Diane Maclagan

[[!redirects Diane Maclagen]]
[[!redirects Diana Maclagen]]
Diane Maclagan is lecturer at Warwick university working in [[tropical geometry]].

* [webpage](http://homepages.warwick.ac.uk/staff/D.Maclagan/)

category:people


Diarmuid Crowley

* [personal page](http://www.dcrowley.net/)

## Selected writings

On [[sphere-fiber bundles]] and specifically on [[3-sphere]]-[[fiber bundles]] over the [[4-sphere]] in relation to [[exotic 7-spheres]]:

* {#CrowleyEscher03} [[Diarmuid Crowley]], [[Christine Escher]], _A classification of $S^3$-bundles over $S^4$_, Differential Geometry and its Applications Volume 18, Issue 3, May 2003, Pages 363-380 (doi:10.1016/S0926-2245(03)00012-3))

category: people


dichotomy between nice objects and nice categories

# Contents
* table of contents
{: toc}

## Idea

> If the decision to let every commutative ring define a scheme gives
standing to bizarre schemes, allowing it gives a category of schemes
with nice properties ([Deligne](#Deligne98))

In the context of [[category theory]], it is a general phenomenon that

 * Nice [[objects]] tend to form non-nice [[categories]].

 * Nice categories tend to contain non-nice objects.

Here we take a category to be the "nicer" the more [[limits]], [[colimits]] etc. it admits, and additionally the more good properties like [[exactness properties]], being a [[topos]], etc. it has.

On the other hand, a "nice object" is, loosely speaking, an object in some context which has more special [[properties]] than the generic object in that context will have. For instance [[manifolds]] are nice objects in the context of [[generalized smooth space|generalized smooth spaces]]. [[field|Fields]] are nice objects in the context of [[commutative rings]].

Clearly, the more extra properties one imposes, the less likely it is that these are preserved under limits and colimits. For instance

* not all [[quotients]] $X/G$ of a manifold $X$ by the action of a group $G$ are again manifolds (this is a colimit which fails to exist);

* not all [[fiber products]] $Y \times_X Y$ of surjections $Y \to X$ of manifolds are again manifolds (this is a limit that fails to exist).

* it is often false that the [[coproduct]] of fields (in the category of commutative rings) is again a field.

## Some nPOV rules of thumb

Without making any sweeping judgments, a general [[nPOV]] heuristic is that faced with a choice between working with a non-nice category of nice objects and a larger nice category of non-nice objects, it is usually preferable to switch attention to the nice category. For example, instead of working with the category of smooth manifolds, work instead with an ambient category with better properties, such as a category of [[diffeological spaces]] or a model of [[synthetic differential geometry]], in which the category of manifolds is fully embedded. [Moerdijk-Reyes](#MR) can be read largely as an implementation of that philosophy.

On the other hand, another methodological heuristic is to move from a nice object to a nice category attached to it. For instance, the category of [[modules]] over a field is about as well behaved as it could possibly be, and similarly one often contemplates the category of [[vector bundles]] over a smooth compact manifold.

## Formalization

The notion of "nice objects" can be formalized to some degree for instance in terms of _Isbell self-duality_ as described in [Lawvere](#Law).

## References

* {#Deligne98} [[Pierre Deligne]], _Quelques idées maîtresses de l'œuvre de A. Grothendieck_, [[Matériaux pour l’histoire des mathématiques au XXe siècle|Matériaux pour l'histoire des mathématiques au XX]]$^e$ siecle (Nice, 1996), Societé Mathématique de France, 1998, pp. 11–19.

* {#MR} [[Ieke Moerdijk]] and [[Gonzalo Reyes]], [[Models for Smooth Infinitesimal Analysis]]

* {#Law}  [[William Lawvere]], _Taking categories seriously_ ([pdf](http://www.emis.de/journals/TAC/reprints/articles/8/tr8.pdf))


Dickey Lie bracket

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## Idea

In [[Lagrangian field theory]] the _Dickey bracket_ is a canonical [[Lie bracket]] on [[conserved currents]]. Under the relation between conserved currents and [[infinitesimal symmetries of the Lagrangian]] the Dickey bracket lifts the canonical Lie bracket on [[infinitesimal symmetries]] to their associated conserved currents ([[Noether's theorem]]), exhibiting a [[Lie algebra extension]].

## References

The Dickey Lie bracket on [[conserved currents]] is due to

* [[Leonid Dickey]], _Soliton equations and Hamiltonian systems_, Advanced Series in Mathematical Physics, Vol. 12 (World Scientific 1991).

and is reviewed in

* [[Glenn Barnich]], [[Marc Henneaux]],  section 3 of _Isomorphisms between the Batalin-Vilkovisky antibracket and the Poisson bracket_, Journal of Mathematical Physics 37, 5273-5296 (1996) ([arXiv:hep-th/9601124](http://arxiv.org/abs/hep-th/9601124), [DOI 10.1063/1.531726](http://dx.doi.org/10.1063/1.531726))

The statement that the Dickey bracket Lie algebra of currents is a central [[Lie algebra extension]] of the algebra of symmetries by [[de Rham cohomology]] of the jet bundle appears as theorem 11.2 in (Part II of)

* {#Vinogradov84a} [[Alexandre Vinogradov]], _The $\mathcal{C}$-spectral sequence, Lagrangian formalism, and conservation laws. I. the linear theory_, Journal of Mathematical Analysis and Applications **100**, 1-40 (1984) ([doi](http://dx.doi.org/10.1016/0022-247x(84)90071-4))

* {#Vinogradov84b} [[Alexandre Vinogradov]], _The $\mathcal{C}$-spectral sequence, Lagrangian formalism, and conservation laws. II. The nonlinear theory_, Journal of Mathematical Analysis and Applications **100**, Issue 1, 30 April 1984, Pages 41-129 ([publisher](http://www.sciencedirect.com/science/article/pii/0022247X84900726))

and is stated as exercise 2.28 on p. 203 of

* [[Alexandre Vinogradov]], [[Joseph Krasil'shchik]] (eds.), _Symmetries and conservation laws for differential equations of mathematical physics_, vol. 182 of Translations of Mathematical Monographs, AMS (1999)

A lift of the Dickey Lie bracket on cohomologically trivial spaces to an equivalent _[[model structure for L-infinity algebras|L-infinity equivalent]]_ [[L-infinity algebra|L-infinity bracket]] is constructed, under some assumptions, in

* {#BarnichFulpLadaStasheff97} [[Glenn Barnich]], [[Ronald Fulp]], [[Tom Lada]], [[Jim Stasheff]], _The sh Lie structure of Poisson brackets in field theory_, Communications in Mathematical Physics 191, 585-601 (1998) ([arXiv:hep-th/9702176](http://arxiv.org/abs/hep-th/9702176))

* [[Martin Markl]], [[Steve Shnider]], _Differential Operator Endomorphisms of an Euler-Lagrange Complex_, Contemporary Mathematics, Volume 231, 1999 ([arXiv:9808105](http://arxiv.org/abs/math/9808105))

The cohomologically non-trivial lift is discussed in

* [[Hisham Sati]], [[Urs Schreiber]], _[[schreiber:Lie n-algebras of BPS charges]]_, J. High Energ. Phys. (2017) 2017: 87 ([arXiv:1507.08692](http://arxiv.org/abs/1507.08692))

[[!redirects Dickey bracket]]
[[!redirects Dickey brackets]]

[[!redirects Dickey Lie bracket]]
[[!redirects Dickey Lie brackets]]
[[!redirects Dickey Lie-bracket]]
[[!redirects Dickey Lie-brackets]]
[[!redirects Dickey-Lie bracket]]
[[!redirects Dickey-Lie brackets]]

[[!redirects Dickey bracket Lie algebra]]
[[!redirects Dickey bracket Lie algebras]]


Dickson invariant

#Contents#
* table of contents
{:toc}

## Definition

Let $q$ be a [[prime number|prime]] power, $V$ a [[vector space]] of [[dimension]] $n$ over the [[finite field]] $\mathbf{F}_q$, and $\mathbf {G L}(V)$, the [[general linear group]] of invertible [[linear transformations]] on $V$.

The ring of **Dickson invariants** is defined as the $GL(V)$-[[invariants]] $Sym(V)^{\mathbf{G L}(V)}$ in the [[symmetric algebra]] on $V$.

This is a [[graded algebra|graded]] [[polynomial algebra]] on $n$ variables. The degrees of the generators are $q^n - q^i$ for $i=0,\dots,n-1$.

See [Wilkerson 83](#Wilkerson83).

## Properties

### Spaces with DI cohomology

The [[infinity-groups]] whose [[classifying spaces]]/[[deloopings]] have mod 2 [[ordinary cohomology]] given by rank $n$ [[Dickson invariants]] are precisely these 4 of which the first three are [[compact Lie groups]] and the last one in a [[2-compact group]]:

| $n=$ | 1 | 2 | 3 | 4 |
|--|--|--|--|--|
| $DI(n)=$ | [[Z/2]] | [[SO(3)]] | [[G2]] | [[G3]] |

([Dwyer-Wilkerson 93, top of p. 38 (2 of 28)](#DwyerWilkerson93))

## References

Named after [[Leonard Dickson]].

* {#Wilkerson83} [[Clarence Wilkerson]], _A primer on Dickson invariants_, in [[Haynes Miller]], [[Stewart Priddy]] (eds.) _Proceedings of the Northwestern Homotopy Theory Conference_, 1983  ([doi:10.1090/conm/019](http://dx.doi.org/10.1090/conm/019)) ([pdf](http://www.math.purdue.edu/~wilker/papers/dickson.pdf))

*  {#DwyerWilkerson93} [[William Dwyer]], [[Clarence Wilkerson]], _A new finite loop space at the prime two_, J. Amer. Math. Soc. 6 (1993), 37-64  ([doi:10.1090/S0894-0347-1993-1161306-9](https://doi.org/10.1090/S0894-0347-1993-1161306-9))

[[!redirects Dickson invariants]]


Diederik Roest

* [webpage](http://thep.housing.rug.nl/people/diederik-roest)

## Selected writings

Formulation of ([[Lagrangian densities]] for) [[type II supergravity]] with "democratic"/"[[pregeometric RR-field|pregeometric]]" [[RR-fields]] subject to [[self-dual higher gauge theory|self-duality]]:

* [[Eric Bergshoeff]], [[Renata Kallosh]], [[Tomas Ortin]], [[Diederik Roest]], [[Antoine Van Proeyen]], *New Formulations of D=10 Supersymmetry and D8-O8 Domain Walls*, Class. Quant. Grav. **18** (2001) 3359-3382 [[arXiv:hep-th/0103233](https://arxiv.org/abs/hep-th/0103233), [doi:10.1088/0264-9381/18/17/303](https://doi.org/10.1088/0264-9381/18/17/303)]

## Related entries

* [[gauged supergravity]]

category: people


dielectric medium

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## Idea

(...)

## Related entries

* [[electromagnetism]]

* [[pre-metric electromagnetism]]

## References

### General

See most references on [[electromagnetism]], e.g. these articles

* {#Russkoff70} G. Russakoff, *A Derivation of the Macroscopic Maxwell Equations*, American Journal of Physics **38** (1970) 1188–1195 [[doi:10.1119/1.1976000](https://doi.org/10.1119/1.1976000)]

* {#deLangeRaab06} O. L. de Lange, R. E. Raab *Surprises in the multipole description of macroscopic electrodynamics*, American Journal of Physics **74** (2006) 301–312 [[doi:10.1119/1.2151213](https://doi.org/10.1119/1.2151213)]

or these textbook accounts:

* [[Lev Landau]], [[Evgeny Lifshitz]], Chapter II of: *Electrodynamics of Continuous Media*, volume 8 of: *[[Course of Theoretical Physics]]*, Pergamon Press (1960, 1984) [[ISBN:9780750626347](https://shop.elsevier.com/books/electrodynamics-of-continuous-media/koch/978-0-08-057060-0), [archive](https://archive.org/details/ost-physics-landaulifshitz-electrodynamicsofcontinuousmedia/mode/2up),  [pdf](https://www.aldebaran.cz/studium/books/1984_Landau-Electrodynamics.pdf)]

* [[Sophocles J. Orfanidis]], Section 1 of: *Electromagnetic Waves and Antennas* (1999-2016) [[web](http://eceweb1.rutgers.edu/~orfanidi/ewa), [pdf](http://eceweb1.rutgers.edu/~orfanidi/ewa/ewa-1up.pdf)]

* [[Walter Greiner]], Part II of: *Classical Electrodynamics*, Springer (1998) [[doi:10.1007/978-1-4612-0587-6](https://doi.org/10.1007/978-1-4612-0587-6)]

See also

* Wikipedia, *[Dielectric](https://en.wikipedia.org/wiki/Dielectric)*

The "constitutive equations" are referred to as the "material-affinor" in:

* {#Dantzig34} [[David van Dantzig]], p. 426 of: *The fundamental equations of electromagnetism, independent of metrical geometry*, Mathematical Proceedings of the Cambridge Philosophical Society **30** 4  (1934) 421-427 [[doi:10.1017/S0305004100012664](https://doi.org/10.1017/S0305004100012664)]

The actual terminology "constitutive relation" appears in

* {#Kottler22b} [[Friedrich Kottler]], *Maxwell'sche Gleichungen und Metrik*, Sitz. Akad. Wien IIa **131** (1922) 119-146 [Engl. transl. by [[David H. Delphenich|Delphenich]]: [pdf](http://www.neo-classical-physics.info/uploads/3/4/3/6/34363841/kottler_-_maxwells_equation.pdf), [[Kottler-MaxwellEquationsAndMetric.pdf:file]]]

and *constitutive map* in:

* [[E. Jan Post]], *The constitutive map and some of its ramifications*, Annals of Physics **71** 2 (1972) 497-518 [doi:10.1016/0003-4916(72)90129-7]

[[!include electromagnetism in terms of differential forms -- references]]

[[!redirects dielectric media]]

[[!redirects dielectric]]
[[!redirects dielectrics]]

[[!redirects constitutive equation]]
[[!redirects constitutive equations]]

[[!redirects constitutive relation]]
[[!redirects constitutive relations]]

[[!redirects mermittivity]]

[[!redirects dielectric tensor]]
[[!redirects dielectric tensors]]


Diers spectrum

## Idea

[…]

## Definition

Suppose $U\colon A\to B$ is a [[functor]] that has a [[left multiadjoint]].

