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09:00 | Polynomial models of type theory ABSTRACT. Polynomials (also known as containers) represent datatypes which, like polynomial functions, can be expressed using sums and products. Extending this analogy, I will describe the category of polynomials in terms of sums and products for fibrations. This category arises from a distributive law between the pseudomonad ‘freely adding’ indexed sums to a fibration, and its dual adding indexed products. A fibration with sums and products is essentially the structure defining a categorical model of dependent type theory. I will show how the process of adding sums to such a fibration is an instance of a general 'gluing' construction for building new models from old ones. In particular we can obtain new models of type theory in categories of polynomials. Finally, I will explore the properties of other type formers in these models, and consider which logical principles are and are not preserved by the construction. |

10:00 | SPEAKER: Maaike Zwart ABSTRACT. While papers showing examples of distributive laws are abundant, papers containing the opposite result - that certain distributive laws cannot exist - are rarely seen. In fact, the only example known to us is in 'Distributing Probability over Nondeterminism' by Varacca and Winskel, which contains a proof by Plotkin that the probability monad does not distribute over the powerset monad. In 2008, Manes and Mulry posed the question of whether the list monad distributes over itself. We solve this question, proving that both the list monad and powerset monad do not distribute over themselves, while there is precisely one distributive law for the multiset monad over itself. |

14:00 | Ornamentation put into practice in ML ABSTRACT. Ornaments have been introduced as a means to lift functions operating on some datatype into new functions operating on an ornamentation of this datatype, i.e., a variant of this datatype carrying additional information. A typical example consists in viewing lists as an ornamentation of natural numbers and then lifting the addition into the concatenation---the user just providing the missing parts of the code. Ornaments have first been introduced in Agda and extensively studied in a categorical setting. We have been exploiting and adapting the idea of ornamentation in the context of ML. Here, we are interested in syntactic liftings where the lifted function is related to the original one as in the general setting but with a closer correspondence: on the one hand, the behavior (and ideally the computation steps) of the lifted function should coincide with the behavior of the original function, except for the additional user-provided code; on the other hand, the lifted code should be as close as possible to the code the user would have manually written. Our approach is to first synthesize a generic version of the base program that can then be specialized to any ornamentation of the base version (including the base version as a particular case), and finally simplified. While the generic version of ML base code requires some form of dependent types to typecheck, we ensure by construction, using staged computation and dependent types, that each instantiation to concrete ornaments can be simplified back into ML code. We use parametricity to show the correspondence between the base and lifted versions. We verified on (small) examples that the lifted code is often close to hand-written code. This approach to ornamentation can be extended to perform disornamentation---with new design challenges and interesting applications. Ornamentation and disornamentation are just two examples of a more general type-based approach to code refactoring and evolution. |

15:00 | Profunctor Optics and the Yoneda Lemma SPEAKER: Jeremy Gibbons ABSTRACT. Profunctor optics are a neat and composable representation of bidirectional data accessors, including lenses, and their dual, prisms. The profunctor representation exploits higher-order functions and higher-kinded type constructor classes. The relationship between the profunctor representation and the familiar representation in terms of "getter" and "setter" functions is not at all obvious. We derive the former from the latter, making the relationship clear. It turns out to be a fairly direct application of the Yoneda Lemma, arguably the most important result in category theory. |