12 research outputs found
Integral Categories and Calculus Categories
Differential categories are now an established abstract setting for differentiation. The paper presents the parallel development for integration by axiomatizing
an integral transformation in a symmetric monoidal category with a coalgebra modality. When integration is combined with differentiation, the two fundamental theorems of calculus are expected to hold (in a suitable sense): a differential category with integration which satisfies
these two theorem is called a calculus category.
Modifying an approach to antiderivatives by T. Ehrhard, it is shown how examples of calculus categories arise as differential categories with antiderivatives in this new sense. Having antiderivatives amounts to demanding that a certain natural transformation K, is invertible. We observe that a differential category having antiderivatives, in this sense, is always a calculus category and we provide examples of such categories
The Difference ?-Calculus: A Language for Difference Categories
Cartesian difference categories are a recent generalisation of Cartesian differential categories which introduce a notion of "infinitesimal" arrows satisfying an analogue of the Kock-Lawvere axiom, with the axioms of a Cartesian differential category being satisfied only "up to an infinitesimal perturbation". In this work, we construct a simply-typed calculus in the spirit of the differential ?-calculus equipped with syntactic "infinitesimals" and show how its models correspond to difference ?-categories, a family of Cartesian difference categories equipped with suitably well-behaved exponentials
Taylor subsumes Scott, Berry, Kahn and Plotkin
The speculative ambition of replacing the old theory of program approximation based on syntactic continuity with the theory of resource consumption based on Taylor expansion and originating from the differential γ-calculus is nowadays at hand. Using this resource sensitive theory, we provide simple proofs of important results in γ-calculus that are usually demonstrated by exploiting Scott's continuity, Berry's stability or Kahn and Plotkin's sequentiality theory. A paradigmatic example is given by the Perpendicular Lines Lemma for the Böhm tree semantics, which is proved here simply by induction, but relying on the main properties of resource approximants: strong normalization, confluence and linearity
Cartesian Differential Comonads and New Models of Cartesian Differential Categories
Cartesian differential categories come equipped with a differential
combinator that formalizes the derivative from multi-variable differential
calculus, and also provide the categorical semantics of the differential
-calculus. An important source of examples of Cartesian differential
categories are the coKleisli categories of the comonads of differential
categories, where the latter concept provides the categorical semantics of
differential linear logic. In this paper, we generalize this construction by
introducing Cartesian differential comonads, which are precisely the comonads
whose coKleisli categories are Cartesian differential categories, and thus
allows for a wider variety of examples of Cartesian differential categories. As
such, we construct new examples of Cartesian differential categories from
Cartesian differential comonads based on power series, divided power algebras,
and Zinbiel algebras.Comment: Accepted and to be published in Cahiers de topologie et g\'eom\'etrie
diff\'erentielle cat\'egorique