arXiv subjects

Enxin Wu

Publications and source records attributed to Enxin Wu.

14 recordsLinked to original sources

arXiv2022

Pushforward and smooth vector pseudo-bundles

In this paper, we study a new operation named pushforward on diffeological vector pseudo-bundles, which is left adjoint to the pullback. We show how to pushforward projective diffeological vector pseudo-bundles to get projective diffeological vector spaces, producing many concrete new examples, together with application to smooth splittings of some projective diffeological vector spaces related to geometry. This brings new objects to diffeology from classical vector bundle theory.

math.DG↗

arXiv2022

Topology on diffeological vector spaces

It is expected that the $D$-topology makes every diffeological vector space into a topological vector space. We show that it is the case for a large class of diffeological vector spaces via $k_ω$-space theory, but not so in general. The paper also proposes the study of a class of almost topological vector spaces.

math.FA↗

arXiv2021

Exterior bundles in diffeology

We explore several notions of $k$-form at a point in a diffeological space, construct bundles of such $k$-forms, and compare sections of these bundles to differential forms. As they are defined locally, our $k$-forms can contain more information than the values of differential forms contain, and we illustrate this with many examples. To organize our work, we develop the basic theory of diffeological vector pseudo-bundles, including a detailed understanding of their limits and colimits, as well as a variety of fibrewise operations such as products, direct sums, tensor products, exterior powers and dual bundles.

math.DG↗

arXiv2020

Smooth classifying spaces

We develop the theory of smooth principal bundles for a smooth group $G$, using the framework of diffeological spaces. After giving new examples showing why arbitrary principal bundles cannot be classified, we define $D$-numerable bundles, the smooth analogs of numerable bundles from topology, and prove that pulling back a $D$-numerable bundle along smoothly homotopic maps gives isomorphic pullbacks. We then define smooth structures on Milnor's spaces $EG$ and $BG$, show that $EG \to BG$ is a $D$-numerable principal bundle, and prove that it classifies all $D$-numerable principal bundles over any diffeological space. We deduce analogous classification results for $D$-numerable diffeological bundles and vector bundles.

math.DG↗

arXiv2019

Diffeological vector spaces

We study the relationship between many natural conditions that one can put on a diffeological vector space: being fine or projective, having enough smooth (or smooth linear) functionals to separate points, having a diffeology determined by the smooth linear functionals, having fine finite-dimensional subspaces, and having a Hausdorff underlying topology. Our main result is that the majority of the conditions fit into a total order. We also give many examples in order to show which implications do not hold, and use our results to study the homological algebra of diffeological vector spaces.

math.DG↗

arXiv2016

Tangent spaces of bundles and of filtered diffeological spaces

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.

math.DG↗

arXiv2016

Convergences and the Intermediate Value Property in Fermat Reals

This paper contains two topics of Fermat reals, as suggested by the title. In the first part, we study the ω-topology, the order topology and the Euclidean topology on Fermat reals, and their convergence properties, with emphasis on the relationship with the convergence of sequences of ordinary smooth functions. We show that the Euclidean topology is best for this relationship with respect to pointwise convergence, and Lebesgue dominated convergence does not hold, among all additive Hausdorff topologies on Fermat reals. In the second part, we study the intermediate value property of quasi-standard smooth functions on Fermat reals, together with some easy applications. The paper is written in the language of Fermat reals, and the idea could be extended to other similar situations.

math.CA↗

arXiv2016

The Fermat Functors, Part I: The theory

In this paper, we use some basic quasi-topos theory to study two functors: one adding infinitesimals of Fermat reals to diffeological spaces (which generalize smooth manifolds including singular spaces and infinite dimensional spaces), and the other deleting infinitesimals on Fermat spaces. We study the properties of these functors, and calculate some examples. These serve as fundamentals for developing differential geometry on diffeological spaces using infinitesimals in a future paper.

