Geoffrey S. H. Cruttwell

This is the foreword for the special issue “ Differential Structures in Computer Science and Mathematics. ” We dedicate this special issue to Phil Scott (1947–2023), who initially organized and then invited us to serve as editors of this special issue.

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 submersions are employed as an ideal class of maps which behaves well under this operation and the tangent bundle functor. This issue is reflected in tangent category theory, which aims to axiomatize the tangent bundle functor of differential geometry categorically. Key constructions such as connections, tangent fibrations, or reverse tangent categories require one to work with pullbacks preserved by the tangent bundle functor. In previous work, this issue has been left as a technicality and solved by introducing extra structure to carry around. This paper gives an alternative to this by focusing on a special class of maps in a tangent category called tangent display maps; such maps are well-behaved with respect to pullbacks and applications of the tangent functor. We develop some of the general theory of such maps, show how using them can simplify previous work in tangent categories, and show that in the tangent category of smooth manifolds, they are the same as the submersions. Finally, we consider a subclass of tangent display maps to define open subobjects in any tangent category, allowing one to build a canonical split restriction tangent category in which the original one naturally embeds.

This paper describes the notion of a (latent) factorization system for a restriction category. Analogous to factorization systems for ordinary categories, a description in terms of a Galois connection based on an orthogonality relation is given. The orthogonality relation involving a unique cross-map (or lifting), however, is between a pair, consisting of a map together with a restriction idempotent, and a map. This gives a significantly different character to the theory. A restriction category in which the idempotents split is precisely a partial map category whose partiality is given by the system of monics which split the restriction idempotents. A restriction factorization on such a partial map category is completely characterized by a factorization on the total maps which is stable with respect to the system of monics; that is, pullbacks of both E and M-maps along these special monics are (respectively) E and M-maps. Examples of latent factorization systems are discussed. In particular, one source of latent factorizations is provided by lifting latent factorizations from the base of a latent fibration into the total category: this is a generalization of the observation that one can lift a factorization from the base of an ordinary fibration into the total category.

In this paper, we explain how the abstract notion of a differential bundle in a tangent category provides a new way of thinking about the category of modules over a commutative ring and its opposite category. MacAdam previously showed that differential bundles in the tangent category of smooth manifolds are precisely smooth vector bundles. Here we provide characterizations of differential bundles in the tangent categories of commutative rings and (affine) schemes. For commutative rings, the category of differential bundles over a commutative ring is equivalent to the category of modules over that ring. For affine schemes, the category of differential bundles over the Spec of a commutative ring is equivalent to the opposite category of modules over said ring. Finally, for schemes, the category of differential bundles over a scheme is equivalent to the opposite category of quasi-coherent sheaves of modules over that scheme.

In this paper, we explain how the abstract notion of a differential bundle in a tangent category provides a new way of thinking about the category of modules over a commutative ring and its opposite category. MacAdam previously showed that differential bundles in the tangent category of smooth manifolds are precisely smooth vector bundles. Here we provide characterizations of differential bundles in the tangent categories of commutative rings and (affine) schemes. For commutative rings, the category of differential bundles over a commutative ring is equivalent to the category of modules over that ring. For affine schemes, the category of differential bundles over the Spec of a commutative ring is equivalent to the opposite category of modules over said ring. Finally, for schemes, the category of differential bundles over a scheme is equivalent to the opposite category of quasi-coherent sheaves of modules over that scheme.

Cartesian reverse differential categories (CRDCs) are a recently defined structure which categorically model the reverse differentiation operations used in supervised learning. Here, we define a related structure called a monoidal reverse differential category, prove important results about its relationship to CRDCs, and provide examples of both structures, including examples coming from models of quantum computation.

Cartesian reverse differential categories (CRDCs) are a recently defined structure which categorically model the reverse differentiation operations used in supervised learning. Here we define a related structure called a monoidal reverse differential category, prove important results about its relationship to CRDCs, and provide examples of both structures, including examples coming from models of quantum computation.

We propose a categorical semantics of gradient-based machine learning algorithms in terms of lenses, parametric maps, and reverse derivative categories. This foundation provides a powerful explanatory and unifying framework: it encompasses a variety of gradient descent algorithms such as ADAM, AdaGrad, and Nesterov momentum, as well as a variety of loss functions such as MSE and Softmax cross-entropy, shedding new light on their similarities and differences. Our approach to gradient-based learning has examples generalising beyond the familiar continuous domains (modelled in categories of smooth maps) and can be realized in the discrete setting of boolean circuits. Finally, we demonstrate the practical significance of our framework with an implementation in Python.

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 existing fibrational notions such as monoidal fibrations and discrete double fibrations, proves a representation theorem for double fibrations, and shows how double fibrations are a type of internal fibration.