Rencontre, June 15, 2023


10:30 – 12:00
Noam Zeilberger (École Polytechnique)

The classical statement of the Chomsky-Schützenberger representation theorem says that any context-free language may be represented as the homomorphic image of the intersection of a Dyck language of balanced parentheses with a regular language.  In the talk I will discuss a fibrational perspective on context-free grammars and finite-state automata that grew out of a long-running project with Paul-André Melliès on type refinement systems, but with a surprising twist that only emerged when we considered the C-S theorem.  It turns out that underlying that theorem is a basic adjunction between categories and colored operads (= multicategories), where the right adjoint $W : Cat \to Oper$ builds a "spliced arrow operad" out of any category, and the left adjoint $C : Oper \to \Cat$ sends any operad to a "contour category" whose arrows have a geometric interpretation as oriented contours of operations.

Based on joint work with Paul-André Melliès that appeared at MFPS 2022, as well as a long version of that article in preparation.

13:45 – 14:45
Aymeric Walch (IRIF)

We extend to general Cartesian categories the idea of Coherent Differentiation recently introduced by Ehrhard in the setting of categorical models of Linear Logic. The first ingredient is a summability structure which induces a partial left-additive structure on the category. Additional functoriality and naturality assumptions on this summability structure implement a differential calculus which can also be presented in a formalism close to Blute, Cockett and Seely's Cartesian differential categories.

15:15 – 16:15
Bryce Clarke (Partout team, INRIA)

What is a lens? As with many questions in mathematics, the answer depends on who you ask! Delta lenses are a particular kind of lens introduced in 2011 to model bidirectional transformations between systems. In elementary terms, a delta lens is a functor equipped with a suitable choice of lifts, generalising the notion of split opfibration. However, despite admitting a simple definition, many of the interesting and natural category-theoretic properties of delta lenses remain hidden from view without alternative characterisations. In this talk, I will reflect upon several perspectives of delta lenses developed in my PhD thesis, The double category of lenses, and demonstrate their richness as a mathematical structure. The talk will explore the connections between these different approaches, while emphasising the relationship between lenses and double categories, and conclude with several avenues for future work.