math.CT↗

arXiv2015

Tangent spaces and tangent bundles for diffeological spaces

We study how the notion of tangent space can be extended from smooth manifolds to diffeological spaces, which are generalizations of smooth manifolds that include singular spaces and infinite-dimensional spaces. We focus on two definitions. The internal tangent space of a diffeological space is defined using smooth curves into the space, and the external tangent space is defined using smooth derivations on germs of smooth functions. We prove fundamental results about these tangent spaces, compute them in many examples, and observe that while they agree for smooth manifolds and many of the examples, they do not agree in general. After this, we recall Hector's definition of the tangent bundle of a diffeological space, and show that both scalar multiplication and addition can fail to be smooth, revealing errors in several references. We then give an improved definition of the tangent bundle, using what we call the dvs diffeology, which ensures that scalar multiplication and addition are smooth. We establish basic facts about these tangent bundles, compute them in many examples, and study the question of whether the fibres of tangent bundles are fine diffeological vector spaces. Our examples include singular spaces, spaces whose natural topology is non-Hausdorff (e.g., irrational tori), infinite-dimensional vector spaces and diffeological groups, and spaces of smooth maps between smooth manifolds (including diffeomorphism groups).

math.DG↗

arXiv2015

The D-topology for diffeological spaces

Diffeological spaces are generalizations of smooth manifolds which include singular spaces and function spaces. For each diffeological space, Iglesias-Zemmour introduced a natural topology called the $D$-topology. However, the $D$-topology has not yet been studied seriously in the existing literature. In this paper, we develop the basic theory of the $D$-topology for diffeological spaces. We explain that the topological spaces that arise as the $D$-topology of a diffeological space are exactly the $Δ$-generated spaces and give results and examples which help to determine when a space is $Δ$-generated. Our most substantial results show how the $D$-topology on the function space $C^{\infty}(M,N)$ between smooth manifolds compares to other well-known topologies.

math.DG↗

arXiv2015

Calculus in the ring of Fermat reals Part I: Integral calculus

We develop the integral calculus for quasi-standard smooth functions defined on the ring of Fermat reals. The approach is by proving the existence and uniqueness of primitives. Besides the classical integral formulas, we show the flexibility of the Cartesian closed framework of Fermat spaces to deal with infinite dimensional integral operators. The total order relation between scalars permits to prove several classical order properties of these integrals and to study multiple integrals on Peano-Jordan-like integration domains.

math.CA↗

arXiv2015

The homotopy theory of diffeological spaces

Diffeological spaces are generalizations of smooth manifolds. In this paper, we study the homotopy theory of diffeological spaces. We begin by proving basic properties of the smooth homotopy groups that we will need later. Then we introduce the smooth singular simplicial set $S^D(X)$ associated to a diffeological space $X$, and show that when $S^D(X)$ is fibrant, it captures smooth homotopical properties of $X$. Motivated by this, we define $X$ to be fibrant when $S^D(X)$ is, and more generally define cofibrations, fibrations and weak equivalences in the category of diffeological spaces using the smooth singular simplicial set functor. We conjecture that these form a model category structure, but in this paper we assume little prior knowledge of model categories, and instead focus on concrete questions about smooth manifolds and diffeological spaces. We prove that our setup generalizes the naive smooth homotopy theory of smooth manifolds by showing that a smooth manifold without boundary is fibrant and that for fibrant diffeological spaces, the weak equivalences can be detected using ordinary smooth homotopy groups. We also show that our definition of fibrations generalizes Iglesias-Zemmour's theory of diffeological bundles. We prove enough of the model category axioms to show that every diffeological space has a functorial cofibrant replacement. We give many explicit examples of objects that are cofibrant, not cofibrant, fibrant and not fibrant, as well as many other examples showing the richness of the theory. For example, we show that the free loop space of a smooth manifold is fibrant. One of the implicit points of this paper is that the language of model categories is an effective way to organize homotopical thinking, even when it is not known that all of the model category axioms are satisfied.

math.AT↗

arXiv2014

Homological Algebra for Diffeological Vector Spaces

Diffeological spaces are natural generalizations of smooth manifolds, introduced by J.M.~Souriau and his mathematical group in the 1980's. Diffeological vector spaces (especially fine diffeological vector spaces) were first used by P. Iglesias-Zemmour to model some infinite dimensional spaces in~\cite{I1,I2}. K.~Costello and O.~Gwilliam developed homological algebra for differentiable diffeological vector spaces in Appendix A of their book~\cite{CG}. In this paper, we present homological algebra of general diffeological vector spaces via the projective objects with respect to all linear subductions, together with some applications in analysis.

math.KT↗

arXiv2014

Categorical frameworks for generalized functions

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.

math.FA↗