Sextic equation - Revision history

en>David Eppstein: /* See also */ Cayley's sextic

2014-10-25T20:09:45Z

← Older revision		Revision as of 22:09, 25 October 2014
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	In [[mathematics]], the [[category (mathematics)\|category]] '''Rel''' has the class of [[Set (mathematics)\|sets]] as [[object (category theory)\|objects]] and [[binary relation]]s as [[morphism (category theory)\|morphisms]].		The writer is recognized by the title of Numbers Lint. California is our birth place. Bookkeeping is her working day occupation now. To play baseball is the hobby he will never quit performing.<br><br>My homepage: std testing at home ([http://www.associazioneitalianafotografi.it/community/gruppi/points-to-know-when-confronted-with-candidiasis/ simply click the next internet site])

	~~A morphism (or arrow) ''R'' : ''A'' → ''B'' in this category is a relation between the sets ''A'' and ''B'', so {{nowrap\| ''R'' ⊆ ''A'' × ''B''}}.~~

	The ~~[[Composition of relations\|composition]] of two relations ''R'': ''A'' → ''B'' and ''S'': ''B'' → ''C''~~ is ~~given~~ by:
	~~:(''a'', ''c'') ∈ ''S'' <small>o</small> ''R'' if (and only if) for some ''b'' ∈ ''B'', (''a'', ''b'') ∈ ''R'' and (''b'', ''c'') ∈ ''S''.~~

	~~==Properties==~~
	~~The category '''Rel''' has~~ the ~~[[category~~ of sets]] '''Set''' as a (wide) [[subcategory]], where the arrow (function) {{nowrap\| ''f'' : ''X'' → ''Y''}} in '''Set''' corresponds to the functional relation {{nowrap\| ''F'' ⊆ ''X'' × ''Y''}} defined by: {{nowrap\|1= (''x'', ''y'') ∈ ''F'' ⇔ ''f''(''x'') = ''y''}}.

	The category '''Rel''' can be obtained from the category '''Set''' as the [[Kleisli category]] for the [[monad (category theory)\|monad]] whose [[functor]] corresponds to [[power set]], interpreted as a covariant functor.

	~~Perhaps a bit surprising at first sight is the fact that [[Product (category theory)\|product]] in '''Rel'''~~ is ~~given by the [[disjoint union]] (rather than the [[cartesian product]] as it is in '''Set'''), and so is the [[coproduct]]~~.

	~~'''Rel'''~~ is ~~[[Closed monoidal category\|monoidal closed]], with both~~ the ~~monoidal product <math>A \otimes B</math> and the [[internal hom]] <math>A \Rightarrow B</math> given by [[cartesian product]] of sets~~.

	~~The [[Involution (mathematics)\|involutary]] operation of taking the [[Inverse relation\|inverse]] (or converse) of a relation, where {{nowrap\| (''b'', ''a'') ∈ ''R''~~<~~sup~~>−1<~~/sup~~> : ~~''B'' → ''A''}} if and only if {{nowrap\|~~ (~~''a'', ''b'') ∈ ''R''~~ : ~~''A'' → ''B''}}, induces a contravariant functor {{nowrap\| '''Rel'''<sup>op<~~/~~sup> → '''Rel'''}} that leaves the objects invariant but reverses the arrows and composition~~. ~~This makes '''Rel''' into a [[dagger category]]~~. ~~In fact, '''Rel''' is a [[dagger compact category]~~].

