2023
DOI: 10.48550/arxiv.2301.05542
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Tangent categories as a bridge between differential geometry and algebraic geometry

Abstract: Discussions of tangent vectors, tangent spaces, and differentials are important in both differential geometry and algebraic geometry. In this paper, we use the abstract notion of a tangent category to make some of these commonalities precise. In particular, we focus on the idea of a differential bundle in a tangent category, which gives a new way to compare smooth vector bundles and modules. The results of this paper also give a new characterization of the opposite category of modules over a commutative ring a… Show more

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Cited by 2 publications
(13 citation statements)
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“…The main objective of this paper is to show that both ALGP and the opposite category ALG op P are tangent categories (Theorem 4.3.3 and Theorem 4.4.4), whose tangent bundles are adjoints to one another (Lemma 4.4.2). This is a generalization of the fact that ALGCom and ALG op Com are both well-known examples of tangent categories, see for example [11] for full details. Briefly, the tangent bundle of a commutative R-algebra A in ALGCom is given by the algebra of dual numbers over A (Example 4.3.5):…”
Section: Introductionmentioning
confidence: 85%
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“…The main objective of this paper is to show that both ALGP and the opposite category ALG op P are tangent categories (Theorem 4.3.3 and Theorem 4.4.4), whose tangent bundles are adjoints to one another (Lemma 4.4.2). This is a generalization of the fact that ALGCom and ALG op Com are both well-known examples of tangent categories, see for example [11] for full details. Briefly, the tangent bundle of a commutative R-algebra A in ALGCom is given by the algebra of dual numbers over A (Example 4.3.5):…”
Section: Introductionmentioning
confidence: 85%
“…The tangent category ALG op Com is closely related to algebraic geometry, as explained in [11]. Indeed, it is famously known that ALG op Com , the opposite category of commutative algebras, is equivalent to the category of affine schemes over R, the building blocks in algebraic geometry.…”
Section: T(a) := A[ǫ]mentioning
confidence: 99%
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