We show how to apply forward and reverse mode Combinatory Homomorphic Automatic Differentiation (CHAD) (Vákár, 2021 ,b) to total functional programming languages with expressive type systems featuring the combination of • tuple types; • sum types; • inductive types; • coinductive types; • function types.We achieve this by analysing the categorical semantics of such types in Σ-types (Grothendieck constructions) of suitable categories. Using a novel categorical logical relations technique for such expressive type systems, we give a correctness proof of CHAD in this setting by showing that it computes the usual mathematical derivative of the function that the original program implements. The result is a principled, purely functional and provably correct method for performing forward and reverse mode automatic differentiation (AD) on total functional programming languages with expressive type systems.
This paper is a contribution towards a two dimensional extension of the basic ideas and results of Janelidze-Galois theory. In the present paper, we give a suitable counterpart notion to that of absolute admissible Galois structure for the lax idempotent context, compatible with the context of lax orthogonal factorization systems. As part of this work, we study lax comma 2categories, giving analogue results to the basic properties of the usual comma categories. We show that each morphism of a 2-category induces a 2-adjunction between lax comma 2-categories and comma 2-categories, playing the role of the usual change of base functors. With these induced 2-adjunctions, we are able to show that each 2-adjunction induces 2-adjunctions between lax comma 2-categories and comma 2-categories, which are our analogues of the usual lifting to the comma categories used in Janelidze-Galois theory. We give sufficient conditions under which these liftings are 2-premonadic and induce a lax idempotent 2-monad, which corresponds to our notion of 2-admissible 2-functor. In order to carry out this work, we analyse when a composition of 2-adjunctions is a lax idempotent 2-monad, and when it is 2-premonadic. We give then examples of our 2-admissible 2-functors (and, in particular, simple 2-functors), specially using a result that says that all admissible (2-)functors in the classical sense are also 2-admissible (and hence simple as well). We finish the paper relating coequalizers in lax comma 2-categories and Kan extensions.
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We give sufficient conditions for effective descent in categories of (generalized) internal multicategories. Two approaches to study effective descent morphisms are pursued. The first one relies on establishing the category of internal multicategories as an equalizer of categories of diagrams. The second approach extends the techniques developed by Ivan Le Creurer in his study of descent for internal essentially algebraic structures.
Given a pseudomonad T , we prove that a lax T -morphism between pseudoalgebras is a T -pseudomorphism if and only if there is a suitable (possibly noncanonical) invertible T -transformation. This result encompasses several results on noncanonical isomorphisms, including Lack's result on normal monoidal functors between braided monoidal categories, since it is applicable in any 2-category of pseudoalgebras, such as the 2-categories of monoidal categories, cocomplete categories, bicategories, pseudofunctors and so on.
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