Categorical Semantics for Dynamically Typed Languages, Notes for History of Programming Languages, 2017

These are the notes for a talk I gave for Matthias Felleisen’s History of Programming Languages class. I’ve tried to avoid recounting detailed descriptions of category theory and domain theory here, which I think the cited papers themselves did a good job of doing. Instead, I’ve tried to take advantage of the connections between syntax and category theory to reframe some of the results in these papers as syntactic translations, especially the theorems in 4, 7. 1 Historical Overview In 1969, Dana Scott wrote a paper in which he said untyped lambda calculus had no mathematical meaning (Scott [1993]), 11 years later he wrote a paper that organized many of the different semantics he and others had since found using the language of category theory (Scott [1980]). This latter paper is really the first deserving of the title “categorical semantics of dynamic typing”, and so I’m going to present some of the theorems and “theorems” presented in that paper, but mingled with the history of the idea and the preceding papers that led to them. In Figure 1 we have a very skeletal timeline: On the left are some foundational categorical logic papers, on the right are some seminal semantics papers by Dana Scott that we’ll go over in some detail throughout this article. I drew them in parallel, but it is clear that there was interaction between the sides, for instance Scott says he received a suggestion by Lawvere in Scott [1972]. But what’s very interesting is that both Lambek and Scott wrote papers on cartesian closed categories and lambda calculus for the 1980 Haskell Curry Festschrift (Lambek [1980] and Scott [1980]), so this seems to have been something of a milestone for the use of category theory in programming languages. For completeness, since I won’t discuss them in detail below, the Lawvere paper is Lawvere [1969] where Lawvere introduced the notion of hyperdoctrine, which allows for the interpretation of higher-order logic. The Lambek papers form a