Smooth Infinitesimal Analysis

Moerdijk, Ieke; Reyes, Gonzalo E.

doi:10.1007/978-1-4757-4143-8_8

Cited by 86 publications

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“…Such examples have been discussed amply in the context of welladapted models of synthetic differential geometry being not aware at all of Weil categories. The reader is referred to [9] and [14] for them. Now we fix a Weil category (K, D) throughout the rest of this section.…”

Section: Microlinearitymentioning

confidence: 99%

Weil diffeology I: Classical differential geometry

Nishimura¹

2017

Math. Appl.

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Abstract. Topos theory is a category-theoretical axiomatization of set theory. Model categories are a category-theoretical framework for abstract homotopy theory. They are complete and cocomplete categories endowed with three classes of morphisms (called fibrations, cofibrations and weak equivalences) satisfying certain axioms. We would like to present an abstract framework for classical differential geometry as an extension of topos theory, hopefully comparable with model categories for homotopy theory. Functors from the category W of Weil algebras to the category Sets of sets are called Weil spaces by Wolfgang Bertram and form the Weil topos after Eduardo J. Dubuc. The Weil topos is endowed intrinsically with the Dubuc functor, a functor from a larger category W of cahiers algebras to the Weil topos standing for the incarnation of each algebraic entity of W in the Weil topos. The Weil functor and the canonical ring object are to be defined in terms of the Dubuc functor. The principal objective of this paper is to present a category-theoretical axiomatization of the Weil topos with the Dubuc functor intended to be an adequate framework for axiomatic classical differential geometry. We will give an appropriate formulation and a rather complete proof of a generalization of the familiar and desired fact that the tangent space of a microlinear Weil space is a module over the canonical ring object.

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Section: Microlinearitymentioning

confidence: 99%

Weil diffeology I: Classical differential geometry

Nishimura¹

2017

Math. Appl.

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“…To this end, we must choose the right geometry to be quantized, and we believe that it is not standard differential geometry and the category of smooth manifolds, but synthetic differential geometry and the category of microlinear spaces that are truly susceptible of quantization. For textbooks on synthetic differential geometry and microlinear space in particular, the reader is referred to Lavendhomme (1996) and Moerdijk and Reyes (1991). For the first attempt to quantize synthetic differential geometry, the reader is referred to Nishimura (1996a,b).…”

Section: Introductionmentioning

confidence: 99%

Theory of microcubes

Nishimura

1997

Int J Theor Phys

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Kock and Lavendhomme have begun to couch the standard theory of iterated tangents within the due framework of synthetic differential geometry. Generalizing their theory of microsquares, we give a general theory of microcubes, its threedimensional generalization, in which an unexpected generalization of the Jacobi identity of vector fields with respect to Lie brackets and a synthetic treatment of Bianchi's first identity are discussed.

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“…Let us remark that [17] also gives a version of the Corollary for the non-commutative case, their Proposition 6.4.1; this, however, seems not correct. In this sense, our Theorem 7.1 is partly meant as a correction to this Proposition, partly a "translation" of it into the pure multiplicative principal bundle calculus, which is our main concern.…”

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confidence: 99%

“…The fact that this "correction term" (or "modification") does not come up in our context can be explained by Theorem 5.4 in [8] (or see [7] Theorem 18.5); here it is proved that the formula dω(x, y, z) = ω(x, y) · ω(y, z) · ω(z, x) already contains this correction term, when translated into "classical" Lie algebra valued forms. The Theorem has the following Corollary, which is essentially what [17] call the infinitesimal version of the Gauss-Bonnet Theorem (for the case where G = SO(2)): Corollary 7.2. Assume P −1 P is commutative, and let the connection ∇ in P P −1 have connection form ω.…”

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confidence: 99%