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This is a preview for the new '''MathML rendering mode''' (with SVG fallback), which is availble in production for registered users.
In [[mathematics]], the '''smash product''' of two [[pointed space]]s (i.e. [[topological space]]s with distinguished basepoints) ''X'' and ''Y'' is the [[quotient space|quotient]] of the [[product space]] ''X'' &times; ''Y'' under the identifications (''x'',&nbsp;''y''<sub>0</sub>)&nbsp;∼&nbsp;(''x''<sub>0</sub>,&nbsp;''y'') for all ''x''&nbsp;∈&nbsp;''X'' and ''y''&nbsp;∈&nbsp;''Y''. The smash product is usually denoted ''X''&nbsp;∧&nbsp;''Y'' or ''X''&nbsp;⨳&nbsp;''Y''. The smash product depends on the choice of basepoints (unless both ''X'' and ''Y'' are [[homogeneous space|homogeneous]]).


One can think of ''X'' and ''Y'' as sitting inside ''X'' &times; ''Y'' as the [[subspace (topology)|subspaces]] ''X'' &times; {''y''<sub>0</sub>} and {''x''<sub>0</sub>} &times; ''Y''. These subspaces intersect at a single point: (''x''<sub>0</sub>, ''y''<sub>0</sub>), the basepoint of ''X'' &times; ''Y''. So the union of these subspaces can be identified with the [[wedge sum]] ''X'' ∨ ''Y''. The smash product is then the quotient
If you would like use the '''MathML''' rendering mode, you need a wikipedia user account that can be registered here [[https://en.wikipedia.org/wiki/Special:UserLogin/signup]]
:<math>X \wedge Y = (X \times Y) / (X \vee Y). \, </math>
* Only registered users will be able to execute this rendering mode.
* Note: you need not enter a email address (nor any other private information). Please do not use a password that you use elsewhere.


The smash product has important applications in [[homotopy theory]], a branch of [[algebraic topology]]. In homotopy theory, one often works with a different [[category (mathematics)|category]] of spaces than the category of all topological spaces. In some of these categories the definition of the smash product must be modified slightly. For example, the smash product of two [[CW complex]]es is a CW complex if one uses the product of CW complexes in the definition rather than the product topology. Similar modifications are necessary in other categories.
Registered users will be able to choose between the following three rendering modes:


==Examples==
'''MathML'''
*The smash product of any pointed space ''X'' with a [[0-sphere]] is homeomorphic to ''X''.
:<math forcemathmode="mathml">E=mc^2</math>
*The smash product of two circles is a quotient of the [[torus]] homeomorphic to the 2-sphere.
*More generally, the smash product of two spheres ''S''<sup>''m''</sup> and ''S''<sup>''n''</sup> is [[homeomorphic]] to the sphere ''S''<sup>''m''+''n''</sup>.
*The smash product of a space ''X'' with a circle is homeomorphic to the [[reduced suspension]] of ''X'':
*:<math> \Sigma X \cong X \wedge S^1. \, </math>
*The ''k''-fold iterated reduced suspension of ''X'' is homeomorphic to the smash product of ''X'' and a ''k''-sphere
*:<math> \Sigma^k X \cong X \wedge S^k. \, </math>
* In [[domain theory]], taking the product of two domains (so that the product is strict on its arguments).


==As a symmetric monoidal product==
<!--'''PNG''' (currently default in production)
For any pointed spaces ''X'', ''Y'', and ''Z'' in an appropriate "convenient" category (e.g. that of [[compactly generated space]]s) there are natural (basepoint preserving) [[homeomorphism]]s
:<math forcemathmode="png">E=mc^2</math>
:<math>\begin{align}
X \wedge Y &\cong Y\wedge X, \\
(X\wedge Y)\wedge Z &\cong X \wedge (Y\wedge Z).
\end{align}</math>
However, for the naive category of pointed spaces, this fails. See the following discussion on MathOverflow.<ref>Omar Antolín-Camarena (mathoverflow.net/users/644), In which situations can one see that topological spaces are ill-behaved from the homotopical viewpoint?, http://mathoverflow.net/questions/76594 (version: 2011-09-28)</ref>


These isomorphisms make the appropriate [[category of pointed spaces]] into a [[symmetric monoidal category]] with the smash product as the monoidal product and the pointed [[0-sphere]] (a two-point discrete space) as the unit object. One can therefore think of the smash product as a kind of [[tensor product]] in an appropriate category of pointed spaces.
'''source'''
:<math forcemathmode="source">E=mc^2</math> -->


==Adjoint relationship==
<span style="color: red">Follow this [https://en.wikipedia.org/wiki/Special:Preferences#mw-prefsection-rendering link] to change your Math rendering settings.</span> You can also add a [https://en.wikipedia.org/wiki/Special:Preferences#mw-prefsection-rendering-skin Custom CSS] to force the MathML/SVG rendering or select different font families. See [https://www.mediawiki.org/wiki/Extension:Math#CSS_for_the_MathML_with_SVG_fallback_mode these examples].
[[Adjoint functors]] make the analogy between the tensor product and the smash product more precise. In the category of [[module (mathematics)|''R''-modules]] over a [[commutative ring]] ''R'', the tensor functor (&ndash; ⊗<sub>''R''</sub> ''A'') is left adjoint to the internal [[Hom functor]] Hom(''A'',&ndash;) so that:
:<math>\mathrm{Hom}(X\otimes A,Y) \cong \mathrm{Hom}(X,\mathrm{Hom}(A,Y)).</math>
In the [[category of pointed spaces]], the smash product plays the role of the tensor product. In particular, if ''A'' is [[locally compact Hausdorff]] then we have an adjunction
:<math>\mathrm{Hom}(X\wedge A,Y) \cong \mathrm{Hom}(X,\mathrm{Hom}(A,Y))</math>
where Hom(''A'',''Y'') is the space of based continuous maps together with the [[compact-open topology]].


