The theory of differential categories uses category theory to provide the foundations of differential structures in both mathematics and computer science. The story of differential categories began in the early 2000s, when Ehrhard and Regnier noticed that many models of Linear Logic had a natural notion of differential operator, in which the logical and mathematical notions of “linear” coincided. This led to their introduction of Differential Linear Logic, the differential $\lambda$ -calculus, and differential proof nets. Following this, Blute, Cockett, and Seely introduced categorical counterparts to these ideas in the form of differential categories and Cartesian differential categories, which were then further expanded upon by many others, including Fiore and Ehrhard. Afterwards, Cockett and Cruttwell connected these structures to existing categorical forms of differential structure via Rosický’s notion of a tangent category, which has led to further connections in many areas of mathematics, including (synthetic) differential geometry, algebraic geometry, commutative algebra, and so on.

The theory of differential categories has been used to study differentiation in a variety of fields and now has a rich literature of its own. Differential categories have been successful in formalizing various key aspects of differential calculus, ranging from the very basic foundational aspects of differentiation, such as the synthetic derivation of commutative algebra, to the more complex notions of differential geometry, such as the tangent bundle. Beyond mathematics, the theory of differential categories is also of interest in computer science, due to its use in the foundations of differentiable programming, as well as providing categorical frameworks for automatic differentiation (AD) and machine learning. The theory of differential categories remains an active and vibrant area of research, with numerous researchers and research groups worldwide working on the subject.

This special issue of Mathematical Structures in Computer Science (MSCS) collects papers on recent developments in the theory of differential categories, from both theoretical and applicative perspectives. We thank the authors for submitting these papers (which we briefly summarize below) and also appreciate the assistance of our anonymous referees in reviewing the submissions.

• • From Differential Linear Logic to Coherent Differentiation, T. Ehrhard. A recent development in the theory of differential categories is the concept of coherent differentiation, where the summations required by differentiation can be controlled and kept deterministic. This survey provides an overview of the recently introduced categorical and syntactic settings of coherent differentiation, showing how the main ideas from Differential Linear Logic and the differential $\lambda$ -calculus are compatible with determinism.

• • Coherent Taylor expansion as a bimonad, T. Ehrhard & A. Walch. This paper extends coherent differentiation by incorporating Taylor expansion. The main idea involves extending summability to an infinitary functor that intuitively maps any object to the object of its countable summable families. They provide a general theory of Taylor expansion, which are analogous to differential categories and Cartesian differential categories.

• • The Grothendieck construction in the context of tangent categories, M. Lanfranchi. The Grothendieck construction establishes an equivalence between fibrations and indexed categories. This paper provides a Grothendieck construction for a suitable notion of fibrations for tangent categories. This then leads to a Grothendieck equivalence between tangent fibrations and tangent indexed categories.

• • An Axiomatics and a Combinatorial Model of Creation/Annihilation Operators, M. Fiore. This work investigates the mathematical structure of creation/annihilation operators on the (symmetric or bosonic) Fock space. From the viewpoint of the theory of differential categories, these operators may be seen as derivations that only need to satisfy the Leibniz rule. This paper also provides a model for a Fock space with creation/annihilation operators that arises in the setting of generalized species of structures.

• • Differentiable Causal Computations via Delayed Trace (Extended Version), D. Sprunger & S. Katsumata. This is an extended version of a LICS2019 paper. This paper investigates causal computations, modeling these in category theory via a construction on Cartesian categories that comes equipped with a novel trace-like operation called “delayed trace.” Applying this construction to a Cartesian differential category yields another Cartesian differential category, where the differential combinator is constructed using an abstract version of backpropagation.

• • AD for ML-family languages: correctness via logical relations, F. Lucatelli Nunes & M. Vakar. This paper presents a simple, direct, and reusable logical relations technique for languages with term and type recursion and partially defined differentiable functions. They demonstrate this by the case of AD, presenting a correctness proof of a dual numbers style AD code transformation for realistic functional languages in the ML family. The key parts of this story are to interpret a functional programming language as a suitable freely generated categorical structure, as well as a powerful monadic logical relations technique for term recursion and recursive types.

• • Pseudolimits for Tangent Categories and Equivariant Tangents for Varieties and Smooth Manifolds, D. Pronk & G. Vooys. This paper studies pseudolimits in the 2-category of tangent categories. The authors first show that the pseudocone of a tangent indexing functor is a tangent category, and then go on to show that the forgetful functor from tangent categories to mere categories creates and preserves pseudolimits (in the appropriate sense). As an application, this allows them to describe how equivariant descent interacts with the tangent structures on the category of smooth manifolds and on categories of algebraic varieties over a field.

Last, we would like to dedicate this special issue to Phil Scott. In fact, the idea for this special issue was his to start with. Indeed, in the summer of 2023, Phil reached out to the four of us, asking if we would be editors for a special issue of MSCS on the topic of differential categories. At that time, Phil was battling cancer, and yet continued his role as editor for MSCS. Unfortunately, Phil passed away in December 2023. We lost a colleague, mentor, and, of course, a dear friend. We thank Phil for initiating this idea, inviting us to be part of it, as well as his support of us and the differential categories.