Cartesian Difference Categories

Cartesian differential categories are categories equipped with a differential combinator which axiomatizes the directional derivative. Important models of Cartesian differential categories include classical differential calculus of smooth functions and categorical models of the differential $\lambda$-calculus. However, Cartesian differential categories cannot account for other interesting notions of differentiation of a more discrete nature such as the calculus of finite differences. On the other hand, change action models have been shown to capture these examples as well as more "exotic" examples of differentiation. But change action models are very general and do not share the nice properties of Cartesian differential categories. In this paper, we introduce Cartesian difference categories as a bridge between Cartesian differential categories and change action models. We show that every Cartesian differential category is a Cartesian difference category, and how certain well-behaved change action models are Cartesian difference categories. In particular, Cartesian difference categories model both the differential calculus of smooth functions and the calculus of finite differences. Furthermore, every Cartesian difference category comes equipped with a tangent bundle monad whose Kleisli category is again a Cartesian difference category.

Categorical Stochastic Processes and Likelihood

We take a category-theoretic perspective on the relationship between probabilistic modeling and gradient based optimization. We define two extensions of function composition to stochastic process subordination: one based on a co-Kleisli category and one based on the parameterization of a category with a Lawvere theory. We show how these extensions relate to the category of Markov kernels Stoch through a pushforward procedure.We extend stochastic processes to parametric statistical models and define a way to compose the likelihood functions of these models. We demonstrate how the maximum likelihood estimation procedure defines a family of identity-on-objects functors from categories of statistical models to the category of supervised learning algorithms Learn.Code to accompany this paper can be found on GitHub (https://github.com/dshieble/Categorical_Stochastic_Processes_and_Likelihood).

Jets and differential linear logic

We prove that the category of vector bundles over a fixed smooth manifold and its corresponding category of convenient modules are models for intuitionistic differential linear logic. The exponential modality is modelled by composing the jet comonad, whose Kleisli category has linear differential operators as morphisms, with the more familiar distributional comonad, whose Kleisli category has smooth maps as morphisms. Combining the two comonads gives a new interpretation of the semantics of differential linear logic where the Kleisli morphisms are smooth local functionals, or equivalently, smooth partial differential operators, and the codereliction map induces the functional derivative. This points towards a logic, and hence a computational theory of non-linear partial differential equations and their solutions based on variational calculus.

Cartesian Difference Categories

Cartesian differential categories are categories equipped with a differential combinator which axiomatizes the directional derivative. Important models of Cartesian differential categories include classical differential calculus of smooth functions and categorical models of the differential $$\lambda $$ λ -calculus. However, Cartesian differential categories cannot account for other interesting notions of differentiation such as the calculus of finite differences or the Boolean differential calculus. On the other hand, change action models have been shown to capture these examples as well as more “exotic” examples of differentiation. However, change action models are very general and do not share the nice properties of a Cartesian differential category. In this paper, we introduce Cartesian difference categories as a bridge between Cartesian differential categories and change action models. We show that every Cartesian differential category is a Cartesian difference category, and how certain well-behaved change action models are Cartesian difference categories. In particular, Cartesian difference categories model both the differential calculus of smooth functions and the calculus of finite differences. Furthermore, every Cartesian difference category comes equipped with a tangent bundle monad whose Kleisli category is again a Cartesian difference category.

On semantics of a term calculus for classical logic

The calculus of Curien and Herbelin was introduced to provide the Curry-Howard correspondence for classical logic. The terms of this calculus represent derivations in the sequent calculus proof system and reduction reflects the process of cut-elimination. We investigate some properties of two well-behaved subcalculi of untyped calculus of Curien and Herbelin, closed under the call-by-name and the call-by-value reduction, respectively. Continuation semantics is given using the category of negated domains and Moggi?s Kleisli category over predomains for the continuation monad. Soundness theorems are given for both versions thus relating operational and denotational semantics. A thorough overview of the work on continuation semantics is given.

A Categorical Axiomatics for Bisimulation

We give an axiomatic category theoretic account of bisimulation in process algebras based on the idea of functional bisimulations as open maps. We work with 2-monads, T, on Cat. Operations on processes, such as nondeterministic sum, prexing and parallel composition are modelled using functors in the Kleisli category for the 2-monad T. We may define the notion of open map for any such 2-monad; in examples of interest, that agrees exactly with the usual notion of functional bisimulation. Under a condition on T, namely that it be a dense KZ-monad, which we define, it follows that functors in Kl(T) preserve open maps, i.e., they respect functional bisimulation. We further<br />investigate structures on Kl(T) that exist for axiomatic reasons,<br />primarily because T is a dense KZ-monad, and we study how those structures help to model operations on processes. We outline how this analysis gives ideas for modelling higher order processes. We conclude by making comparison with the use of presheaves and profunctors to model process calculi.