Differential Structure, Tangent Structure, and SDG

Cockett, J. R. B.; Cruttwell, G. S. H.

doi:10.1007/s10485-013-9312-0

Cited by 61 publications

(281 citation statements)

References 12 publications

Supporting

Mentioning

275

Contrasting

Order By: Relevance

“…The keen eyed reader will notice that the axioms of a Cartesian difference category are very similar to the axioms of a Cartesian differential category. Indeed, 7] it may be useful to write them out using element-like notation.…”

Section: Cartesian Difference Categoriesmentioning

confidence: 99%

See 1 more Smart Citation

Cartesian Difference Categories

Alvarez-Picallo

Lemay

2020

Lecture Notes in Computer Science

View full text Add to dashboard Cite

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 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.In this section we briefly review Cartesian differential categories, so that the reader may compare Cartesian differential categories with the new notion of Cartesian difference categories which we introduce in the next section. For a full detailed introduction on Cartesian differential categories, we refer the reader to the original paper [4]. Cartesian Left Additive CategoriesHere we provide the definition of Cartesian left additive categories [4] -where "additive" is meant being skew enriched over commutative monoids. In particular this means that we do not assume the existence of additive inverses, i.e., "negative elements". Definition 1. A left additive category [4] is a category X such that each hom-set X(A, B) is a commutative monoid with addition operation + : X(A, B)× X(A, B) → X(A, B) and zero element (called the zero map) 0 ∈ X(A, B), such that pre-composition preserves the additive structure: (f + g) • h = f • h + g • h and 0 • f = 0. A map k in a left additive category is additive if post-composition by k preserves the additive structure: k • (f + g) = k • f + k • g and k • 0 = 0.By a Cartesian category we mean a category X with chosen finite products where we denote the binary product of objects A and B by A×B with projection maps π 0 : A × B → A and π 1 : A × B → B and pairing operation −, − , and the chosen terminal object as ⊤ with unique terminal maps ! A : A → ⊤.Definition 2. A Cartesian left additive category [4] is a Cartesian category X which is also a left additive category such all projection maps π 0 : A × B → A and π 1 : A × B → B are additive 3 .By [4, Proposition 1.2.2], an equivalent axiomatization is of a Cartesian left additive category is that ...

show abstract

Section: Cartesian Difference Categoriesmentioning

confidence: 99%

“…Finally, in Section 5 we provide some examples of Cartesian difference categories; notably, the calculus of finite differences and the stream calculus. We highlight that by [7,Proposition 4.2], the last two axioms [CD.6] and [CD.7] have an equivalent alternative expression.…”

Section: Introductionmentioning

confidence: 99%

Cartesian Difference Categories

Alvarez-Picallo

Lemay

2020

Lecture Notes in Computer Science

View full text Add to dashboard Cite

show abstract

“…In particular, let C and D be subcategories of the category LRS of locally ringed spaces or slice categories of subcategories of LRS. [8], [9], [10], amongst others), one may notice the similarity of a join restriction functor between join restriction categories, or perhaps more readily with adhesive functors between adhesive categories (cf. [17], for instance), and our given definition of a geometrically adhesive functor above.…”

Section: In Particular This Gives An Isomorphism Of Categoriesmentioning

confidence: 99%

The Greenberg Functor is Site Cocontinuous

Geoff

2020

2018 MATRIX Annals

View full text Add to dashboard Cite

In this paper we show that it is possible to define a topology on the category of formal schemes over a ring of p-adic integers such that the left adjoint of the Greenberg Transform is a site cocontinuous functor when we equip the category of schemes over the residue field with theétale topology. We show furthermore that this topology allows us to give an isomorphism between the corresponding fundamental groups, and use this isomorphism to show that it is possible to geometrize the quasicharacters of a p-adic torus by a local system on a formal scheme over the ring of p-adic integers.

show abstract

“…This definition of a Cartesian differential category is not exactly that of [28], but it is mostly straightfoward to check that they are equivalent. The biggest changes are in axioms CD6 and CD7, for which we have taken alternate forms given in [6,Proposition 4.2].…”

Section: Lemma 28 If C Is a Left Additive Category So Is St(c)mentioning

confidence: 99%

Differentiable Causal Computations via Delayed Trace

Sprunger

Katsumata

2019

2019 34th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS)

View full text Add to dashboard Cite

We investigate causal computations taking sequences of inputs to sequences of outputs where the nth output depends on the first n inputs only. We model these in category theory via a construction taking a Cartesian category C to another category St(C) with a novel trace-like operation called "delayed trace", which misses yanking and dinaturality axioms of the usual trace. The delayed trace operation provides a feedback mechanism in St(C) with an implicit guardedness guarantee.When C is equipped with a Cartesian differential operator, we construct a differential operator for St(C) using an abstract version of backpropagation through time, a technique from machine learning based on unrolling of functions. This obtains a swath of properties for backpropagation through time, including a chain rule and Schwartz theorem. Our differential operator is also able to compute the derivative of a stateful network without requiring the network to be unrolled.

show abstract

Differential Structure, Tangent Structure, and SDG

Cited by 61 publications

References 12 publications

Cartesian Difference Categories

Cartesian Difference Categories

The Greenberg Functor is Site Cocontinuous

Differentiable Causal Computations via Delayed Trace

Contact Info

Product

Resources

About