	~~==See also==~~
	*[[Allegory (category theory)~~]]. The category of relations is the paradigmatic example of an allegory.~~

	~~[[Category:Monoidal categories\|Relations]]~~

en>Addbot: Bot: Migrating 3 interwiki links, now provided by Wikidata on d:q2083446

2013-03-02T15:45:39Z

Bot: Migrating 3 interwiki links, now provided by Wikidata on d:q2083446

← Older revision		Revision as of 17:45, 2 March 2013
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	~~Hi there~~. ~~Let me start by introducing~~ the ~~author~~, ~~her name~~ is ~~Myrtle Cleary~~. ~~In her expert lifestyle she is~~ a ~~payroll clerk but she~~'~~s always wanted her own business~~. ~~North Dakota~~ is ~~where me~~ and ~~my spouse reside~~. ~~To collect coins~~ is 1 of the ~~things I adore most.~~<br><br>~~Also visit my webpage std testing at home; [http~~:/~~/drupal~~.~~12thirty4~~.~~com/gmaps/node/4991 take~~ a ~~look at~~ the ~~site here~~],		In [[mathematics]], the [[category (mathematics)\|category]] '''Rel''' has the class of [[Set (mathematics)\|sets]] as [[object (category theory)\|objects]] and [[binary relation]]s as [[morphism (category theory)\|morphisms]].

			A morphism (or arrow) ''R'' : ''A'' → ''B'' in this category is a relation between the sets ''A'' and ''B'', so {{nowrap\| ''R'' ⊆ ''A'' × ''B''}}.

			The [[Composition of relations\|composition]] of two relations ''R'': ''A'' → ''B'' and ''S'': ''B'' → ''C'' is given by:
			:(''a'', ''c'') ∈ ''S'' <small>o</small> ''R'' if (and only if) for some ''b'' ∈ ''B'', (''a'', ''b'') ∈ ''R'' and (''b'', ''c'') ∈ ''S''.

			==Properties==
			The category '''Rel''' has the [[category of sets]] '''Set''' as a (wide) [[subcategory]], where the arrow (function) {{nowrap\| ''f'' : ''X'' → ''Y''}} in '''Set''' corresponds to the functional relation {{nowrap\| ''F'' ⊆ ''X'' × ''Y''}} defined by: {{nowrap\|1= (''x'', ''y'') ∈ ''F'' ⇔ ''f''(''x'') = ''y''}}.

			The category '''Rel''' can be obtained from the category '''Set''' as the [[Kleisli category]] for the [[monad (category theory)\|monad]] whose [[functor]] corresponds to [[power set]], interpreted as a covariant functor.

			Perhaps a bit surprising at first sight is the fact that [[Product (category theory)\|product]] in '''Rel''' is given by the [[disjoint union]] (rather than the [[cartesian product]] as it is in '''Set'''), and so is the [[coproduct]].

			'''Rel''' is [[Closed monoidal category\|monoidal closed]], with both the monoidal product <math>A \otimes B</math> and the [[internal hom]] <math>A \Rightarrow B</math> given by [[cartesian product]] of sets.

			The [[Involution (mathematics)\|involutary]] operation of taking the [[Inverse relation\|inverse]] (or converse) of a relation, where {{nowrap\| (''b'', ''a'') ∈ ''R''<sup>−1</sup> : ''B'' → ''A''}} if and only if {{nowrap\| (''a'', ''b'') ∈ ''R'' : ''A'' → ''B''}}, induces a contravariant functor {{nowrap\| '''Rel'''<sup>op</sup> → '''Rel'''}} that leaves the objects invariant but reverses the arrows and composition. This makes '''Rel''' into a [[dagger category]]. In fact, '''Rel''' is a [[dagger compact category]].

			==See also==
			*[[Allegory (category theory)]]. The category of relations is the paradigmatic example of an allegory.

			[[Category:Monoidal categories\|Relations]]

en>Giftlite: /* See also */ −wlink already included in the main article

2012-08-10T17:09:53Z

See also: −wlink already included in the main article

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Hi there. Let me start by introducing the author, her name is Myrtle Cleary. In her expert lifestyle she is a payroll clerk but she's always wanted her own business. North Dakota is where me and my spouse reside. To collect coins is 1 of the things I adore most.<br><br>Also visit my webpage std testing at home; [http://drupal.12thirty4.com/gmaps/node/4991 take a look at the site here],