The<i>D</i>-topology for diffeological spaces

Christensen, J. Daniel; Sinnamon, Gordon; Wu, Enxin

doi:10.2140/pjm.2014.272.87

Cited by 68 publications

(97 citation statements)

References 7 publications

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“…This makes them convenient for many applications, ranging from mathematical physics to homological algebra [Wu] and homotopy theory [CW1]. In [CW2], we studied tangent spaces of diffeological spaces, focusing on the internal tangent space T x (X).…”

Section: Introductionmentioning

confidence: 99%

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Tangent spaces of bundles and of filtered diffeological spaces

Christensen

2017

Proc. Amer. Math. Soc.

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104

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Abstract. We show that a diffeological bundle gives rise to an exact sequence of internal tangent spaces. We then introduce two new classes of diffeological spaces, which we call weakly filtered and filtered diffeological spaces, whose tangent spaces are easier to understand. These are the diffeological spaces whose categories of pointed plots are (weakly) filtered. We extend the exact sequence one step further in the case of a diffeological bundle with filtered total space and base space. We also show that the tangent bundle T H X defined by Hector is a diffeological vector space over X when X is filtered or when X is a homogeneous space, and therefore agrees with the dvs tangent bundle introduced by the authors in a previous paper.

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Section: Introductionmentioning

confidence: 99%

“…The descriptions of limits, colimits and function spaces are quite simple, and are concisely described in [CSW,Section 2]. The underlying set of a (co)limit is the (co)limit of the underlying sets.…”

Section: Introductionmentioning

confidence: 99%

Tangent spaces of bundles and of filtered diffeological spaces

Christensen

2017

Proc. Amer. Math. Soc.

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104

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“…In fact, being ∆-generated is the same as being R-generated, see [CSW13], and so to define d(X) is is sufficient to consider instead all continuous curves p : R → X. The category of ∆-generated spaces has may desirable properties, even beyond the category of compactly generated spaces, see [Dug].…”

Section: A Comparison With Others Spaces Of Embedded Manifoldsmentioning

confidence: 99%

“…In section Section 2.3 we provide a diffeology for the space ψ d (R m ), that is a collection of smooth plots U → ψ d (R m ) parametrized by finite dimensional smooth manifolds. Topologies induced by diffeologies have been studied before, for example in [CSW13] where they are called the D-topology. In this context we have the following: We also have:…”

Section: A Comparison With Others Spaces Of Embedded Manifoldsmentioning

confidence: 99%

Tori detect invertibility of topological field theories

Schommer-Pries

2018

Geom. Topol.

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A d-dimensional invertible topological field theory is a functor from the symmetric monoidal (∞, n)-category of d-bordisms (embedded into R ∞ and equipped with a tangential (X, ξ)-structure) which lands in the Picard subcategory of the target symmetric monoidal (∞, n)-category. We classify these field theories in terms of the cohomology of the (n − d)-connective cover of the Madsen-Tillmann spectrum. This is accomplished by identifying the classifying space of the (∞, n)-category of bordisms with Ω ∞−n M T ξ as an E ∞ -spaces. This generalizes the celebrated result of Galatius-Madsen-Tillmann-Weiss [Gal+09] in the case n = 1, and of Bökstedt-Madsen [BM14] in the n-uple case. We also obtain results for the (∞, n)-category of d-bordisms embedding into a fixed ambient manifold M , generalizing results of Randal-Williams [Ran11] in the case n = 1. We give two applications: (1) We completely compute all extended and partially extended invertible TFTs with target a certain category of n-vector spaces (for n ≤ 4), and (2) we use this to give a negative answer to a question raised by Gilmer and Masbaum in [GM13].7 Examples and applications 52 7.

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“…For more detailed discussion about the D-topology of diffeological spaces, see [26,Chapter 2] and [8]. Now, let us turn to Frölicher spaces.…”

Section: Both Functors Act As Identity On Arrows)mentioning

confidence: 99%

Categorical frameworks for generalized functions

Giordano

2015

Arab. J. Math.

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We tackle the problem of finding a suitable categorical framework for generalized functions used in mathematical physics for linear and non-linear PDEs. We are looking for a Cartesian closed category which contains both Schwartz distributions and Colombeau generalized functions as natural objects. We study Frölicher spaces, diffeological spaces and functionally generated spaces as frameworks for generalized functions. The latter are similar to Frölicher spaces, but starting from locally defined functionals. Functionally generated spaces strictly lie between Frölicher spaces and diffeological spaces, and they form a complete and cocomplete Cartesian closed category. We deeply study functionally generated spaces (and Frölicher spaces) as a framework for Schwartz distributions, and prove that in the category of diffeological spaces, both the special and the full Colombeau algebras are smooth differential algebras, with a smooth embedding of Schwartz distributions and smooth pointwise evaluations of Colombeau generalized functions. Mathematics Subject Classification

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TheD-topology for diffeological spaces

Cited by 68 publications

References 7 publications

Tangent spaces of bundles and of filtered diffeological spaces

Tangent spaces of bundles and of filtered diffeological spaces

Tori detect invertibility of topological field theories

Categorical frameworks for generalized functions

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