The __Diers spectrum__ of $U$ is the functor $B^{op}\to Set$
that sends an object $X\in B$ to the set $I$
that indexes the value
$(g_i\colon B\to U(A_i))_{i\in I}$
of the [[left multiadjoint]] of $U$ at $X$.
A morphism $f\colon X\to X'$ is sent to the induced map $I'\to I$
that assigns to $i'\in I'$ the unique $i\in I$
such that $g_{i'}\circ f\colon X\to U(A'_i)$ factors through $g_i$.

## Example

The category of [[integral domains]] and injective homomorphisms
is a [[multireflexive subcategory]] of the category of [[commutative rings]]
and ring homomorphisms.

The associated Diers spectrum functor is a functor
$$CRing^{op} \to Set$$
that sends a [[commutative ring]] to its [[prime spectrum]].

## Example

The category of [[reduced rings]] and injective homomorphisms
is a [[multireflexive subcategory]] of the category of [[commutative rings]]
and ring homomorphisms.

The associated Diers spectrum functor is a functor
$$CRing^{op} \to Set$$
that sends a [[commutative ring]] to its poset of [[radical ideals]].
This is precisely the underlying poset of the localic [[Zariski spectrum]].

## Reference

* [[Yves Diers]], _Some spectra relative to functors_.  Journal of Pure and Applied Algebra 22:1 (1981), 57–74.  [doi][1]

    [1]: https://doi.org/10.1016/0022-4049(81)90082-7

* [[Axel Osmond]]: _On Diers theory of Spectrum I: Stable functors and right multi-adjoints_, [arXiv](https://arxiv.org/abs/2012.00853).

* [[Axel Osmond]]: _On Diers theory of Spectrum II Geometries and dualities_, [arXiv](https://arxiv.org/abs/2012.02167).


Dieter Degrijse

* [Mathematics Genealogy page](https://www.genealogy.math.ndsu.nodak.edu/id.php?id=171966)

## Selected writings

On [[proper equivariant homotopy theory]]:

* {#DHLPS19} [[Dieter Degrijse]], [[Markus Hausmann]], [[Wolfgang Lück]], [[Irakli Patchkoria]], [[Stefan Schwede]], _Proper equivariant stable homotopy theory_ ([arXiv:1908.00779](https://arxiv.org/abs/1908.00779))

category: people


Dieter Lüst

* [webpage](http://homepages.physik.uni-muenchen.de/~Dieter.Luest/)

## Selected writings

Relation of [[Niemeier lattices]] to [[heterotic string theory]]:

* [[Wolfgang Lerche]], [[Dieter Lüst]], [[Adrian Norbert Schellekens]], *Ten-dimensional heterotic strings from Niemeier lattices*, Physics Letters B **181** 1–2 (1986) 71-75 [doi:10.1016/0370-2693(86)91257-8, [inspire:233203](https://inspirehep.net/literature/233203)]

On [[heterotic M-theory on ADE-orbifolds]]:

* Michael Faux, [[Dieter Lüst]], [[Burt Ovrut]], _Intersecting Orbifold Planes and Local Anomaly Cancellation in M-Theory_, Nucl. Phys. B554: 437-483, 1999 ([arXiv:hep-th/9903028](https://arxiv.org/abs/hep-th/9903028))

* Michael Faux, [[Dieter Lüst]], [[Burt Ovrut]], _Local Anomaly Cancellation, M-Theory Orbifolds and Phase-Transitions_, Nucl. Phys. B589: 269-291, 2000 ([arXiv:hep-th/0005251](https://arxiv.org/abs/hep-th/0005251))

* Michael Faux, [[Dieter Lüst]], [[Burt Ovrut]], _An M-Theory Perspective on Heterotic K3 Orbifold Compactifications_, Int. J. Mod. Phys. A18:3273-3314, 2003 ([arXiv:hep-th/0010087](https://arxiv.org/abs/hep-th/0010087))

* Michael Faux, [[Dieter Lüst]], [[Burt Ovrut]], _Twisted Sectors and Chern-Simons Terms in M-Theory Orbifolds_, Int. J. Mod. Phys. A18: 2995-3014, 2003 ([arXiv:hep-th/0011031](https://arxiv.org/abs/hep-th/0011031))

Textbook on [[string theory]]:

* {#BlumenhagenLustTheisen13} [[Ralph Blumenhagen]], [[Dieter Lüst]], [[Stefan Theisen]], _Basic Concepts of String Theory_ Part of the series Theoretical and Mathematical Physics, Springer 2013

On [[string phenomenology]] and the [[landscape of perturbative string theory vacua]]:

* _Seeing through the String Landscape(a string hunter’s companion)_, 2008 ([pdf](http://theo2.inrne.bas.bg/~dobrev/Luest.pdf))

## Related $n$Lab entries

* [[string theory]]

* [[string phenomenology]]

* [[black holes in string theory]]

* [[Riemannian orbifold]]

* [[intersecting D-brane model]]

category: people

[[!redirects Dieter Luest]]


Dieter Pumplün

* [Wikipedia entry](https://de.wikipedia.org/wiki/Dieter_Pumpl%C3%BCn)

* [Obituary](https://www.fernuni-hagen.de/universitaet/aktuelles/2021/07/am-28-prof-dieter-pumpluen-verstorben.shtml)

* [MathematicsGenealogy page](https://www.mathgenealogy.org/id.php?id=21603)

## Selected writings

Early discussion of ([[comonad|co]])[[monads]], their [[Kleisli categories]] and [[Eilenberg-Moore categories]] and given apparently the first statement of [morphisms of monads](monad#BicategoryOfMonads):

* {#Pumplün70} [[Dieter Pumplün]], *Eine Bemerkung über Monaden und adjungierte Funktoren*, Mathematische Annalen **185** (1970) 329-337 [[eudml:161964](https://eudml.org/doc/161964), [[Pumpluen-Monaden.pdf:file]]]

On [[totally convex spaces]]:

* [[Dieter Pumplün]], [[Helmut Röhrl]]: _Banach Spaces and totally convex spaces I_, Comm. Alg. **12** 8 (1984) 953-1019 [[doi:10.1080/00927878408823035](https://doi.org/10.1080/00927878408823035)]

* [[Dieter Pumplün]], [[Helmut Röhrl]], _Banach Spaces and totally convex spaces II_, Comm. Alg. **13** 5 (1985) 1047-1113 [[doi:10.1080/00927878508823205](https://doi.org/10.1080/00927878508823205)]

On [[totally convex algebras]]:

* [[Dieter Pumplün]], [[Helmut Röhrl]], *Totally convex algebras*, Comment. Math. Univ. Carolin. **33** 2 (1992) 205–235
[[pdf](http://emis.maths.tcd.ie/journals/CMUC/pdf/cmuc9202/pump.pdf), [dml:10338.dmlcz/118490](https://dml.cz/handle/10338.dmlcz/118490)]

Introducing [[superconvex spaces]]:

* [[Dieter Pumplün]], *Banach spaces and superconvex modules*, in *Symposia Gaussiana -- Conference A: Mathematics and Theoretical Physics*, De Gruyter (1995) [[doi:10.1515/9783110886726.323](https://doi.org/10.1515/9783110886726.323)]

[[!redirects Dieter Pumplun]]

category: people


Dieter Puppe

* [wikipedia entry](http://de.wikipedia.org/wiki/Dieter_Puppe)

**Siegmund Dieter Puppe** (1930-2005) was a German mathematician who made important contributions to [[algebraic topology]] and [[homotopy theory]] in particular. He held a chair at Heidelberg University from 1968 until his retirement in 1996. Among other publications he notably published in 1966 together with H.-B. Brinkmann _Kategorien und Funktoren_ , one of the first German language textbooks[^rem] on [[category theory]].

[^rem]: In the same year Hasse and Michler published _Theorie der Kategorien_ , Deutscher Verlag der Wissenschaften Berlin 1966.

* Klaus Volkert, _Vier Heidelberger Topologen 1935-1996_, Jahresbericht der Deutschen Mathematiker-Vereinigung **124**:215–238 (2022). ([doi](https://doi.org/10.1365/s); [pdf](https://link.springer.com/content/pdf/10.1365/s13291-022-00254-8.pdf))

## Selected publications

On [[homological algebra]]:

* [[Albrecht Dold]], [[Dieter Puppe]], _Homologie nicht-additiver Funktoren. Anwendungen_, Ann. Inst. Fourier **11** (1961) pp. 201-312. ([numdam:AIF_1961__11__201_0](http://archive.numdam.org/item/AIF_1961__11__201_0), [pdf](http://archive.numdam.org/article/AIF_1961__11__201_0.pdf))

On [[abelian categories]]:

* Dieter Puppe, _Korrespondenzen in abelschen Kategorien_ , Math. Ann. **148** (1962) pp.1-30. ([gdz](http://gdz.sub.uni-goettingen.de/dms/load/pdf/?PPN=GDZPPN002290812))

On [[homotopy theory]]:

* {#Homotopietheorie} [[Tammo tom Dieck]], [[Klaus Heiner Kamps]], [[Dieter Puppe]], *Homotopietheorie*, Lecture Notes in Mathematics **157** Springer 1970 ([doi:10.1007/BFb0059721](https://link.springer.com/book/10.1007/BFb0059721))

On [[dualizable objects]], [[traces]] and the [[Becker-Gottlieb transfer]]:

* [[Albrecht Dold]], [[Dieter Puppe]], *Duality, Trace and Transfer*, Proceedings of the Steklov Institute of Mathematics, **154** (1984) 85–103 [[mathnet:tm2435](http://mi.mathnet.ru/tm2435), [pdf](https://www.maths.ed.ac.uk/~v1ranick/papers/doldpup2.pdf)]

## Related $n$Lab entries

* [[stable homotopy category]], [[smash product of spectra]]

* [[Puppe sequence]]

* [[Eilenberg-Zilber theorem]]

category: people

[[!redirects D. Puppe]]


Dieter Schildknecht

* [webpage](https://www2.physik.uni-bielefeld.de/1419.html)

## Selected writings

On [[vector meson dominance]]:

* {#Schildknecht72} [[Dieter Schildknecht]], _Vector meson dominance, photo- and electroproduction from nucleons_,  in: _Photon-Hadron Interactions II_ Springer Tracts in Modern Physics, vol 63. Springer 1972 ([doi:10.1007/BFb0041507](https://doi.org/10.1007/BFb0041507))

* {#Schildknecht05} [[Dieter Schildknecht]], _Vector Meson Dominance_, Acta Phys. Polon. B37:595-608, 2006 ([arXiv:hep-ph/0511090](https://arxiv.org/abs/hep-ph/0511090))

category: people


Dieter Wandschneider

* [Wikipedia entry](https://de.wikipedia.org/wiki/Dieter_Wandschneider)

## related $n$Lab entries

* [[dialectic]]

category: people


Dietmar Salamon

**Dietmar Salamon** is a German mathematician at ETH, Zurich (earlier at University of Warwick), where he retired as prof. emeritus in 2018. His main directions of research include dynamical systems and symplectic and contact geometry and topology.

* [homepage](http://www.math.ethz.ch/~salamon/)

## Selected writings

* [[Dusa McDuff]], D.A. Salamon, _J-holomorphic curves and symplectic topology_, AMS Colloquium Publications __52__, 2004.

* [[Dusa McDuff]], Dietmar Salamon, _Introduction to symplectic topology_, 2 ed. Oxford Mathematical Monographs 1998. x+486 pp.

