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Submersions, immersions, and étale maps in diffeology
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Abstract
One of the most important challenges in diffeology is providing suitable analogous for submersions, immersions, and étale maps (i.e., local diffeomorphisms) consistent with the classical versions of these maps between manifolds. In this paper, we introduce diffeological submersions, immersions, and étale maps as an adaptation of these maps in diffeology by a nonlinear approach. We explore their behaviors from different aspects in a systematized manner with respect to plots, and especially, show that for manifolds, there is no difference between the classical versions and diffeological versions of these maps. Moreover, we introduce a class of diffeological spaces, called diffeological étale manifolds, and establish the rank theorem for them. It is also proved that a diffeological étale space on a diffeological orbifold is again a diffeological orbifold.
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A "Chen space" is a set X equipped with a collection of "plots" - maps from convex sets to X - satisfying three simple axioms. While an individual Chen space can be much worse than a smooth manifold, the category of all Chen spaces is much better behaved than the category of smooth manifolds. For example, any subspace or quotient space of a Chen space is a Chen space, and the space of smooth maps between Chen spaces is again a Chen space. Souriau's "diffeological spaces" share these convenient properties. Here we give a unified treatment of both formalisms. Following ideas of Dubuc, we show that Chen spaces, diffeological spaces, and even simplicial complexes are examples of "concrete sheaves on a concrete site". As a result, the categories of such spaces are locally cartesian closed, with all limits, all colimits, and a weak subobject classifier. For the benefit of differential geometers, our treatment explains most of the category theory we use. Comment: 43 pages, version to be published; includes corrected definition of "concrete site"
In this paper, we introduce sheaves on diffeological spaces as sheaves on the site of plots. We also define quasi-sheaves by presheaves that respect the limit over covering generating families. It is shown that every sheaf on diffeological spaces is a quasi-sheaf. Moreover, every sheaf on a diffeological space gives a sheaf on the D-topological structure. Finally, we characterize a class of sheaves, so-called comma sheaves, and prove that it is equivalent to the comma category of 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 a filtered
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, and therefore agrees
with the tangent bundle introduced by the authors in a previous paper.
Tangent spaces and tangent bundles of smooth manifolds are the building
blocks for differential geometry. We study how these structures can be extended
to diffeological spaces, which are generalizations of smooth manifolds that
include singular spaces and infinite-dimensional spaces. There are several
equivalent ways of defining the tangent space of a smooth manifold at a point
which give rise to inequivalent definitions in the case of diffeological
spaces. In this paper we focus on two ways. 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. After proving basic results about these tangent spaces, we compare
them by calculating many examples and observe that while they agree for smooth
manifolds and most of the examples, they do not agree in general.
After this, we recall Hector's definition of the tangent bundle of a
diffeological space as the union of the internal tangent spaces with a certain
diffeology. We 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 the 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).
I. Categories, Functors and Natural Transformations.- 1. Axioms for Categories.- 2. Categories.- 3. Functors.- 4. Natural Transformations.- 5. Monics, Epis, and Zeros.- 6. Foundations.- 7. Large Categories.- 8. Hom-sets.- II. Constructions on Categories.- 1. Duality.- 2. Contravariance and Opposites.- 3. Products of Categories.- 4. Functor Categories.- 5. The Category of All Categories.- 6. Comma Categories.- 7. Graphs and Free Categories.- 8. Quotient Categories.- III. Universals and Limits.- 1. Universal Arrows.- 2. The Yoneda Lemma.- 3. Coproducts and Colimits.- 4. Products and Limits.- 5. Categories with Finite Products.- 6. Groups in Categories.- IV. Adjoints.- 1. Adjunctions.- 2. Examples of Adjoints.- 3. Reflective Subcategories.- 4. Equivalence of Categories.- 5. Adjoints for Preorders.- 6. Cartesian Closed Categories.- 7. Transformations of Adjoints.- 8. Composition of Adjoints.- V. Limits.- 1. Creation of Limits.- 2. Limits by Products and Equalizers.- 3. Limits with Parameters.- 4. Preservation of Limits.- 5. Adjoints on Limits.- 6. Freyd's Adjoint Functor Theorem.- 7. Subobjects and Generators.- 8. The Special Adjoint Functor Theorem.- 9. Adjoints in Topology.- VI. Monads and Algebras.- 1. Monads in a Category.- 2. Algebras for a Monad.- 3. The Comparison with Algebras.- 4. Words and Free Semigroups.- 5. Free Algebras for a Monad.- 6. Split Coequalizers.- 7. Beck's Theorem.- 8. Algebras are T-algebras.- 9. Compact Hausdorff Spaces.- VII. Monoids.- 1. Monoidal Categories.- 2. Coherence.- 3. Monoids.- 4. Actions.- 5. The Simplicial Category.- 6. Monads and Homology.- 7. Closed Categories.- 8. Compactly Generated Spaces.- 9. Loops and Suspensions.- VIII. Abelian Categories.- 1. Kernels and Cokernels.- 2. Additive Categories.- 3. Abelian Categories.- 4. Diagram Lemmas.- IX. Special Limits.- 1. Filtered Limits.- 2. Interchange of Limits.- 3. Final Functors.- 4. Diagonal Naturality.- 5. Ends.- 6. Coends.- 7. Ends with Parameters.- 8. Iterated Ends and Limits.- X. Kan Extensions.- 1. Adjoints and Limits.- 2. Weak Universality.- 3. The Kan Extension.- 4. Kan Extensions as Coends.- 5. Pointwise Kan Extensions.- 6. Density.- 7. All Concepts are Kan Extensions.- Table of Terminology.
Diffeological spaces are generalizations of smooth manifolds which include
singular spaces and function spaces. For each diffeological space, P.
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 Delta-generated spaces and give results
and examples which help to determine when a space is Delta-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.
We study the diffeology of the quotient of a 2-torus by an irrational flow
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- P Iglesias
- Y Karshon
- M Zadka
P. Iglesias, Y. Karshon, M. Zadka, Orbifolds as diffeologies, Trans. Amer. Math. Soc. 362(6), (2010) 2811-2831.
- P Iglesias-Zemmour
- E Prato
- Quasifolds
P. Iglesias-Zemmour, E. Prato, Quasifolds, diffeology and noncommutative geometry, J. Noncommut.
Geom., 15(2), (2021) 735-759.
Introduction to Smooth Manifolds
- J M Lee
J.M. Lee, Introduction to Smooth Manifolds, Graduate Texts in Mathematics 218, 2nd edition, Springer,
2013.
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I. Moerdijk, D.A. Pronk, Orbifolds, sheaves and groupoids, K-Theory 12(1) (1997), 3-21.
- A Schmeding
A. Schmeding, The Diffeomorphism Group of a Non-Compact Orbifold, arXiv:1301.5551v4.
- J.-M Souriau
- Groupes Différentiels
J.-M. Souriau, Groupes différentiels, in: Differential geometrical methods in mathematical physics, Proc.
Conf., Aix-en-Provence/Salamanca, 1979, in: Lecture Notes in Math. 836, Springer Verlag, 1980, 91-128.
Email address: ahmadi@stu.yazd.ac.ir, alirezaahmadi13@yahoo.com