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Kazuhisa Shimakawa

Publications and source records attributed to Kazuhisa Shimakawa.

7 recordsLinked to original sources
arXiv2025

Nonstandard diffeology and generalized functions

We introduce a nonstandard extension of the category of diffeological spaces, and demonstrate its application to the study of generalized functions. Just as diffeological spaces are defined as concrete sheaves on the site of Euclidean open sets, our nonstandard diffeological spaces are defined as concrete sheaves on the site of open subsets of nonstandard Euclidean spaces, i.e. finite dimensional vector spaces over (the quasi-asymptotic variant of) Robinson's hyperreal numbers. It is shown that nonstandard diffeological spaces form a category which is enriched over the category of diffeological spaces, is closed under small limits and colimits, and is cartesian closed. Furthermore, it can be shown that the space of nonstandard functions on the extension of a Euclidean open set is a smooth differential algebra that admits an embedding of the differential vector space of Schwartz distributions. Since our algebra of generalized functions comes as a hom-object in a category, it enables not only the multiplication of distributions but also the composition of them. To illustrate the usefulness of this property we show that the homotopy extension property can be established for smooth relative cell complexes by exploiting extended maps.

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arXiv2024

Generalized maps between diffeological spaces

By utilizing the idea of Colombeau's generalized function, we introduce a notion of asymptotic map between arbitrary diffeological spaces. The category consisting of diffeological spaces and asymptotic maps is enriched over the category of diffeological spaces, and inherits completeness and cocompleteness. In particular, the set of asymptotic functions on a Euclidean open set include Schwartz distributions and form a Colombeau type smooth differential algebra over Robinson's field of asymptotic numbers. To illustrate the usefulness of our machinery, we show that homotopy extension property can be established for smooth relative cell complexes if we exploit asymptotic maps instead of smooth ones.

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arXiv2019

On homotopy types of diffeological cell complexes

We introduce the notion of smooth cell complexes and its subclass consisting of gathered cell complexes within the category of diffeological spaces (cf. Definitions 1 and 3). It is shown that the following hold. (1) With respect to the D-topology, every smooth cell complex is a topological cell complex (cf. Proposition 2). It is paracompact and Hausdorff if it is countable (cf. Proposition 8). (2) Every continuous map between gathered cell complexes is continuously homotopic to a smooth map (cf. Theorem 5). (3) Any topological cell complex is continuously homotopy equivalent to a gathered (hence smooth) cell complex (cf. Theorem 6). (4) Every D-open cover of a smooth countable cell complex has a subordinate partition of unity by smooth functions (cf. Theorem 9).

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arXiv2019

Configuration space of intervals with partially summable labels

A configuration space of intervals in $\mathbb R^1$ with partially summable labels is constructed. It is a kind of an extension of the configuration space with partially summable labels constructed by the second author and at the same time a generalization of the configuration space of intervals with labels in a based space constructed by the first author. An approximation theorem of the preceding configuration space is generalized to our case. When partially summable labels are given by a partial abelian monoid $M,$ we prove that it is weakly homotopy equivalent to the space of based loops on the classifying space of $M$ under some assumptions on $M$.

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arXiv2018

A model structure on the category of diffeological spaces

We construct a model category structure on the category of diffeological spaces which is Quillen equivalent to the model structure on the category of topological spaces based on the notions of Serre fibrations and weak homotopy equivalences.

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arXiv2009

Interactions of strings and equivariant homology theories

We introduce the notion of the space of parallel strings with partially summable labels, which can be viewed as a geometrically constructed group completion of the space of particles with labels. We utilize this to construct a machinery which produces equivariant generalized homology theories from such simple and abundant data as partial monoids.

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