* [[Dusa McDuff]], Dietmar Salamon, _J-holomorphic curves and quantum cohomology_, AMS, University Lecture Series __6__, 1994.; [revised pdf](http://www.barnard.edu/math/mcduff/survey/6manjl26.pdf)

* [[Joel W. Robbin]], D. Salamon, _[[Maslov index]] for paths_, Topology __32__  (1993), no. 4, 827--844, [doi](http://dx.doi.org/10.1016/0040-9383(93)90052-W), [pdf](http://www.math.ethz.ch/~salamon/PREPRINTS/maslov.pdf), [MR94i:58071](http://www.ams.org/mathscinet-getitem?mr=94i:58071)

* Joel Robbin, Dietmar Salamon, _The spectral flow and the Maslov index_, Bull. London Math. Soc. 27 (1995), no. 1, 1--33 [pdf](https://people.math.ethz.ch/~salamon/PREPRINTS/spec.pdf)

* [[Joel W. Robbin]], Dietmar A. Salamon, _A construction of the Deligne--Mumford [[orbifold]]_, J. Eur. Math. Society, ISSN1435-9855, Vol. 8, Nº 4, 2006, 611--699, arXiv:[math/0407090](https://arxiv.org/abs/math/0407090) [MR2009d:32012](http://www.ams.org/mathscinet-getitem?mr=2009d:32012), _Corrigendum_, J. Eur. Math. Soc. (JEMS)  __9__  (2007),  no. 4, 901--905, [doi](https://doi.org/10.4171/JEMS/101)

* [[Joel W. Robbin]], Dietmar A. Salamon,  _Lyapunov maps, simplicial complexes and the Stone functor_, Ergodic Theory Dynam. Systems __12__ (1992), no. 1, 153--183, [doi](http://dx.doi.org/10.1017/S0143385700006647), [MR93h:58091](http://www.ams.org/mathscinet-getitem?mr=93h:58091)

*  [[Joel W. Robbin]], Dietmar A. Salamon, _Dynamical systems, Shape Theory and the Conley  index_, Ergodic Theory Dynam. Systems 8 (1988) 375--393

* [[Joel W. Robbin]], [[Yongbin Ruan]], Dietmar A. Salamon,  _The moduli space of regular stable maps_, Math. Z. __259__ (2008), no. 3, 525--574, [doi](https://doi.org/10.1007/s00209-007-0237-x), [MR2010a:58014](http://www.ams.org/mathscinet-getitem?mr=2010a:58014)

* Joel W. Robbin, Dietmar A. Salamon, _Feynman path integrals and the metaplectic representation_, Math. Z. __221__ (1996), no. 2, 307--335, [MR98f:58051](http://www.ams.org/mathscinet-getitem?mr=98f:58051), [doi](https://doi.org/10.1007/BF02622118)

* [[Joel W. Robbin]], Dietmar A. Salamon, _Phase functions and path integrals_, Symplectic geometry (Proc., ed. D. Salamon), 203--226, London Math. Soc. Lecture Note Ser. __192__, Cambridge Univ. Press 1993, [[RobbinSalamonPhaseFunctionsPathIntegrals.djvu:file]].

On [[complex Lie groups]]:

* [[Dietmar Salamon]], *Notes on complex Lie groups*, 2019 ([pdf](https://people.math.ethz.ch/~salamon/PREPRINTS/cx-lie.pdf), [[Salamon_ComplexLieGroups.pdf:file]])

On [[compact Lie groups]]:

* [[Dietmar Salamon]], *Notes on compact Lie groups*, 2021 ([pdf](https://people.math.ethz.ch/~salamon/PREPRINTS/liegroup.pdf), [[Salamon_CompactLieGroups.pdf:file]])

category: people

[[!redirects Dietmar A. Salamon]]


Dietmar Schumacher

* [MathGenealogy page](https://www.mathgenealogy.org/id.php?id=56975)

## Selected writings

On [[indexed categories]]:

* [[P. T. Johnstone]], [[R. Paré]], [[R. D. Rosebrugh]], [[D. Schumacher]], [[R. J. Wood]], [[G. C. Wraith]], _Indexed Categories and Their Applications_, Lecture Notes in Mathematics **661** (1978) [[doi:10.1007/bfb0061362](https://doi.org/10.1007/bfb0061362)]

On the [[indexed adjoint functor theorem]]:

* [[Robert Paré]], [[Dietmar Schumacher]], *Abstract families and the adjoint functor theorems*, in _Indexed categories and their applications_, Lecture Notes in Math. **661** Springer (1978) [[doi:10.1007/BFb0061361](https://doi.org/10.1007/BFb0061361)]

category: people

[[!redirects D. Schumacher]]


Dietmar Vogt

* [webpage](http://www2.math.uni-wuppertal.de/~vogt/)

## related $n$Lab entries

* [[Fréchet space]]

category: people


Dieudonne module

#Contents#
* table of contents
{:toc}

## Idea

It is often easier to study formal group schemes using the associated Dieudonne module.

## Definition

A *Dieudonné module* is defined to be a [[module]] over the [[Dieudonné ring]] $D_k$.

If $G$ is an [[affine scheme|affine]] [[commutative ring|commutative]] [[unitary]] [[group scheme]] over $k$ we construct (in [Proposition (codir)](#thmcodir)) a codirected diagram of Witt modules such that the codirected system $M(G)_n:= Hom(G, W_n(k))$ and its limit $M(G):=lim_n Hom(G, W_n(k))$ become Dieudonné modules.

This construction can be executed similarly in the [[ind-object|Ind category]] so we have Dieudonné modules for [[formal group]] schemes and [[p-divisible groups]].

## Properties

There is a contravariant equivalence $p.tor.for.Gr\simeq D_k .Mod.fin$ between the category of $p$-torsion formal groups and that of finitely generated $D_k$-modules.

## Examples

### Codirected diagrams of Witt modules

+-- {: .num_prop}
###### Proposition (codir){#thmcodir}
Let $k$ be a perfect field of prime characteristic $p$. Let $W_n(-):kRing\to Ring$ be the [[functor]]  assigning to a $k$-ring its (p-adic) [[Dieudonné ring]].

Let $\underline W$ be the codirected diagram

$$W_{1k}\stackrel{V}{\to}W_{2k}\stackrel{V}{\to}W_{3k}\stackrel{V}{\to}\dots\label{codir}$$

where $V:\begin{cases}W_{n} (k)\to W_{n+1} (k)
\\
(x_0,x_1,\dots,x_{n-1})\mapsto (0,x_0,x_1,\dots, x_{n-1})\end{cases}$ denotes the translation. $V$ is sometimes called [[Verschiebung morphism]] and satisfies $VF=id$ and $FV=p$ where $F$ is the [[Frobenius morphism]].

With the multiplication obtained by the algebra map $k\hookrightarrow A$ for $A\in kRing$ and the induced map $W(k)\to W_n(A)\hookrightarrow W(A)$ the above diagram (1) is *not* a diagram in the category of Dieudonné modules since $V$ is not $W(k)$-linear but satisfies $T(\lambda a)=\lambda^{1/p}T(a)$ hence we have to redefine the multiplication by scalars in $W(k)$ by

$$*:\begin{cases}
W (k)\times W_{n} (R)\to W_{n} (R)
\\
(a,w)\mapsto a^{p^{1-n}} R\cdot w
\end{cases}$$

with this (where we recall that $W_n:=coker( T^n:W_k\to W_k)$) we have $W(k)$-linear map since

$$\displaystyle V(\lambda\star a)=V(\overline{\lambda}^{p^{1-n}}a)=V(F(\overline{\lambda}^{p^{-n}})a)=\overline{\lambda}^{p^{-n}}V(a)=\lambda\star V(a)$$
=--

For an (affine commutative unipotent) group scheme the codirected systems of Witt modules endows the limit of the hom-spaces $hom(G,W_n(k))$ with the structure of a Dieudonné module:

+-- {: .num_theorem #thmAcu}
###### Theorem
([III.5](#Demazure), $Acu_k\simeq Tor_V D_kMod$)

(see also [[group scheme]] for more context concerning this theorem)

Let $k$ be a [[perfect field]] of prime characteristic $p$. Since $k$ is perfect Frobenius is an automorphism.

On the left we have the category of [[affine scheme|affine]] commutative [[unipotent group scheme|unipotent group schemes]]. On the right we have the category of all [[Dieudonné ring|D_k]][[Dieudonné module|-modules]] of $V$-torsion. The (contravariant) equivalence is given by

$$M:\begin{cases}
Acu_k&\to& Tor_V D_kMod
\\
G&\mapsto&colim_n Acu_k(G,W_{nk})
\end{cases}$$

=--

+-- {: .proof}
If we recall that the Dieudonné ring of $k$ is a $\mathbb{Z}$-[[graded ring]] where the degree $n$-part is the $1$-[[dimension|dimensional]] [[free object|free module]] generated by $V^{-n}$ if $n\lt 0$ and by $F^n$ if $n\gt 0$, we see that morphisms in $colim_n Acu_k(G,W_{nk})$ can be multiplied by powers of $V$ reps. powers of $F$ by postcomposition.
=--

## Properties

Since all the $V$ operators are are monomorphisms, we get that $Hom(G, W_n(k))\to Hom(G, W_{n+1}(k))$ are all injective and hence we can identify $Hom(G, W_n(k))$ with a submodule of $D(G)$ or explicitly we know that $Hom(G, W_n(k))=\{m \in D(G): V^n(m)=0\}$. Thus every element of $D(G)$ is killed by a power of $V$.

If we introduce a bit of abstraction we can see the beauty of all this. Let $D=\Lambda\{F, V\}$ be the noncommutative polynomial ring over the Witt vectors on two indeterminates that satisfy the commutation laws $Fw=w^p F$, $w^p V=Vw$, and $FV=VF=p$. This is called the [[Dieudonne ring]]. We have a canonical way to consider $D(G)$ as a left $D$-module. Thus $G\mapsto D(G)$ is a [[contravariant functor]] from affine unitary group schemes to the category of $D$-[[module]]s with $V$ torsion. This turns out to be an [[equivalence of categories|anti-equivalence]] of categories.

## References

* Michel Demazure, Lectures on $p$-Divisible Groups

* Jesse Kass, notes on Dieudonné modules

* (expositional) Aaron Mazel-Gee, _Dieudonné modules and the classification of formal groups_, [pdf](https://math.berkeley.edu/~aaron/xkcd/dieudonne-modules.pdf)

[[!redirects Dieudonné module]][[!redirects Dieudonne modules]][[!redirects Dieudonné modules]]


Dieudonné determinant

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Linear algebra
+-- {: .hide}
[[!include homotopy - contents]]
=--
#### Algebra
+--{: .hide}
[[!include higher algebra - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

The _Dieudonné determinant_ is a variant of the notion of [[determinant]] which applies to [[matrices]] with values in [[division rings]], such as the [[quaternions]]. It is crucial for making sense of the notion of "[[special linear group]]" over such coefficients, such as the group [[SL(2,H)]] over the [[quaternions]].

## Definition

[Dieudonné 43](#Dieudonne) showed that for $K$ a [[division ring]], there is a [[group homomorphism]] of the form

\[
  \label{TheGroupHomomorphism}
  \alpha
    \;\colon\;
  GL(n,K)/[GL(n,K), GL(n,K)]
   \longrightarrow
  K^\times/[K^\times, K^\times]
  \,,
\]

where

* $GL(n,K)$ is the [[group]] of $n \times n$ [[invertible  matrices]] with entries in $K$,

* $K^\times = GL(1,K)$ is the [[group of units]] of $K$, and

* $G/[G,G]$ is the [[abelianization]] of the group $G$.

Moreover, this group homomorphism (eq:TheGroupHomomorphism)
is uniquely determined by the following properties:

* If a row of the matrix $T$ is left multiplied by $a \in K^\times$ then $\alpha(T)$ is left multiplied by $[a] \in K^\times/[K^\times K^\times]$.

* If a multiple of one row of $T$ is added to another row of $T$, then $\alpha(T)$ is unchanged.

Composing the homomorphism $\alpha$ (eq:TheGroupHomomorphism)
with the [[quotient]] map

$$
  q
    \,\colon\,
  GL(n,K)
    \longrightarrow
  GL(n,K) / [GL(n,K), GL(n,K)]
$$

gives the **Dieudonné determinant**

\[
  \label{TheDieudonneDeterminant}
  det_{D}
  \,\coloneqq\,
  q \circ \alpha
   \;\colon\;
  GL(n,K)
   \longrightarrow
  K^\times / [K^\times, K^\times]
  \,.
\]

## The quaternionic case

In the case that $K$ is the ring of [[quaternions]], $\mathbb{H}$, we have a group isomorphism

$$
  \mathbb{H}^\times
  /
  [
    \mathbb{H}^\times,
    \mathbb{H}^\times
  ]
   \longrightarrow
 \mathbb{R}_{\gt 0}
$$

where $\mathbb{R}_{\gt 0}$ is made into a group using [[multiplication]] of [[real numbers]], coming from the fact that now the [[commutator subgroup]] $[\mathbb{H}^\times, \mathbb{H}^\times] $ is the group [[Sp(1)]] of quaternions $q$ with $|q| = 1$, and that any invertible quaternion can uniquely be written as $a q$ where $a \in \mathbb{R}_{\gt 0}$ and $\left\vert q \right\vert = 1$.

Thus, in this case we can write the Dieudonné determinant (eq:TheDieudonneDeterminant)
more concretely as a [[group homomorphism]] of the form

$$
  det_{D}
    \,\colon\,
  GL(n,\mathbb{H})
    \longrightarrow
  \mathbb{R}_{\gt 0}
  \,.
$$

Furthermore, for any $n \times n$ quaternionic matrix $T$ that is not invertible, setting $det_D(T) \coloneqq 0$  [[extension|extends]] the Dieudonné determinant to a [[function]] of the form

\[
  \label{DieudonneDetOverQuaternions}
  det_{D}
    \;\colon\;
  M_n(\mathbb{H})
    \longrightarrow
  \mathbb{R}_{\ge 0}
  \,,
\]

where $\mathrm{M}_n(\mathbb{H})$ is now the full [[matrix algebra]] of $n \times n$ matrices with [[quaternion]] entries.  This function (eq:DieudonneDetOverQuaternions) is a [[monoid]] [[homomorphism]], in that:

\[
  \label{QuaternionicDieudonneIsMonoidHomomorphism}
  det_{D}(S T)
    \,=\,
  det_{D}(S)
  det_{D}(T),
   \qquad
  det_{D}(1) = 1
  \,.
\]

##  Relation to the Study determinant

There is another notion of determinant for [[quaternion|quaternionic]] [[matrices]], the **Study determinant**.  This turns out to be just the square of the Dieudonné determinant, but it often gives a more convenient way to compute the Dieudonné determinant.

