21 citations · 41 across the 10 of their papers we have counts for
11 papers
Ehresmann connections in tangent categories
Geoffrey Cruttwell, Marcello Lanfranchi
The theory of connections is at the very core of differential geometry. Discovered by Levi-Civita and Christoffel and later studied by Cartan, Koszul, and others, connections appea…
Pullbacks in tangent categories and tangent display maps
Geoffrey Cruttwell, Marcello Lanfranchi
In differential geometry, the existence of pullbacks is a delicate matter, since the category of smooth manifolds does not admit all of them. When pullbacks are required, often sub…
A Tangent Category Perspective on Connections in Algebraic Geometry
G. S. H. Cruttwell, Jean-Simon Pacaud Lemay, Elias Vandenberg
There is an abstract notion of connection in any tangent category. In this paper, we show that when applied to the tangent category of affine schemes, this recreates the classical…
Double Fibrations
Geoffrey Cruttwell, Michael Lambert, Dorette Pronk +1
This paper defines double fibrations (fibrations of double categories) and describes their key examples and properties. In particular, it shows how double fibrations relate to exis…
Monoidal Reverse Differential Categories
Geoffrey Cruttwell, Jonathan Gallagher, Jean-Simon Pacaud Lemay +1
Cartesian reverse differential categories (CRDCs) are a recently defined structure which categorically model the reverse differentiation operations used in supervised learning. Her…
Categorical semantics of a simple differential programming language
Geoffrey Cruttwell, Jonathan Gallagher, Dorette Pronk
With the increased interest in machine learning, and deep learning in particular, the use of automatic differentiation has become more wide-spread in computation. There have been t…
Latent Fibrations: Fibrations for Categories of Partial Maps
Robin Cockett, Geoff Cruttwell, Jonathan Gallagher +1
Latent fibrations are an adaptation, appropriate for categories of partial maps (as presented by restriction categories), of the usual notion of fibration. The paper initiates the…
Integral and differential structure on the free -ring modality
G. S. H. Cruttwell, J. -S. P. Lemay, R. B. B. Lucyshyn-Wright
Integral categories were recently developed as a counterpart to differential categories. In particular, integral categories come equipped with an integration operator, known as an…
Affine geometric spaces in tangent categories
R. F. Blute, G. S. H. Cruttwell, R. B. B. Lucyshyn-Wright
We continue the program of structural differential geometry that begins with the notion of a tangent category, an axiomatization of structural aspects of the tangent functor on the…
Cartesian differential categories revisited
G. S. H. Cruttwell
We revisit the definition of Cartesian differential categories, showing that a slightly more general version is useful for a number of reasons. As one application, we show that the…
Differential restriction categories
J. R. B. Cockett, G. S. H. Cruttwell, J. D. Gallagher
We combine two recent ideas: cartesian differential categories, and restriction categories. The result is a new structure which axiomatizes the category of smooth maps defined on o…