Corpus ID: 198162011

# Towards formalizing and extending differential programming using tangent categories

@inproceedings{Cruttwell2019TowardsFA,
title={Towards formalizing and extending differential programming using tangent categories},
author={Geoff S. H. Cruttwell and Jonathan Gallagher and Benjamin MacAdam},
year={2019}
}
• Published 2019
This paper gives an interpretation of a simple differential programming language (over finite dimensional R vector spaces), into a setting derived from synthetic differential geometry (SDG). The main theorem of this paper is Theorem 5.6, where we establish that there is always an interpretation of our simple differential programming language into a category of partial maps of a well-adapted smooth topos that preserves the derivative and all the control structures of the differential programming… Expand
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#### References

SHOWING 1-10 OF 40 REFERENCES
Differential Structure, Tangent Structure, and SDG
• Mathematics, Computer Science
• Appl. Categorical Struct.
• 2014
It is shown that tangent structures appropriately span a very wide range of definitions, from the syntactic and structural differentials arising in computer science and combinatorics, through the concrete manifolds of algebraic and differential geometry, and finally to the abstract definitions of synthetic differential geometry. Expand
Differential categories
• Mathematics, Computer Science
• Mathematical Structures in Computer Science
• 2006
The notion of a categorical model of the differential calculus is introduced, and it is shown that it captures the not-necessarily-closed fragment of Ehrhard–Regnier differential $\lambda$-calculus. Expand
A convenient differential category
• Mathematics, Computer Science
• ArXiv
• 2010
It is shown that the category of Mackey-complete, separated, topological convex bornological vector spaces and bornological linear maps is a differential category and will ultimately yield a wide variety of models of differential logics. Expand
Finiteness spaces
• T. Ehrhard
• Computer Science, Mathematics
• Mathematical Structures in Computer Science
• 2005
A new denotational model of linear logic based on the purely relational model, where webs are equipped with a notion of ‘finitary’ subsets satisfying a closure condition and proofs are interpreted as finitary sets is investigated. Expand
Tangent spaces and tangent bundles for diffeological spaces
• Mathematics
• 2014
We study how the notion of tangent space can be extended from smooth manifolds to diffeological spaces, which are generalizations of smooth manifolds that include singular spaces andExpand
Differential Calculus over General Base Fields and Rings
• Mathematics
• 2004
Abstract We present an axiomatic approach to finite- and infinite-dimensional differential calculus over arbitrary infinite fields (and, more generally, suitable rings). The corresponding basicExpand
Synthetic Differential Geometry
Preface to the second edition (2005) Preface to the first edition (1981) Part I. The Synthetic Ttheory: 1. Basic structure on the geometric line 2. Differential calculus 3. Taylor formulae - oneExpand
Restriction categories as enriched categories
• Computer Science, Mathematics
• Theor. Comput. Sci.
• 2014
It is shown that restriction categories can be seen as a kind of enriched category; this allows their theory to be studied by way of the enrichment and varying the base of this enrichment also allows the important notions of join and range restriction category to be understood in the same manner. Expand
On Köthe sequence spaces and linear logic
• T. Ehrhard
• Computer Science, Mathematics
• Mathematical Structures in Computer Science
• 2002
This work provides a simple setting in which typed λ-calculus and differential calculus can be combined and gives a few examples of computations. Expand
CARTESIAN DIFFERENTIAL CATEGORIES
• Mathematics
• 2009
This paper revisits the authors' notion of a dierential category from a dierent perspective. A dierential category is an additive symmetric monoidal category with a comonad (a \coalgebra modality")Expand