Any matrix $T \in \mathrm{M}_n(\mathbb{H})$ determines by right multiplication a [[homomorphism]] of left $\mathbb{H}$-[[modules]] $T \colon \mathbb{H}^n \to \mathbb{H}^n$.  Choosing any element $i \in \mathbb{H}$ with $i^2 = -1$ gives $\mathbb{H}^n$ the structure of a left $\mathbb{C}$-module: indeed, a [[complex vector space]] of dimension $2n$.  In this way we can identify $T \colon \mathbb{H}^n \to \mathbb{H}^n$ with a [[complex numbers|complex]]-[[linear transformation]] of a complex vector space, and define its [[determinant]] in the usual way for such a transformation.

This determinant turns out not to depend on the choice of $i \in \mathbb{H}$ with $i^2 = -1$, and it is called the **Study determinant** $det_S(T)$.

It clearly obeys

$$  det_S(S T) = det_S(S) det_S(T), \qquad det_S(1) = 1
  \,.
$$

*A priori* it is complex-valued, but in fact it takes nonnegative real values, because one can show ([Arlaksen 96](#Arlaksen96)) that

$$
  det_{S}(T) = det_{D}(T)^2
  \,,
$$

where the determinant on the right is the Dieudonné determinant (eq:DieudonneDetOverQuaternions).

Thus, by (eq:QuaternionicDieudonneIsMonoidHomomorphism), the Study determinant on $n \times n$ quaternionic matrices defines a monoid homomorphism

$$
  det_{S}
    \;\colon\;
  M_n(\mathbb{H})
   \longrightarrow
  \mathbb{R}_{\ge 0}
$$

which contains exactly as much information as the Dieudonné determinant.

## Examples and applications

Over any division ring, the Dieudonné determinant of an invertible $2 \times 2$ matrix has the following form:

$$ \det \left({\begin{array}{cc} a & b \\ c & d \end{array}}\right) =
\left\lbrace{\begin{array}{ccc} [-c b] & \text{if } \; a = 0 \\ [a d - a c a^{-1}b] & \text{if } \; a \ne 0 \end{array}}\right. \, .$$

Here $[x]$ stands for the equivalence class of $x \in K^\times$ in the abelianization $K^\times/[K^\times, K^\times]$.

The [[special linear group]] $\mathrm{SL}(n,\mathbb{H})$ over the [[quaternions]] (e.g. [[SL(2,H)]]) is defined to be the group of $n \times n$ quaternionic matrices for which the Dieudonné determinant equals 1, or equivalently for which the Study determinant equals 1.

We can show that the [[quaternionic unitary group]] [[Sp(n)]] is a subgroup of $\mathrm{SL}(n,\mathbb{H})$, as follows.    So, unlike the real and complex cases, there is no additional concept of "special unitary group" in the quaternionic case.

Every quaternionic unitary matrix is conjugate to a diagonal one since every element is conjugate to one in a maximal torus, $\mathrm{U}(n)$ contains a maximal torus of $\mathrm{Sp}(n)$, and every matrix in $\mathrm{U}(n)$ is conjugate to a diagonal one.   Thus, every $g \in Sp(n)$ is conjugate to one of the form $\mathrm{diag}(q_1, \dots, q_n)$ with $q_i \in \mathbb{C} \subset \mathbb{H}$ and $|q_i| = 1$ for all $i$.  By the defining properties of the Dieudonné determinant we have $det_D(\mathrm{diag}(q_1, \dots, q_n)) = 1$, and since this determinant is invariant under conjugation we have $det_D(g) = 1$ for all $g \in Sp(n)$.

## Draxl's pre-determinant
  {#DraxlPredeterminant}

[Draxl 83](#Draxl83) introduces a more primitive notion, the _Dieudonné predeterminant_:

Given an invertible matrix $T$ over a [[skew-field]] (and in some other cases) there is a _strict Bruhat normal form_ of $T$ ([Draxl 83, Sec. 19, Thm. 1, Def. 1 (p. 128)](#Draxl83)) $T = U D P L$ where

* $P$ is a permutation matrix

* $U$ is upper triangular unidiagonal,

* $D$ diagonal

* $L$ lower triangular unidiagonal matrix.

The case when for $P$ the identity matrix can be taken may be viewed as generic as such matrices are dense in a number of meanings and contexts. This is the case of belonging to the _big_ Bruhat cell or equivalently to the _main_ Gauss cell, and the decomposition $T = U D L$ is the [[Gauss decomposition]]. Recall that other Bruhat cells are in commutative case of higher codimension, hence not dense, and similar statements can be made in a number of noncommutative contexts. Shifted Gauss cells, which correspond to a decomposition of matrices in the form $P U D L$ for $P$ fixed, are also dense in the same sense as they are simply the shifts (by multiplication by an invertible matrix $P$) of the main cell.

The _Dieudonné predeterminant_ $\delta\epsilon\tau(T)$, introduced by Draxl, is the product of the entries of the diagonal part $D$ upside down ([Draxl 83, Sec. 20, Def. 1 (p. 133)](#Draxl83)) if the matrix is invertible and zero otherwise.

The Dieudonné determinant is then the image of $\delta\epsilon\tau(T)$ under the projection to the [[abelianization]] ([Draxl 83, Sec. 20, Cor. 1 (p. 135)](#Draxl83))

It is well known that the Gauss decomposition of matrices over a noncommutative ring  has a simple expression in terms of [[quasideterminants]], as shown by [Gelfand-Retakh 02](#GelfandRetakh02) (and in their earlier references, around 1990), from which it can be inferred that the Dieudonné predeterminant can be generically presented as a signed product of quasideterminants.

## References

The concept is due to

* {#Dieudonne43}  [[Jean Dieudonné]], _Les déterminants sur un corps non commutatif_,  Bulletin de la Société Mathématique de France, Volume 71  (1943), p. 27-45  ([numdam:BSMF_1943__71__27_0](http://www.numdam.org/item/?id=BSMF_1943__71__27_0))

Exposition and review:

* {#Arlaksen96} Helmer Aslaksen, _Quaternionic determinants_,  The Mathematical Intelligencer 18, 57–65 (1996) ([doi:10.1007/BF03024312](https://doi.org/10.1007/BF03024312), [pdf](https://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.65.877&rep=rep1&type=pdf))

* {#CohenDeLeo99} Nir Cohen, Stefano De Leo, _The quaternionic determinant_, El. J. Lin. Alg. 7, 100-111 (2000) ([arXiv:math-ph/9907015](https://arxiv.org/abs/math-ph/9907015))

* [[Nigel Hitchin]], _$SL(2)$ over the octonions_ ([arXiv:1805.02224](https://arxiv.org/abs/1805.02224))

* [[Joás Venâncio]], [[Carlos Batista]], Section 2 of: _Two-Component Spinorial Formalism using Quaternions for Six-dimensional Spacetimes_ ([arXiv:2007.04296](https://arxiv.org/abs/2007.04296))

See also:

* Wikipedia, _[Dieudonné determinant](https://en.wikipedia.org/wiki/Dieudonn%C3%A9_determinant)_

Comparison to [[quasideterminants]] is in

* {#GelfandRetakh02} [[Israel Gelfand]], Vladimir Retakh, Robert Lee Wilson, _Quaternionic quasideterminants and determinants_, 2002 ([math.QA/0206211](https://arxiv.org/abs/math/0206211))

* {#Draxl83} [[Peter Draxl]], Section III.20 of: _Skew fields_, London Math. Soc. Lecture notes __81__, Cambridge University Press 1983 ([doi:10.1017/CBO9780511661907](https://doi.org/10.1017/CBO9780511661907), ISBN:9780511661907)

For lectures on quasideterminants see

* [[V. Retakh]], R. Wilson, _Advanced course on quasideterminants and universal localization_,  124 pp, CRM, Barcelona, 2007 ([pdf](http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.123.3499&rep=rep1&type=pdf))

[[!redirects Dieudonné determinants]]
[[!redirects Dieudonne determinant]]
[[!redirects Dieudonne determinants]]


Dieudonné ring

The *Dieudonné ring $D_k$ of a field $k$ of prime [[characteristic]] $p$* is defined to be the ring generated by two objects $F,V$ subject to the relations

$$FV=VF=p$$

$$Fw=w^\sigma F$$

$$w V=V w^\sigma$$

where

$$\sigma:\begin{cases}
W(k)\to W(k)
\\
(w_1,w_2,\dots)\mapsto (w_1^p,w_2^p,\dots)
\end{cases}$$

denotes the endomorphism of the [[Witt ring]] $W(k)$ of $k$ given by raising each component of the [[Witt vectors]] to the $p$-th power; this means that $\sigma$ is component-wise given by the [[Frobenius endomorphism]] of the field $k$.

More precisely an element of $D_k$ can uniquely be written as a finite sum

$$\sum_{i\gt 0} a_{-i} V^i + a_0 + \sum_{i\gt 0} a_i F^i$$

The Dieudonné ring is a $\mathbb{Z}$-[[graded ring]] where the degree $n$-part is the $1$-[[dimension|dimensional]] [[free object|free module]] generated by $V^{-n}$ if $n\lt 0$ and by $F^n$ if $n\gt 0$

## References

* [[Pierre Cartier]], [[algebraic group|Groupes algébriques]] et [[formal group|groupes formels]], in Théorie des Groupes Algébriques (Bruxelles, 1962) [pdf](http://www.famaf.unc.edu.ar/andrus/papers/cartier1.pdf)

* [[Jean Dieudonné]], _Lie groups and Lie [[hyperalgebra]]s over a field of characteristic $p\gt 0$. IV_, American Journal of Mathematics

* Manin, Ju. I., Theory of commutative formal groups over fields of finite characteristic, 1963


Diff

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Differential geometry
+--{: .hide}
[[!include synthetic differential geometry - contents]]
=--
=--
=--

The term _$Diff$_ is often used as a name for one of the following [[categories]]:

* the category of [[smooth manifolds]] with [[smooth functions]] between them (see at *[[SmoothManifolds]]*),

* the category of [[differentiable manifolds]] for some degree of differentiation (which authors may or may not indicate by a subscript),

* the category of [[diffeological spaces]].

Beware that all these categories are different (not [[equivalence of categories|equivalent]]), albeit closely related.

category: disambiguation


diffeological groupoid

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Differential geometry
+--{: .hide}
[[!include synthetic differential geometry - contents]]
=--
#### $\infty$-Lie theory
+--{: .hide}
[[!include infinity-Lie theory - contents]]
=--
=--
=--

# Contents
* table of contents
{:toc}

## Idea

Like a [[Lie groupoid]] is an [[internal groupoid]] in the [[category]] [[Diff]] of [[smooth manifolds]], a _diffeological groupoid_ is more generally an internal groupoid in the larger category of [[diffeological spaces]].

If we regard Lie groupoids as special cases of [[stacks]] on [[Diff]] ([[smooth stacks]]), then diffeological groupoids are a little more general special cases.

## Examples

* The [[path groupoid]] of a smooth manifold (and indeed of a diffeological space) $X$ is a diffeological groupoid.

* A pointed connected diffeological groupoid is a _diffeological group_, generalizing the notion of [[Lie group]].

## Related concepts

* [[smooth groupoid]]

## References

Diffeological groups were intoduced in

* {#Souriau79} [[Jean-Marie Souriau]], _Groupes différentiels_, in _Differential Geometrical Methods in Mathematical Physics_ (Proc. Conf., Aix-en-Provence/Salamanca, 1979), Lecture Notes in Math. 836, Springer, Berlin, (1980), pp. 91–128. ([doi:10.1007/BFb0089728](https://doi.org/10.1007/BFb0089728), [mr:607688](http://www.ams.org/mathscinet-getitem?mr=607688))

* {#Souriau84} [[Jean-Marie Souriau]], _Groupes différentiels et physique mathématique_,  In: Denardo G., Ghirardi G., Weber T. (eds.) Group Theoretical Methods in Physics. Lecture Notes in Physics, vol 201. Springer 1984 ([doi:10.1007/BFb0016198](https://doi.org/10.1007/BFb0016198))

motivated by the examples appearing in [[geometric quantization]], such as the (Hamiltonian) [[diffeomorphism group]] and its [[quantomorphism group]] extension.

