# Differential Geometry of Microlinear Frolicher Spaces I

@inproceedings{Nishimura2010DifferentialGO, title={Differential Geometry of Microlinear Frolicher Spaces I}, author={Hirokazu Nishimura}, year={2010} }

The central object of synthetic differential geometry is microlinear spaces. In our previous paper [Microlinearity in Frölicher spaces -beyond the regnant philosophy of manifolds-, International Journal of Pure and Applied Mathematics, 60 (2010), 15-24] we have emancipated microlinearity from within well-adapted models to Frölicher spaces. Therein we have shown that Frölicher spaces which are microlinear as well as Weil exponentiable form a cartesian closed category. To make sure that such Fr…

## 10 Citations

Differential Geometry of Microlinear Frolicher Spaces IV-1

- Mathematics
- 2011

The central object of synthetic differential geometry is microlinear spaces. In our previous paper [Microlinearity in Frolicher spaces -beyond the regnant philosophy of manifolds-, International…

Inflnite Dimensional Spaces and Cartesian Closedness

- Mathematics
- 2011

Infinite dimensional spaces frequently appear in physics; there are several approaches to obtain a good categorical framework for this type of space, and cartesian closedness of some category,…

AXIOMATIC DIFFERENTIAL GEOMETRY II-1 { VECTOR FIELDS

- Mathematics
- 2012

In our previous paper entitled \Axiomatic dierential geometry I { to- wards model categories of dierential geometry", we have given a category-theoretic framework of dierential geometry. As the rst…

Axiomatic Differential Geometry II-2: Differential Forms

- Mathematics
- 2012

We refurbish our axiomatics of differential geometry introduced in [Mathematics for Applications,, 1 (2012), 171-182]. Then the notion of Euclideaness can naturally be formulated. The principal…

eu AXIOMATIC DIFFERENTIAL GEOMETRY II-4 – ITS DEVELOPMENTS –

- Philosophy
- 2013

In our previous paper (Axiomatic Differential Geometry II-3) we have discussed the general Jacobi identity, from which the Jacobi identity of vector fields follows readily. In this paper we derive…

Axiomatic Differential Geometry III-1

- Mathematics
- 2012

In this paper is proposed a kind of model theory for our axiomatic differential geometry. It is claimed that smooth manifolds, which have occupied the center stage in differential geometry, should be…

Axiomatic Differential Geometry ?-1

- Mathematics
- 2012

In this paper is proposed a kind of model theory for our axiomatic differential geometry. It is claimed that smooth manifolds, which have occupied the center stage in differential geometry, should be…

3 1 O ct 2 01 2 Axiomatic Differential Geometry II-1-Its Developments-Chapter 1 : Vector Fields

- Mathematics
- 2014

In our previous paper entitled ”Axiomatic differential geometry -towards model categories of differential geometry-, we have given a categorytheoretic framework of differential geometry. As the first…

N ov 2 01 2 Axiomatic Differential Geometry I-1-Towards Model Categories of Differential Geometry-Hirokazu Nishimura

- 2014

AXIOMATIC DIFFERENTIAL GEOMETRY II-3 - ITS DEVELOPMENTS - CHAPTER 3: THE GENERAL JACOBI IDENTITY

- Mathematics
- 2013

As the fourth paper of our series of papers concerned with ax- iomatic differential geometry, this paper is devoted to the general Jacobi iden- tity supporting the Jacobi identity of vector fields.…

## References

SHOWING 1-10 OF 32 REFERENCES

Differential Geometry of Microlinear Frolicher Spaces IV-1

- Mathematics
- 2011

The central object of synthetic differential geometry is microlinear spaces. In our previous paper [Microlinearity in Frolicher spaces -beyond the regnant philosophy of manifolds-, International…

MICROLINEARITY IN FRöLICHER SPACES - BEYOND THE REGNANT PHILOSOPHY OF MANIFOLDS -

- Mathematics
- 2010

Frolicher spaces and smooth mappings form a Cartesian closed category. It was shown in our previous paper (Far East Journal of Mathe- matical Sciences, 35 (2009), 211-223) that its full subcategory…

A much larger class of Frolicher spaces than that of convenient vector spaces may embed into the Cahier topos

- Mathematics
- 2009

It is well known that the category of Frolicher spaces and smooth mappings is Cartesian closed. The principal objective in this paper is to show that the full subcategory of Frolicher spaces that…

Synthetic differential geometry of jet bundles

- Mathematics
- 2001

The theory of infinite jet bundles provides the very foundation for the geometric theory of nonlinear partial differential equations, but it is hard to say that orthodox differential geometry is an…

The affine bundle theorem in synthetic differential geometry of jet bundles

- Mathematics
- 2006

In our [Higher-order preconnections in synthetic differential geometry of jet bundles, Beitr\"{a}ge zur Algebra und Geometrie, 45 (2004), 677-696] we have established the affine bundle theorem in the…

Theory of microcubes

- Mathematics
- 1997

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…

Basic Concepts of Synthetic Differential Geometry

- Mathematics
- 1996

Introduction. 1. Differential Calculus and Integrals. 2. Weil Algebras and Infinitesimal Linearity. 3. Tangency. 4. Differential Forms. 5. Connections. 6. Global Actions. 7. On the Algebra of the…

General Jacobi Identity Revisited

- Mathematics
- 1999

In a previous paper (Nishimura, 1997) we probedthe deeper structure of the Jacobi identity of vectorfields with respect to Lie brackets within the realm ofsynthetic differential geometry to find what…

Higher-Order Preconnections in Synthetic Differential Geometry of Jet Bundles

- Mathematics
- 2004

In our previous papers (Nishimura (2001 and 2003)) we dealt with jet bundles from a synthetic perch by regarding a 1-jet as something like a pin- pointed (nonlinear) connection (called a…

The Convenient Setting of Global Analysis

- Mathematics
- 1997

Introduction Calculus of smooth mappings Calculus of holomorphic and real analytic mappings Partitions of unity Smoothly realcompact spaces Extensions and liftings of mappings Infinite dimensional…