In particular, taking ''A'' to be the [[unit circle]] ''S''<sup>1</sup>, we see that the suspension functor Σ is left adjoint to the [[loop space]] functor Ω.
==Demos==
:<math>\mathrm{Hom}(\Sigma X,Y) \cong \mathrm{Hom}(X,\Omega Y).</math>


==References==
Here are some [https://commons.wikimedia.org/w/index.php?title=Special:ListFiles/Frederic.wang demos]:
{{Reflist}}
*{{Hatcher AT}}


{{DEFAULTSORT:Smash Product}}
 
[[Category:Topology]]
* accessibility:
[[Category:Homotopy theory]]
** Safari + VoiceOver: [https://commons.wikimedia.org/wiki/File:VoiceOver-Mac-Safari.ogv video only], [[File:Voiceover-mathml-example-1.wav|thumb|Voiceover-mathml-example-1]], [[File:Voiceover-mathml-example-2.wav|thumb|Voiceover-mathml-example-2]], [[File:Voiceover-mathml-example-3.wav|thumb|Voiceover-mathml-example-3]], [[File:Voiceover-mathml-example-4.wav|thumb|Voiceover-mathml-example-4]], [[File:Voiceover-mathml-example-5.wav|thumb|Voiceover-mathml-example-5]], [[File:Voiceover-mathml-example-6.wav|thumb|Voiceover-mathml-example-6]], [[File:Voiceover-mathml-example-7.wav|thumb|Voiceover-mathml-example-7]]
[[Category:Binary operations]]
** [https://commons.wikimedia.org/wiki/File:MathPlayer-Audio-Windows7-InternetExplorer.ogg Internet Explorer + MathPlayer (audio)]
** [https://commons.wikimedia.org/wiki/File:MathPlayer-SynchronizedHighlighting-WIndows7-InternetExplorer.png Internet Explorer + MathPlayer (synchronized highlighting)]
** [https://commons.wikimedia.org/wiki/File:MathPlayer-Braille-Windows7-InternetExplorer.png Internet Explorer + MathPlayer (braille)]
** NVDA+MathPlayer: [[File:Nvda-mathml-example-1.wav|thumb|Nvda-mathml-example-1]], [[File:Nvda-mathml-example-2.wav|thumb|Nvda-mathml-example-2]], [[File:Nvda-mathml-example-3.wav|thumb|Nvda-mathml-example-3]], [[File:Nvda-mathml-example-4.wav|thumb|Nvda-mathml-example-4]], [[File:Nvda-mathml-example-5.wav|thumb|Nvda-mathml-example-5]], [[File:Nvda-mathml-example-6.wav|thumb|Nvda-mathml-example-6]], [[File:Nvda-mathml-example-7.wav|thumb|Nvda-mathml-example-7]].
** Orca: There is ongoing work, but no support at all at the moment [[File:Orca-mathml-example-1.wav|thumb|Orca-mathml-example-1]], [[File:Orca-mathml-example-2.wav|thumb|Orca-mathml-example-2]], [[File:Orca-mathml-example-3.wav|thumb|Orca-mathml-example-3]], [[File:Orca-mathml-example-4.wav|thumb|Orca-mathml-example-4]], [[File:Orca-mathml-example-5.wav|thumb|Orca-mathml-example-5]], [[File:Orca-mathml-example-6.wav|thumb|Orca-mathml-example-6]], [[File:Orca-mathml-example-7.wav|thumb|Orca-mathml-example-7]].
** From our testing, ChromeVox and JAWS are not able to read the formulas generated by the MathML mode.
 
==Test pages ==
 
To test the '''MathML''', '''PNG''', and '''source''' rendering modes, please go to one of the following test pages:
*[[Displaystyle]]
*[[MathAxisAlignment]]
*[[Styling]]
*[[Linebreaking]]
*[[Unique Ids]]
*[[Help:Formula]]
 
*[[Inputtypes|Inputtypes (private Wikis only)]]
*[[Url2Image|Url2Image (private Wikis only)]]
==Bug reporting==
If you find any bugs, please report them at [https://bugzilla.wikimedia.org/enter_bug.cgi?product=MediaWiki%20extensions&component=Math&version=master&short_desc=Math-preview%20rendering%20problem Bugzilla], or write an email to math_bugs (at) ckurs (dot) de .

Latest revision as of 22:52, 15 September 2019

This is a preview for the new MathML rendering mode (with SVG fallback), which is availble in production for registered users.

If you would like use the MathML rendering mode, you need a wikipedia user account that can be registered here [[1]]

  • Only registered users will be able to execute this rendering mode.
  • Note: you need not enter a email address (nor any other private information). Please do not use a password that you use elsewhere.

Registered users will be able to choose between the following three rendering modes:

MathML

E=mc2


Follow this link to change your Math rendering settings. You can also add a Custom CSS to force the MathML/SVG rendering or select different font families. See these examples.

Demos

Here are some demos:


Test pages

To test the MathML, PNG, and source rendering modes, please go to one of the following test pages:

Bug reporting

If you find any bugs, please report them at Bugzilla, or write an email to math_bugs (at) ckurs (dot) de .

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