Textbook account:

* {#PIZ} [[Patrick Iglesias-Zemmour]], Chapter 7 of: _Diffeology_, Mathematical Surveys and Monographs, AMS (2013) ([web](http://math.huji.ac.il/~piz/Site/The%20Book.html), [ ISBN:978-0-8218-9131-5](https://bookstore.ams.org/surv-185))

[[!redirects diffeological groupoid]]
[[!redirects diffeological groupoids]]
[[!redirects diffeological n-groupoid]]
[[!redirects diffeological n-groupoids]]

[[!redirects diffeological group]]
[[!redirects diffeological groups]]


diffeological singular simplicial set -- definition

+-- {: .num_defn #DiffeologicalSingularSimplicialSet}
###### Definition
**(diffeological singular simplicial set)**

Consider the [[simplicial object|simplicial]] [[diffeological space]]

$$
  \array{
    \Delta
    &
    \overset{
      \Delta^\bullet_{diff}
    }{
      \longrightarrow
    }
    &
    DiffeologicalSpaces
    \\
    [n]
      &\mapsto&
    \Delta^n_{diff}
    \mathrlap{
    \coloneqq
    \big\{
      \vec x \in \mathbb{R}^{n+1}
      \;\vert\;
      \underset{i}{\sum} x^i = 1
    \big\}
    }
  }
$$

which in degree $n$ is the standard extended [[n-simplex]] inside [[Cartesian space]] $\mathbb{R}^{n+1}$, equipped with its sub-[[diffeology]].

This induces a [[nerve and realization]] [[adjunction]] between [[diffeological spaces]] and [[simplicial sets]]:

\[
  \label{NerveAndRealizationForDiffeologicalSpaces}
  DiffeologicalSpaces
  \underoverset
    {
      \underset{Sing_{\mathrlap{diff}}}{\longrightarrow}
    }
    {
      \overset{ \left\vert - \right\vert_{\mathrlap{diff}} }{\longleftarrow}
    }
    {
      \phantom{AA}\bot\phantom{AA}
    }
  SimplicialSets
  \,,
\]

where the [[right adjoint]] is the _diffeological [[singular simplicial set]] [[functor]]_ $Sing_{diff}$.

=--

(e.g. [Christensen-Wu 13, Def. 4.3](diffeological+space#ChristensenWu13))

+-- {: .num_remark #DiffeologicalSingularSetIsPathGroupoid}
###### Remark
**(diffeological singular simplicial set as [[path ∞-groupoid]])**

Regarding [[simplicial sets]] as [[classical model structure on simplicial sets|presenting]] [[∞-groupoids]], we may think of $Sing_{diff}(X)$ (Def. \ref{DiffeologicalSingularSimplicialSet}) as the [[path ∞-groupoid]] of the [[diffeological space]] $X$.

In fact, by the discussion at [[shape via cohesive path ∞-groupoid]] we have that $Sing_{diff}$ is [[equivalence in an (∞,1)-category|equvialent]] to the [[shape modality|shape]] of [[diffeological spaces]] regarded as [[objects]] of the [[cohesive (∞,1)-topos]] of [[smooth ∞-groupoids]]:

$$
  Sing_{diff}
  \;\simeq\;
  Shp \circ i
  \;\;\colon\;\;
  DiffeologicalSpaces
  \overset{i}{\hookrightarrow}
  SmoothGroupoids_{\infty}
  \overset{Shape}{\longrightarrow}
   Groupoids_\infty
$$

=--


diffeological space

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Differential geometry
+-- {: .hide}
[[!include synthetic differential geometry - contents]]
=--
#### Cohesive toposes
+-- {: .hide}
[[!include cohesive infinity-toposes - contents]]
=--
=--
=--

{:mynumdef: .num_defn style="border:solid #cccccc;border-width:2px 1px;padding:0 1em;margin:0 1em;"}

# Contents
* table of contents
{: toc}

## Idea

A **diffeological space** is a type of [[generalized smooth space]].  As with the other variants, it subsumes the notion of [[smooth manifold]] but also naturally captures other spaces that one would like to think of as smooth spaces but aren't manifolds; for example, the space of all smooth maps between two smooth manifolds can be made into a diffeological space. (These mapping spaces are rarely manifolds themselves, see [[manifolds of mapping spaces]].)

In a little more detail, a **diffeology**, $\mathcal{D}$ on a set $X$ is a [[presheaf]] on the [[category of open subsets]] of Euclidean spaces with smooth maps as morphisms.  To each open set $U \subseteq \mathbb{R}^n$, it assigns a subset of $\Set(U,X)$.  The functions in $\Set(U,X)$ are to be regarded as the "smooth functions" from $U$ to $X$.  A **diffeological space** is then a set together with a diffeology on it.

Diffeological spaces were originally introduced in ([Souriau 79](#Souriau79)). They have subsequently been developed in the textbook ([Iglesias-Zemmour 13](#PIZ))

## Definition

+-- {: mynumdef #DiffSp}
###### Definition
Let $\mathcal{Op}$ denote the [[site]] whose objects are the open subsets of the Euclidean spaces $\mathbb{R}^n$  and whose morphisms are [[smooth map]]s between these.
The [[Grothendieck topology]] on $\mathcal{Op}$
is generated by the [[coverage]] of open covers,
i.e., a family of maps $\{U_i\to X\}_{i\in I}$
is a covering family if every map $U_i\to X$ is an open embedding and the union of the images of $U_i$ in $X$ equals $X$.

A **diffeological space** is a pair $(X,\mathcal{D})$ where

* $X$ is a set

* and $\mathcal{D} \in Sh(\mathcal{Op})$ is a **[[diffeology]]** on $X$:

  * a [[subobject|subsheaf]] of the sheaf $U \mapsto Hom_{Set}(U,X)$ with $\mathcal{D}(*) = X$

  * equivalently: a [[concrete sheaf]] on the [[site]] $\mathcal{Op}$ such that $\mathcal{D}(*) = X$ - a [[concrete object|concrete]] [[smooth space]] (see there for more details).

A [[morphism]] of diffeological spaces is a morphism of the corresponding [[sheaves]]: we take $DiffeologicalSp \hookrightarrow Sh(CartSp)$ to be the full [[subcategory]] on the diffeological spaces in the sheaf topos.

=--

For $(X,\mathcal{D})$ a diffeological space, and for any $U \in \mathcal{Op}$, the set $\mathcal{D}(U)$ is also called the set of **[[plots]]** in $X$ on $U$. This is to be thought of as the set of ways of mapping $U$ smoothly into the would-be space $X$. This assignment _defined_ what it means for a map $U \to X$ of sets to be smooth.

For some comments on the reasoning behind this kind of definition of generalized [[space]]s see [[motivation for sheaves, cohomology and higher stacks]].

A sheaf on the site $\mathcal{Op}$ of open subsets of Euclidean spaces is completely specified by its restriction to [[CartSp]], the full subcategory of [[Cartesian space]]: The [[fully faithful functor]] $CartSp \hookrightarrow \mathcal{Op}$ is a [[dense subsite]]-inclusion.  Therefore in the sequel we shall often restrict our attention to [[CartSp]].

One may define a _[[smooth sets]]_ to be _any_ sheaf of [[CartSp]].
A diffeological space is equivalently a [[concrete sheaf]] on the [[concrete site]] [[CartSp]]. (For details see [this Prop.](geometry+of+physics+--+smooth+sets#DiffeologicalSpacesAreTheConcreteSmoothSets) at _[[geometry of physics -- smooth sets]]_.)

The [[full subcategory]]

$$
  DiffeologicalSpaces \hookrightarrow Sh(CartSp)
$$

on all concrete sheaves is not a [[topos]], but is a [[quasitopos]].

This is Prop. \ref{DiffeologicalSpacesAreTheConcreteSmoothSets} below.

The concreteness condition on the sheaf is a reiteration of the fact that a diffeological space is a subsheaf of the sheaf $U \mapsto X^{|U|}$.  In this way, one does not have to explicitly mention the underlying set $X$ as it is determined by the sheaf on the one-point open subset of $\mathbb{R}^0$.

## Examples

* Every [[smooth manifold]] $X$, i.e. every object of [[Diff]], becomes a diffeological space by defining the [[plots]] on $U \in CartSp$ to be the ordinary [[smooth function]]s from $U$ to $X$, i.e. the morphisms in [[Diff]]:

  $$
    X : U \mapsto Hom_{Diff}(U,X)
    \,.
  $$

* For $X$ and $Y$ two diffeological spaces, their [[product]] as sets $X \times Y$ becomes a diffeological space whose [[plots]] are pairs consisting of a [[plot]] into $X$ and one into $Y$

  $$
    X \times Y : U \mapsto Hom_{DiffSp}(U,X) \times Hom_{DiffSp}(U,Y)
    \,.
  $$

* Given any two diffeological spaces $X$ and $Y$, the set of
  morphisms $Hom_{DiffSp}(X,Y)$ becomes a smooth space by taking
  the [[plots]] on some $U$ to be the smooth morphisms $X \times U \to Y$,
  i.e. the smooth $U$-parameterized families of smooth maps from $X$ to $Y$:

  $$
    [X,Y] : U \mapsto Hom_{DiffSp}(X \times U, Y)
    \,.
  $$

  In this formula we regard $U \in CartSp \hookrightarrow Diff$
  as a diffeological space according to the above example.
  In fact, we apply secretly here the [[Yoneda embedding]]
  and use the general formula for the
  cartesian [[closed monoidal structure on presheaves]].

## Properties

### Relation to topological and $\Delta$-generated spaces
 {#RelationToTopologicalSpaces}

[[!include adjunction between topological spaces and diffeological spaces]]

Further discussion of the [[D-topology]] is in [CSW 13](#CSW13).

### Topological homotopy type and diffeological shape
 {#OnTopologicalHomotopyTypeAndDiffeologicalShape}

[[!include diffeological singular simplicial set -- definition]]

[[!include topological homotopy type is cohesive shape of continuous diffeology -- proposition]]

\begin{proposition}\label{InternalHomOnDTopSpacesHasCorrectWeakType}
**([[diffeological space|diffeological]] [[internal hom]] on [[D-topological spaces]] has correct [[shape via cohesive path ∞-groupoid|cohesive shape]])**
\linebreak
  For $X, A \,\in\, DTopSp \hookrightarrow TopSp$ a [[pair]] of [[D-topological spaces]], their [[internal hom]] formed in diffeological spaces has diffeological [[singular simplicial complex]] $Sing_{diff}(-)$ (eq:NerveAndRealizationForDiffeologicalSpaces)
 [[weak homotopy equivalence|weakly homotopy equivalent]] to the ordinary [[singular simplicial set]] $Sing(-)$ of the ordinary [[mapping space]] $Maps_{Top}$ with its [[compact-open topology]]:

$$
  X,\, A
  \;\in\;
  k TopSp
  \;\;\;\;\;\;
  \vdash
  \;\;\;\;\;\;
  Sing_{diff}
  \left(
  Maps_{dfflg}
  \big(
     X
     ,\,
     A
  \big)
  \right)
  \underoverset
    {\in \mathrm{W}_{wh}}
    {}
    {\longrightarrow}
  Sing
  \left(
  Maps_{Top}
  \big(
     X
     ,\,
     A
  \big)
  \right)
$$
\end{proposition}
\begin{proof}
By [SYH 10](#SYH10) we have the following morphism:
\[
  \label{AComparisonMoprphismForShapeOfDiffeologicalMappingSpace}
  Cdfflg
  \big(
    Maps_{Top}(X,\,A)
  \big)
  \underoverset
    { \in Cdfflg(\mathrm{W}_{wh}) }
    { Cdfflg(\phi) }
    {\longrightarrow}
  Cdfflg
  \big(
    \mathbf{smap}(X ,\, A)
  \big)
  \;\;
  \simeq
  \;\;
  Maps_{dfflg}
  (X,\, A)
  \,,
\]
where $\mathbf{smap}$ is some topologization of the set of maps (defined on their [p. 6](https://arxiv.org/pdf/1010.3336.pdf#page=6) ) of which all we need to know is that:

1. (shown on the right of (eq:AComparisonMoprphismForShapeOfDiffeologicalMappingSpace)) its image under $Cdfflg$ is [[isomorphism|isomorphic]] to the internal hom $Maps_{dfflg}$ in diffeological spaces, according to their Prop. 4.7 ([p. 7](https://arxiv.org/pdf/1010.3336.pdf#page=7)),

1. (shown on the left of (eq:AComparisonMoprphismForShapeOfDiffeologicalMappingSpace)) it is [[weak homotopy equivalence|weak homotopy equivalent]], via some map $\phi$ according to their Prop. 5.4 ([p. 9](https://arxiv.org/pdf/1010.3336.pdf#page=9)) to the compact-open topology.

Hence the claim follows by using [[2-out-of-3]] in the [[naturality square]] of the natural weak homotopy equivalence from Prop. \ref{TopologicalHomotopyTypeIsCohesiveShapeOfCdfflg}
$$
  Sing_{diff} \circ Cdfflg
  \;
  \underoverset
    {\in \mathrm{W}_{wh}}
    {}
    {\longrightarrow}
  \;
  Sing
$$
applied to (eq:AComparisonMoprphismForShapeOfDiffeologicalMappingSpace).
\end{proof}

### Embedding of smooth manifolds into diffeological spaces
 {#EmbeddingOfSmoothManifoldsIntoDiffeoloticalSpaces}

+-- {: .num_prop}
###### Proposition

The obvious functor from the [[category]] [[SmoothManifolds]] of [[smooth manifolds]] to the [[category]] [[DiffeologicalSpaces]] of diffeological spaces is a [[full and faithful functor]]

$$
  SmoothManifolds \hookrightarrow  DiffeologicalSpacess
  \,.
$$

=--

+-- {: .proof}
###### Proof

This is a direct consequence of the fact that [[CartSp]]$_{smooth}$ is a [[dense sub-site]] of [[SmoothManifolds]] and the [[Yoneda lemma]].

It may nevertheless be useful to spell out the elementary proof directly:

To see that the functor is faithful, notice that if $f,g : X \to Y$ are two [[smooth function]]s that differ at some point, then they must differ in some [[open neighbourhood]] of that point. This [[open ball]] is a [[plot]], hence the corresponding diffeological spaces differ on that [[plot]].

To see that the functor is full, we need to show that a map of sets $f : X \to Y$ that sends [[plots]] to [[plots]] is necessarily a [[smooth function]], hence that all its [[derivative]]s exist. This can be tested already on all smooth curves $\gamma : (0,1) \to X$ in $X$. By [[Boman's theorem]], a function that takes all smooth curves to smooth curves is necessarily a smooth function. But curves are in particular [[plots]], so a function that takes all [[plots]] of $X$ to [[plots]] of $Y$ must be smooth.

=--

+-- {: .num_remark}
###### Remark

The proof shows that we could restrict attention to the full sub-site $CartSp_{dim \leq 1} \subset CartSp$ on the objects $\mathbb{R}^0$ and $\mathbb{R}^1$ and still have a full and faithful embedding

$$
  Diff \hookrightarrow Sh(CartSp_{dim \leq 1})
  \,.
$$

This fact plays a role in the definition of [[Frölicher space]]s, which are [[generalized smooth space]]s defined by [[plots]] by curves into and out of them.

While the site $CartSp_{dim \leq 1}$ is more convenient for some purposes, it is not so useful for other purposes, mostly when diffeological spaces are regarded from the point of view of the full sheaf topos: the sheaf topos $Sh(CartSp_{dim \leq 1})$ lacks some non-[[concrete sheaf|concrete]] sheaves of interest, such as the sheaves of differential forms of degree $\geq 2$.

=--

### Embedding of smooth manifolds with boundary into diffeological spaces

+-- {: .num_prop #SmoothManifoldsWithBoundaryEmbedIntoDiffeologicalSpaces}
###### Proposition
**([[manifolds with boundaries and corners]] form [[full subcategory]] of [[diffeological spaces]])**

The evident [[functor]]

$$
  SmthMfdWBdrCrn \overset{\phantom{AAAA}}{\hookrightarrow} DiffeologicalSpaces
$$

from the [[category]] of [[smooth manifold|smooth]] [[manifolds with boundaries and corners]] to that of [[diffeological spaces]] is [[fully faithful functor|fully faithful]], hence is a [[full subcategory]]-embedding.

=--

([Iglesias-Zemmour 13, 4.16](#PIZ), [Gürer & Iglesias-Zemmour 19](#GurerIZ19))

### Embedding of Banach manifolds into diffeological spaces
 {#EmbeddingOfBanachManifoldsIntoDiffeologicalSpaces}

Also [[Banach manifolds]] embed [[full and faithful functor|fully faithfully]] into the category of diffeological spaces.  In ([Hain](#Hain)) this is discussed in terms of Chen smooth spaces.

### Embedding of Fréchet manifolds into diffeological spaces {#RelationBetweenDeffeologicalAndFrechetStructure}

We discuss a natural embedding of [[Fréchet manifolds]] into the category of  diffeological spaces.

+-- {: .num_defn }
###### Definition

Define a [[functor]]

$$
  \iota \colon FrechetManifolds \to DiffeologicalSpaces
$$

in the evident way by taking for $X$ a [[Fréchet manifold]] for any $U \in $ [[CartSp]] the set of $U$-[[plots]] of $\iota(X)$ to be the set of smooth functions $U \to X$.

=--

+-- {: .num_prop #FrechetEmbedding}
###### Proposition

The functor $\iota \colon FrechetManifolds \hookrightarrow DiffeologicalSpaces$ is a [[full and faithful functor]].

=--

This appears as ([Losik 94, theorem 3.1.1](#Losik94), following [Losik 92](#Losik92)), as variant of the analogous statement for [[Banach manifolds]] in ([Hain](#Hain)). The fact that maps between Fréchet spaces are smooth if and only if they send smooth curves to smooth curves was proved earlier in ([Frölicher 81, théorème 1](#Frolicher))

The statement is also implied by ([Kriegl-Michor 97, cor. 3.14](#KrieglMichor)) which states that functions between [[locally convex vector spaces]] are diffeologically smooth precisely if they send smooth [[curves]] to smooth curves. This is not true if one uses [[Michal-Bastiani smooth map|Michal-Bastiani smoothness]] ([Glöckner 06](#Glockner06)), in which case one merely has a [[faithful functor]] $lctvs \to DiffeologicalSpaces$. Notice that the choice of topology in ([Kriegl-Michor 97](#KrieglMichor)) is such that this equivalence of notions reduces to the above just for Fréchet manifolds.

+-- {: .num_prop }
###### Proposition

Let $X, Y \in SmoothManifold$ with $X$ a [[compact manifold]].

Then under this embedding, the diffeological mapping space structure $C^\infty(X,Y)_{diff}$ on the mapping space coincides with the Fréchet manifold structure $C^\infty(X,Y)_{Fr}$:

$$
  \iota(C^\infty(X,Y)_{Fr})
  \simeq
  C^\infty(X,Y)_{diff}
  \,.
$$

=--

This appears as ([Waldorf 09, lemma A.1.7](#Waldorf09)).

$\,$

### Embedding of diffeological spaces into smooth sets
 {#EmbeddingOfDiffeologicalSpacesIntoTheSheafTopos}

We discuss how diffeological spaces are equivalently the [[concrete objects]] in the [[cohesive topos]] of [[smooth sets]] (see [there](smooth+set#Cohesion)).

+-- {: .num_prop #DiffeologicalSpacesAreTheConcreteSmoothSets}
###### Proposition
**(diffeological spaces are the concrete smooth sets)**

The [[full subcategory]] on the [[concrete objects]] in the [[topos]] $SmoothSet \coloneqq Sh(Cart)$ of [[smooth sets]] is [[equivalence of categories|equivalent]] to the category of diffeological spaces

=--

+-- {: .proof}
###### Proof

The [[concrete sheaves]] for the [[local topos]] $Sh(CartSp)$ are by definition those objects $X$ for which the
$(\Gamma \dashv CoDisc)$-[[unit of an adjunction|unit]]

$$
  X \to CoDisc \Gamma X
$$

is a [[monomorphism]]. Monomorphisms of sheaves are tested objectwise, so that means equivalently that for every $U \in CartSp$ we have that

$$
  X(U) \simeq Hom_{Sh}(U,X) \to Hom_{Sh}(U, Codisc \Gamma X)
  \simeq Hom_{Set}(\Gamma U, \Gamma X)
$$

is a monomorphism. This is precisely the condition on a sheaf to be a diffeological space.

=--

For a fully detailed proof see [this Prop.](geometry+of+physics+--+smooth+sets#DiffeologicalSpacesAreTheConcreteSmoothSets) at _[[geometry of physics -- smooth sets]]_.

+-- {: .num_cor}
###### Corollary

The category of diffeological spaces is a [[quasitopos]].

=--

+-- {: .proof}
###### Proof

This follows from the discussion at [Locality](#Locality).

=--

This has some immediate general abstract consequences

+-- {: .num_cor}
###### Corollary

The category of diffeological spaces is

* a [[cartesian closed category]]

* a [[closed monoidal category]].

=--

### Embedding of diffeological spaces into higher differential geometry
 {#EmbeddingOfDiffeologicalSpacesIntoHigherDifferentialGeometry}

In the last section we saw the embedding of diffeological spaces as precisely the [[concrete objects]] is the [[sheaf topos]] $Sh(CartSp) \simeq Sh(SmthMfd)$ of [[smooth sets]]. This is a general context for [[differential geometry]]. From there one can pass further to [[higher differential geometry]]: the topos of smooth sets in turn embeds

$$
  Sh(CartSp) \hookrightarrow
  Smooth \infty Grpd \coloneqq Sh_\infty(CartSp)
$$

into the [[(∞,1)-topos]] [[Smooth∞Grpd]] of "higher [[smooth sets]]" --[[smooth ∞-groupoids]] -- as precisely the [[truncated object in an (∞,1)-category|0-truncated objects]].

### Distribution theory

Since a space of [[smooth functions]] on a [[smooth manifold]] is canonically a diffeological space, it is natural to consider the _smooth_ [[linear functionals]] on such [[mapping spaces]]. These turn out to be equivalent to the [[continuous linear functionals]], hence to [[distributional densities]]. See at _[[distributions are the smooth linear functionals]]_ for details.

## Related concepts

* [[model structure on diffeological spaces]]

* [[smooth set]]

* [[Diff]]

* [[diffeological groupoid]], [[diffeological ∞-groupoid]]

* [[connectology]]

## References
 {#References}

### Foundations
  {#ReferencesGeneral}

The basic idea of understanding [[generalized smooth spaces]] as [[concrete sheaves]] on a [[site]] of smooth test spaces originates in work of [[Kuo Tsai Chen]] (see also at _[[Chen space]]_):

* {#Chen73} [[Kuo Tsai Chen]], _Iterated integrals of differential forms and loop space homology_, Ann. Math. 97 (1973), 217–246 ([jstor:1970846](https://www.jstor.org/stable/1970846))

Chen considered (apart from [[iterated integrals]]) effectively [[presheaves]] on a site of [[convex subsets]] of [[Cartesian spaces]]. In

* {#Chen75} [[Kuo Tsai Chen]], _Iterated integrals, fundamental groups and covering spaces_, Trans. Amer. Math. Soc. 206 (1975), 83–98 ([jstor:1997148](https://www.jstor.org/stable/1997148))

roughly the [[sheaf]] condition was added (without using any of this sheaf-theoretic terminology). The definition of _[[Chen spaces]]_ stabilized in

* {#Chen77} [[Kuo Tsai Chen]], _Iterated path integrals_, Bull. Amer. Math. Soc. 83, (1977), 831–879 ([euclid:1183539443](https://projecteuclid.org/euclid.bams/1183539443))

and served as the basis of a celebrated theorem on the [[de Rham cohomology]] of [[loop spaces]].

A brief review is in

* [[Kuo-Tsai Chen]], _On differentiable spaces_, in:  [[William Lawvere]], [[Stephen Schanuel]] (eds.), _[[Categories in Continuum Physics]]_, Lectures given at a Workshop held at SUNY, Buffalo 1982, Lecture Notes in Mathematics 1174, 1986 ([doi:10.1007/BFb0076928](https://link.springer.com/book/10.1007/BFb0076928))

(which, curiously, still does not make the connection to the theory of sheaves).

{#However} However, Chen does not require the domains of his plots to be [[open subsets]], which makes Chen spaces be closely related to but different from [[diffeological spaces]] (see [Stacey 11, p. 32](#Stacey11))

The proper concept of diffeological spaces was introduced, under the name _difféologie_ and apparently independently from Chen, in:

* {#Souriau79} [[Jean-Marie Souriau]], _Groupes différentiels_, in _Differential Geometrical Methods in Mathematical Physics_ (Proc. Conf., Aix-en-Provence/Salamanca, 1979), Lecture Notes in Math. 836, Springer, Berlin, (1980), pp. 91–128. ([doi:10.1007/BFb0089728](https://doi.org/10.1007/BFb0089728), [mr:607688](http://www.ams.org/mathscinet-getitem?mr=607688))

* {#Souriau84} [[Jean-Marie Souriau]], _Groupes différentiels et physique mathématique_,  In: Denardo G., Ghirardi G., Weber T. (eds.) Group Theoretical Methods in Physics. Lecture Notes in Physics, vol 201. Springer 1984 ([doi:10.1007/BFb0016198](https://doi.org/10.1007/BFb0016198))

motivated there by [[diffeological groups]] arising in [[geometric quantization]].

The article

* {#BaezHoffnung11} [[John Baez]], [[Alexander Hoffnung]], _Convenient Categories of Smooth Spaces_, Transactions of the American Mathematical Society **363** 11 (2011) [[arXiv:0807.1704](http://arxiv.org/abs/0807.1704), [jstor:41307457](https://www.jstor.org/stable/41307457)]

proved that diffeological spaces are [[concrete sheaves]] forming a [[quasi-topos]].

A discussion of the relations of variants of the definition is in

* {#Stacey11} [[Andrew Stacey]], _Comparative Smootheology_, Theory and Applications of Categories,  Vol. 25, 2011, No. 4, pp 64-117. ([tac:25-04](http://www.tac.mta.ca/tac/volumes/25/4/25-04abs.html))

### Differential geometry of diffeological spaces

Following Souriau, a comprehensive textbook account of [[differential geometry]] formulated in terms of [[diffeological spaces]] (and coining that term) is

* {#PIZ} [[Patrick Iglesias-Zemmour]], _Diffeology_, Mathematical Surveys and Monographs, AMS (2013) ([web](http://math.huji.ac.il/~piz/Site/The%20Book.html), [ ISBN:978-0-8218-9131-5](https://bookstore.ams.org/surv-185))

following the thesis

* {#IglesiasZemmour85} [[Patrick Iglesias-Zemmour]], _Fibrations difféologiques et Homotopie_,  Dissertation (1985) ([web](http://math.huji.ac.il/~piz/Site/The%20Articles/D9DD15EE-6993-4CA3-8B9B-4FC1DEF4A418.html), [pdf](http://math.huji.ac.il/~piz/documents/TheseEtatPI.pdf), [[IglesiasZemmourFibrationsDiffeologiques1985.pdf:file]])

which contains some useful material that may not yet have made it into the book.

Further exposition and lecture notes are in

* [[Patrick Iglesias-Zemmour]], _Diffeologies_, talk at _[[New Spaces for Mathematics and Physics]]_, IHP Paris 2015 ([video recording](https://www.youtube.com/watch?v=4sZDmiVOhaA))

* {#IglesiasZemmour18} [[Patrick Iglesias-Zemmour]], _An introduction to diffeology_, lecture at _[Modern Mathematics Methods in Physics: Diffeology, Categories and Toposes and Non-commutative Geometry Summer School](http://www.turkmath.org/beta/konferans.php?id_konferans=365)_, 2018, to appear in _[[New Spaces for Mathematics and Physics]]_ ([pdf](http://math.huji.ac.il/~piz/documents/AITD.pdf))

[[Cartan calculus]] for [[diffeological spaces]] is developed in

* [[Christian Blohmann]], _Elastic diffeological spaces_, [arXiv:2301.02583](https://arxiv.org/abs/2301.02583).

Application to [[classifying spaces]] and [[universal connections]]:

* [[Mark Mostow]], _The differentiable space structures of Milnor classifying spaces, simplicial complexes, and geometric realizations_, J. Differential Geom. Volume 14, Number 2 (1979), 255-293 ([euclid:jdg/1214434974](https://projecteuclid.org/euclid.jdg/1214434974))

More pointers and monthly seminars at:

* [diffeology.net](http://diffeology.net/)

### Full subcategories

The [[full subcategory]]-inclusion of [[Banach manifolds]] into the category of diffeological spaces is due to

* {#Hain} [[Richard Hain]], _A characterization of smooth functions defined on a Banach space_,  Proc. Amer. Math. Soc. 77 (1979), 63-67 ([web](http://www.ams.org/journals/proc/1979-077-01/S0002-9939-1979-0539632-8/home.html), [pdf](http://www.ams.org/journals/proc/1979-077-01/S0002-9939-1979-0539632-8/S0002-9939-1979-0539632-8.pdf))

The (non-full) embedding of [[locally convex vector spaces]] and [[Michal-Bastiani smooth maps]] into diffeological spaces is discussed around corollary 3.14 in

* {#KrieglMichor} [[Andreas Kriegl]], [[Peter Michor]]: _[[The convenient setting of global analysis]]_, AMS (1997)

That there are diffeologically-smooth maps between locally convex vector spaces that are not continuous, and a fortiori not smooth in the sense of Michal-Bastiani is given, for instance, in

* {#Glockner06} Helge Glöckner, _Discontinuous non-linear mappings on locally convex direct limits_, Publ. Math. Debrecen 68 (2006) 1-13, [arXiv:math/0503387](http://arxiv.org/abs/math/0503387).

The [[full subcategory]]-inclusion of [[Fréchet manifolds]] into diffeological spaces is discussed in

* {#Losik92} [[Mark Losik]], _Fréchet manifolds as diffeologic spaces_, Russian Mathematics 36:5 (1992), 36–42.  English translation: [PDF](https://dmitripavlov.org/scans/losik-frechet-manifolds-as-diffeologic-spaces.pdf).  Russian original: ([mathnet:ivm4812](http://mi.mathnet.ru/eng/ivm4812))

and reviewed in

* {#Losik94} [[Mark Losik]], Section 3 of: _Categorical Differential Geometry_. Cah. Topol. Géom. Différ. Catég., 35(4):274–290, 1994 ([numdam:CTGDC_1994__35_4_274_0](http://www.numdam.org/item/CTGDC_1994__35_4_274_0))

The proof can in fact be deduced from théorème 1 of

* {#Frolicher} [[Alfred Frölicher]], _Applications lisses entre espaces et variétés de Fréchet_, C. R. Acad. Sci. Paris Sér. I Math. **293** (1981), no. 2, 125–127. [BnF](http://gallica.bnf.fr/ark:/12148/bpt6k5533894s/f31.image)

The preservation of [[mapping spaces]] under this embedding is due to

* {#Waldorf09} [[Konrad Waldorf]], _Transgression to Loop Spaces and its Inverse I_, Cah. Topol. Geom. Differ. Categ., 2012, Vol. LIII, 162-210 ([arXiv:0911.3212](http://arxiv.org/abs/0911.3212))

The largest [[topological space|topology]] on the set which underlies a diffeological space with respect to which all [[plots]] are [[continuous functions]] (the "[[D-topology]]") is studied in

* {#SYH10} [[Kazuhisa Shimakawa]], K. Yoshida, [[Tadayuki Haraguchi]], _Homology and cohomology via enriched bifunctors_, Kyushu Journal of Mathematics, 2018 Volume 72 Issue 2 Pages 239-252 ([arXiv:1010.3336](https://arxiv.org/abs/1010.3336), [doi:10.2206/kyushujm.72.239]( https://doi.org/10.2206/kyushujm.72.239))

* {#CSW13} [[J. Daniel Christensen]], Gord Sinnamon, [[Enxin Wu]], _The $D$-topology for diffeological spaces_, Pacific Journal of Mathematics 272 (1), 87-110, 2014 ([arXiv:1302.2935](http://arxiv.org/abs/1302.2935), [doi:10.2140/pjm.2014.272.87](https://msp.org/pjm/2014/272-1/p04.xhtml))

Some [[homotopy theory]] modeled on diffeological spaces (instead of on [[topological spaces]]) via their [[cohesion|cohesive]] [[shape modality|shape]] is discussed in

* {#ChristensenWu13} [[J. Daniel Christensen|J. Daniel Christensen]],  [[Enxin Wu]], _The homotopy theory of diffeological spaces, I. Fibrant and cofibrant objects_, New York J. Math. 20 (2014), 1269-1303 ([arXiv:1311.6394](http://arxiv.org/abs/1311.6394))

The [[full subcategory]]-inclusion of [[manifolds with boundaries and corners]] is discussed in

* {#GurerIZ19} [[Serap Gürer]],  [[Patrick Iglesias-Zemmour]], _Differential forms on manifolds with boundary and corners_, Indagationes Mathematicae, Volume 30, Issue 5, September 2019, Pages 920-929 ([doi:10.1016/j.indag.2019.07.004](https://doi.org/10.1016/j.indag.2019.07.004))

### For orbifolds
 {#ReferencesForOrbifolds}

On [[orbifolds]] regarded as naive local [[quotient spaces]] (instead of [[homotopy quotients]]/[[Lie groupoids]]/[[differentiable stacks]]) but as such formed in [[diffeological spaces]]:

* {#IKZ10} [[Patrick Iglesias-Zemmour]], [[Yael Karshon]], Moshe Zadka, _Orbifolds as diffeologies_, Transactions of the American Mathematical Society 362 (2010), 2811-2831 ([arXiv:math/0501093](https://arxiv.org/abs/math/0501093))

* {#Watts15} [[Jordan Watts]], _The Differential Structure of an Orbifold_, Rocky Mountain Journal of Mathematics, Vol. 47, No. 1 (2017), pp. 289-327 ([arXiv:1503.01740](https://arxiv.org/abs/1503.01740))

On this approach seen in the broader context of [[higher differential geometry]]:

* [[Hisham Sati]], [[Urs Schreiber]], _[[schreiber:Proper Orbifold Cohomology]]_ ([arXiv:2008.01101](https://arxiv.org/abs/2008.01101))

[[!redirects diffeological space]]
[[!redirects diffeological spaces]]

[[!redirects diffeological structure]]
[[!redirects diffeological structures]]

[[!redirects diffeology]]
[[!redirects diffeologies]]

[[!redirects DiffeologicalSpaces]]

[[!redirects Chen smooth space]]
[[!redirects Chen smooth spaces]]


diffeology > history

< [[diffeology]]


diffeomorphism

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#### Differential geometry
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#Contents#
* table of contents
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## Definition

Given two $k$-times [[differentiable manifolds]] (or [[smooth manifolds]]), then a _diffeomorphism_

$$
  f
   \;\colon\;
  X
    \longrightarrow
  Y
$$

is a [[differentiable function]]
such that there exists an [[inverse]] differentiabe function $f^{-1}$ (a function which is an [[inverse function]] on the underlying sets and is itself [[differentiable function|differentiable]] to the given degree).

Diffeomorphisms are the [[isomorphisms]] in the corrresponding [[category]] [[Diff]] of [[differentiable manifolds]]/[[smooth manifolds]].

## Properties

### Relation to homeomorphisms {#RelationHomeomorphism}

It is clear that

+-- {: .num_lemma}
###### Observation

Every diffeomorphism is in particular a [[homeomorphism]] between the underlying [[topological spaces]].

=--

The converse in general fails. There exist differentiable maps with only continuous inverse.  There are also differentiable bijections whose inverse is not even continuous.

+-- {: .num_example}
###### Example

The function $f : \mathbb{R} \to \mathbb{R}$ given by $x \mapsto x^3$ is a homeomorphism but not a diffeomorphism. The diffeomorphism property fails at the origin, where the differential $d f : T_0 \mathbb{R} \to T_0 \mathbb{R}$ is not onto.

=--

But there is a rich collection of theorems about cases when the converse is true after all.

+-- {: .num_defn}
###### Definition

For $n \in \mathbb{N}$, the **[[open n-ball]]** $\mathbb{B}^n$ is the open subset

$$
  \mathbb{B}^n = \{ \vec x \in \mathbb{R}^n | \sum_{i = 1}^n (x^i)^2 \lt 1 \}
  \subset \mathbb{R}^n
$$

of the [[Cartesian space]] $\mathbb{R}^n$ of all points of [[distance]] lower than 1 from the origin. This inherits the structure of a [[smooth manifold]] from the embedding into $\mathbb{R}^n$.

=--

+-- {: .num_theorem}
###### Theorem

In [[dimension]] $d \in \mathbb{N}$ for $d \neq 4$ we have:

every open subset of $\mathbb{R}^d$ which is homeomorphic to $\mathbb{B}^d$ is also diffeomorphic to it.

=--

See the first page of ([Ozols](#Ozols)) for a list of references.

> What's a good/canonical textbook reference for this?

+-- {: .num_remark}
###### Remark

In dimension 4 the analog statement fails due to the existence of [[exotic smooth structure]]s on $\mathbb{R}^4$.

=--

+-- {: .num_theorem}
###### Theorem

For $X$ and $Y$ smooth manifolds of dimension $d = 1$, $d = 2$ or $d = 3$ we have:

if there is a homeomorphism from $X$ to $Y$, then there is also a diffeomorphism.

=--

See the corollary on p. 2 of ([Munkres](#Munkres)).

### Relation to homotopy equivalences
 {#RelationToHomotopyEquivalences}

For the following kinds of manifolds $\Sigma$ it is true that every [[homotopy equivalence]]

$$
  \alpha \colon \Pi(\Sigma) \stackrel{\simeq}{\longrightarrow} \Pi(\Sigma)
$$

(hence every [[equivalence of infinity-groupoids|equivalence]] of their [[fundamental infinity-groupoids]]) is [[homotopy|homotopic]] to a [[diffeomorphism]]

$$
  a \colon \Sigma \stackrel{\simeq}{\longrightarrow} \Sigma
$$

i.e. that given $\alpha$ there is $a$ with

$$
  \alpha \simeq \Pi(a)
  \,.
$$

* for $\Sigma$ any [[surface]] ([Zieschang-Vogt-Coldeway](#ZieschangVogtColdeway))

* for $\Sigma$ a [[Haken 3-manifold]] ([Waldhausen](#Waldhausen68))

* for $\Sigma$ any  [[hyperbolic manifold]] of finite [[volume]] and of [[dimension]] $\geq 3$ (by [[Mostow rigidity theorem]]) (check)

A review of results and relevant literature is also on the first page of ([Hass-Scott 92](#HassScott92))-

## Related concepts

* [[diffeomorphism group]]

* [[coordinate transformation]]

* [[general covariance]]

* [[super-diffeomorphism]]

## References

* {#Ozols} V. Ozols, _Largest normal neighbourhoods_
Proceedings of the American Mathematical Society
Vol. 61, No. 1 (Nov., 1976), pp. 99-101  ([jstor](http://www.jstor.org/stable/2041672)) ([AMS](http://www.ams.org/journals/proc/1976-061-01/home.html): [pdf](http://www.ams.org/journals/proc/1976-061-01/S0002-9939-1976-0431220-4/S0002-9939-1976-0431220-4.pdf))

* {#Munkres} [[James Munkres]], _Obstructions to the smoothing of piecewise-differentiable homeomorphisms_  Bull. Amer. Math. Soc. Volume 65, Number 5 (1959), 332-334.  ([Euclid](http://projecteuclid.org/euclid.bams/1183523321))([AMS](http://www.ams.org/journals/bull/1959-65-05/): [pdf](http://www.ams.org/journals/bull/1959-65-05/S0002-9904-1959-10345-1/S0002-9904-1959-10345-1.pdf))

* {#ZieschangVogtColdeway} Zieschang, Vogt and Coldeway, _Surfaces and planar discontinuous groups_

* {#Waldhausen68} [[Friedhelm Waldhausen]], _On Irreducible 3-Manifolds Which are Sufficiently Large_, Annals of Mathematics
Second Series, Vol. 87, No. 1 (Jan., 1968), pp. 56-88 ([JSTOR](http://www.jstor.org/stable/1970594))

* {#HassScott92} Joel Hass, Peter Scott, _Homotopy equivalence and homeomoprhism of 3-manifolds_, Topology, Vol. 31, No. 3 (1992) pp. 493-517  ([pdf](http://deepblue.lib.umich.edu/bitstream/handle/2027.42/29969/0000331.pdf?sequence=1))

[[!redirects diffeomorphisms]]
[[!redirects diffeomorphic]]


diffeomorphism group

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#### Differential geometry
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#### Manifolds and cobordisms
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[[!include manifolds and cobordisms - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Definition

The _diffeomorphism group_ $Diff(X)$ of a [[smooth manifold]] $X$ is the [[group]] of its [[diffeomorphism]]s: the [[automorphism group]] of $X$ as an [[object]] of the [[category]] [[SmoothMfd]].

Beware that when $X$ is assumed [[orientable]] then sometimes, but not always, $Diff(X)$ is implicitly taken to be the group of orientation-preserving diffeomorphisms.

## Properties

### Relation to homotopy equivalences

For the following kinds of manifolds $\Sigma$ it is true that every [[homotopy equivalence]]

$$
  \alpha \colon \Pi(\Sigma) \stackrel{\simeq}{\longrightarrow} \Pi(\Sigma)
$$

(hence every [[equivalence of infinity-groupoids|equivalence]] of their [[fundamental infinity-groupoids]]) is [[homotopy|homotopic]] to a [[diffeomorphism]]

$$
  a \colon \Sigma \stackrel{\simeq}{\longrightarrow} \Sigma
$$

i.e. that given $\alpha$ there is $a$ with

$$
  \alpha \simeq \Pi(a)
  \,.
$$

* for $\Sigma$ any [[surface]] ([Zieschang-Vogt-Coldeway](#ZieschangVogtColdeway))

* for $\Sigma$ a [[Haken 3-manifold]] ([Waldhausen](#Waldhausen68))

* for $\Sigma$ any  [[hyperbolic manifold]] of finite [[volume]] and of [[dimension]] $\geq 3$ (by [[Mostow rigidity theorem]]) (check)

### Homotopy type and mapping class group
 {#HomotopyTypeAndMappingClassGroup}

The [[homotopy type]] $\Pi(Diff(\Sigma))$ of the diffeomorphism group $Diff(\Sigma)$ is of interest (e.g. [Hatcher 12](#Hatcher12)).

For instance this is the [[automorphism ∞-group]] of a manifold, regarded as a [[k-morphism]] in an [[(∞,n)-category of cobordisms]].

Specifically, the group of connected components is the [[mapping class group]]

$$
  \pi_0(\Pi(Diff(\Sigma))) = MCG(\Sigma)
  \,.
$$

#### For 1-manifolds

$$
  \Pi(Diff(S^1))\simeq \Pi(O(2))
$$

$$
  \Pi(Diff(D^1))\simeq \Pi(O(1))
$$

#### For 2-manifolds (surfaces)
 {#HomotopyTypeForSurfaces}

+-- {: .num_prop #HomotopyTypeOfDiffClosedOrientableSurface}
###### Proposition

For $\Sigma$ a [[closed manifold|closed]] [[orientation|orientable]] [[surface]], then the [[fundamental ∞-groupoid|bare homotopy type]] of its diffeomorphism group is

1. if $\Sigma$ is the [[sphere]] then

   $$
     \begin{aligned}
        \Pi(Diff(S^2)) & \simeq \Pi(O(3))
        \\
        & \simeq MCG(S^2)\times \Pi(SO(3))
        \\
        & \simeq \mathbb{Z}_2 \times \Pi(SO(3))
     \end{aligned}
   $$

1. if $\Sigma$ is the [[torus]] then

   $$
     \begin{aligned}
       \Pi(Diff(S^1 \times S^1))
       & \simeq
       MCG(S^1 \times S^1)\times \Pi(S^1 \times S^1 )
       \\
       & \simeq GL_2(\mathbb{Z}) \times B(\mathbb{Z} \times\mathbb{Z})
     \end{aligned}
   $$

1. in all other cases all higher [[homotopy groups]] vanish:

   $$
     \Pi(Diff(\Sigma)) \simeq MCG(\Sigma)
   $$

=--

The first statement is due to ([Smale 58](#Smale58)), see also at _[[sphere eversion]]_. The second and third are due to ([Earle-Eells 67](#EarleEells67), [Gramain 73](#Gramain73)).

#### For 3-manifolds

+-- {: .num_prop}
###### Proposition

$$
  \Pi(Diff(S^1 \times S^2))
  \simeq
  \Pi(O(2) \times O(3))
  \times
  \Omega \Pi(SO(3))
  \,.
$$

=--

([Hatcher 81](#Hatcher81))

\begin{theorem}
\label{SmaleConjecture}
**(Smale conjecture)**
\linebreak
The bare [[homotopy type]] of the [[diffeomorphism group]] of the [[3-sphere]] is that of the [[orthogonal group]] $O(4)$

$$
  \esh\big(
    Diff(S^3)
  \big)
  \;\simeq\;
  \esh
  \,
   O(4))
  \,,
$$

the equivalence being exhibited by the canonical inclusion

$$
  O(4) \hookrightarrow Diff(S^3)
  \,.
$$

Also

$$
  \esh
   \,
   Diff(D^3)
   \;\simeq\;
   \esh
   \,
   O(3)
   \,.
$$
\end{theorem}

After being conjectured by Smale, this was proven in ([Hatcher 1983](#Hatcher)).

Generally:

+-- {: .num_theorem}
###### Theorem

For every [[smooth manifold|smooth]] [[3-manifold]] the canonical map

$$
  \Pi(Diff(X)) \to \Pi(Homeo(X))
$$

sending [[diffeomorphisms]] to their underlying [[homeomorphisms]] of [[topological spaces]] is a [[weak homotopy equivalence]].

=--

That this follows from the Smale cojecture, theorem \ref{SmaleConjecture}, was shown in ([Cerf](#Cerf)). For discussion see ([Hatcher, 1978](#Hatcher78)).

+-- {: .num_prop}
###### Proposition

If a 3-manifold $X$ is not a [[Seifert 3-manifold]] via an $S^1$-action then

$$
  \Pi(Diff(X)) \simeq MCG(X)
  \,.
$$

=--

If $X$ is Seifert via an $S^1$-action, then the component of $Diff(X)$ are typically $\Pi(S^1)$-s.

## References

### Smooth structure

The observation that infinite-dimensional  [[smooth groups]] such as diffeomorphism groups (and [[quantomorphism groups]] etc.) are naturally regarded as [[internal groups]] in [[diffeological spaces]] -- [[diffeological groups]] -- is due to

* {#Souriau79} [[Jean-Marie Souriau]], _Groupes différentiels_, in _Differential Geometrical Methods in Mathematical Physics_ (Proc. Conf., Aix-en-Provence/Salamanca, 1979), Lecture Notes in Math. 836, Springer, Berlin, (1980), pp. 91–128. ([MathScinet](http://www.ams.org/mathscinet-getitem?mr=607688))

### For 2-manifolds (surfaces)

* {#EarleEells67} C.J. Earle,  J. Eells, _The diffeomorphism group of a compact Riemann surface,
Bulletin of the American Mathematical Society 73(4) 557--559, 1967

* {#ZieschangVogtColdeway} Zieschang, Vogt and Coldeway, _Surfaces and planar discontinuous groups_

* J. S. Dowker, *Note on the structure constants for the diffeomorphisms of the two-sphere* [[arXiv:2301.09487](https://arxiv.org/abs/2301.09487)]

### For 3-manifolds

* {#Cerf} J. Cerf, _Sur les difféomorphismes de la sphère de dimension trois ($\Gamma_4 = 0$)_, Lecture Notes in Math., vol. 53, Springer-Verlag, Berlin and New York, 1968_

* {#Hatcher78} [[Allen Hatcher]], _Linearization in 3-dimensional topology_, Proceedings of the international congress of Mathematicians, Helsinki (1978)

* {#Hatcher81} [[Allen Hatcher]], _On the diffeomorphism group of $S^1\times S^2$_, Proceedings of the AMS 83 (1981), 427-43 ([pdf](http://www.math.cornell.edu/~hatcher/Papers/newDiffS1xS2.pdf))

* {#Hatcher} [[Allen Hatcher]], _A proof of the Smale conjecture, $Diff(S^3) \simeq O(4)$, Annals of Mathematics 117 (1983) ([jstor](http://www.jstor.org/pss/2007035))

* {#Waldhausen68} [[Friedhelm Waldhausen]], _On Irreducible 3-Manifolds Which are Sufficiently Large_, Annals of Mathematics
Second Series, Vol. 87, No. 1 (Jan., 1968), pp. 56-88 ([JSTOR](http://www.jstor.org/stable/1970594))

### For 4-manifolds

For 4-manifolds the analogue of the Smale conjecture fails:

* {#Watanabe18} [[Tadayuki Watanabe]], *Some exotic nontrivial elements of the rational homotopy groups of $Diff(S^4)$* ([arXiv:1812.02448](https://arxiv.org/abs/1812.02448))

* {#Watanabe21} [[Tadayuki Watanabe]], *Addendum to: Some exotic nontrivial elements of the rational homotopy groups of $Diff(S^4)$ (homological interpretation)* ([arXiv:2109.01609](https://arxiv.org/abs/2109.01609))

  > (via  [[graph complexes]])

### General

* {#Hatcher12} [[Alan Hatcher]], _A 50-Year View of Diffeomorphism Groups_, talk at the 50th Cornell Topology Festival in May 2012 ([pdf](http://www.math.cornell.edu/~hatcher/Papers/Diff%28M%292012.pdf), [[HatcherDiffeomorphismReview.pdf:file]])

[[!redirects diffeomorphism groups]]
[[!redirects Smale conjecture]]


difference

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### Context
#### Algebra
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[[!include higher algebra - contents]]
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#### Arithmetic
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[[!include arithmetic geometry - contents]]
=--
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#Contents#
* table of contents
{:toc}

## Definition

### For abelian groups

For $A$ an [[abelian group]] and $(a,b) \in A \times A$ a pair of [[elements]], their **difference** is the element

$$
 b - a \coloneqq b + a^{-1} \in A
 \,.
$$

This is often considerd for the case that $A$ is the abelian group underlying a [[vector space]] $V$, which which case one typically denotes the elements instead as $(\vec x, \vec y) \in V \times V$ and their difference as

$$
  \vec y - \vec x \in V
  \,.
$$

Specifically if $V = \mathbb{R}$ is the [[real line]] or the [[rational numbers]] of just the [[integers]], one just writes $y-x$. Etc.

### For commutative semigroups and magmas

For $(A,+)$ a [[commutative]] [[semigroup]] or [[magma]] and $(a,b) \in A \times A$ a pair of [[elements]], their **difference**, if it exists, is an element $c \in A$ such that

$$
 c + a = b
 \,.
$$

$a$ is called the __subtrahend__ of $b$ and $b$ is called the __minuend__ of $a$.

For example, in the ordered commutative semigroup of positive integers $\mathbb{N}^+$, two positive integers $m,n \in \mathbb{N}^+$ have a difference $n - m$ if $m \lt n$.

If the magma is a multiplicative monoid $\cdot$ of a [[commutative ring]], then the difference is usually called a __quotient__ (see [[divisor (ring theory)]].

A commutative [[quasigroup]] is a commutative magma such that every pair of elements $(a,b) \in A \times A$ has a difference.

## Related concepts

* [[subtraction]]

* [[monus]]

* A _[[derivative]]_ is a limiting ratio of differences.

[[!redirects differences]]

[[!redirects subtrahend]]
[[!redirects subtrahends]]
[[!redirects minuend]]
[[!redirects minuends]]


difference protoring

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# Contents
* table of contents
{: toc}

## Definition ##

A [[protoring]] $M$ is a **difference protoring** if for all terms $a \in M$, there exists a strictly positive term $b \in M_+$ such that $0 \lt a + b$.

## Examples ##

* Every [[ordered integral domain]] is a difference protoring.

## See also ##

* [[protoring]]

* [[integral domain]]

* [[archimedean difference protoring]]

## References

* [[Davorin Lešnik]], *Synthetic Topology and Constructive Metric Spaces*, ([arxiv:2104.10399](https://arxiv.org/abs/2104.10399))

[[!redirects difference protorings]]


difference quotient

#Contents#
* table of contents
{:toc}

## Definition ##

Given a [[Heyting field]] $F$ and a subtype $S \subseteq F$, let us define the subtype $\Delta_{\#}(S) \subseteq S \times S$ of pairs of elements apart from the [[diagonal]] as

$$\Delta_{\#}(S) \coloneqq \sum_{(x,y):S \times S} x \# y$$

### In a field ###

Given a function $f:S \to F$, the __difference quotient__ $q:\Delta_{\#}(S) \to F$ is the partial binary function

$$q(x, y) := \frac{f(x) - f(y)}{x - y}$$

### In a vector space ###

Given a $F$-[[vector space]] $V$ and a function $f:S \to V$, the __difference quotient__ $q:\Delta_{\#}(S) \to V$ is the partial binary function

$$q(x, y) := \frac{1}{x - y} (f(x) - f(y))$$

## See also ##

* [[limit of a binary function approaching a diagonal]]

* [[differentiable function]]

* [[Newton-Leibniz operator]]

* [[rational function]]

* [[reciprocal function]]