Boolean domain - Revision history

en>Octahedron80 at 04:01, 22 October 2014

2014-10-22T04:01:10Z

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	~~In [[category theory]], a '''Kleisli category''' is a [[category (mathematics)\|category]] naturally associated to any [[monad (category theory)\|monad]] ''T''. It~~ is ~~equivalent to~~ the [[Eilenberg–Moore category\|category of free ''T''-algebras]]. The Kleisli category is one of two extremal solutions to the question ''Does every monad arise from an [[Adjunction (category theory)\|adjunction]]?'' The other extremal solution is the [[Eilenberg–Moore category]]. Kleisli categories are named for the mathematician [[Heinrich Kleisli]].		Wilber Berryhill is the title his mothers and fathers gave him and he totally digs that title. To perform lacross is the factor I love most of all. Kentucky is where I've always been residing. For many years she's been working as a journey agent.<br><br>My website :: [http://jplusfn.gaplus.kr/xe/qna/78647 tarot readings]

	~~==Formal definition==~~

	~~Let〈''T'', η, μ〉be a [[monad (category theory)\|monad]] over a category ''C''. The '''Kleisli category''' of ''C'' is the category ''C''<sub>''T''</sub> whose objects~~ and ~~morphisms are given by~~
	~~:<math>\begin{align}\mathrm{Obj}({\mathcal{C}_T}) &= \mathrm{Obj}({\mathcal{C}}), \\~~
	~~\mathrm{Hom}_{\mathcal{C}_T}(X,Y) &= \mathrm{Hom}_{\mathcal{C}}(X,TY).\end{align}</math>~~
	That is, every morphism ''f: X → T Y'' in ''C'' (with codomain ''TY'') can also be regarded as a morphism in ''C''<sub>''T''</sub> (but with codomain ''Y''). Composition of morphisms in ''C''<sub>''T''</sub> is given by
	~~:<math>g\circ_T f = \mu_Z \circ Tg \circ f</math>~~
	~~where ''f: X → T Y''~~ and ~~''g: Y → T Z''~~. ~~The identity morphism~~ is ~~given by~~ the ~~monad unit η:~~
	~~:<math>\mathrm{id}_X = \eta_X</math>~~.

	~~An alternative way of writing this, which clarifies the category in which each object lives,~~ is used by Mac Lane.<ref>Mac Lane(1998) p.147</ref> We use very slightly different notation for this presentation. Given the same monad and category <math>C</math> as above, we associate with each object <math>X</math> in <math>C</math> a new object <math>X_T</math>, and for each morphism <math>f:X\to TY</math> in <math>C</math> a morphism <math>f^*:X_T\to Y_T</math>. Together, these objects and morphisms form our category <math>C_T</math>, where ~~we define~~
	~~:<math>g^\circ_T f^ = (\mu_Z \circ Tg \circ f)^*.</math>~~
	~~Then the identity morphism in <math>C_T</math> is~~
	~~:<math>\mathrm{id}_{X_T} = (\eta_X)^*.</math>~~

	~~==Extension operators and Kleisli triples==~~

	~~Composition of Kleisli arrows can be expressed succinctly by means of the ''extension operator'' (-)* : Hom(''X'', ''TY'') → Hom(''TX'', ''TY'~~'). ~~Given a monad 〈~~'~~'T'', η, μ〉over~~ a ~~category ''C'' and a morphism ''f'' : ''X'' → ''TY'' let~~
	~~:<math>f^* = \mu_Y\circ Tf.</math>~~
	~~Composition in the Kleisli category ''C''<sub>''T''</sub> can then be written~~
	~~:<math>g\circ_T f = g^* \circ f~~.<~~/math~~>
	~~The extension operator satisfies the identities:~~
	:<~~math~~>~~\begin{align}\eta_X^* &= \mathrm{id}_{TX}\\~~
	~~f^*\circ\eta_X &= f\\~~
	~~(g^\circ f)^ &= g^* \circ f^*\end{align}</math>~~
	~~where ''f''~~ : ~~''X'' → ''TY'' and ''g''~~ : ~~''Y'' → ''TZ''. It follows trivially from these properties that Kleisli composition is associative and that η<sub>''X''</sub> is the identity.~~

	~~In fact, to give a monad is to give a ''Kleisli triple'', i.e.~~
	* A function <math>T:\mathrm{ob}(C)\to \mathrm{ob}(C)</math>;
	* For each object <math>A</math> in <math>C</math>, a morphism <math>\eta_A:A\to T(A)</math>;
	* For each morphism <math>f:~~A\to T(B)<~~/~~math> in <math>C</math>, a morphism <math>f^*:T(A)\to T(B)<~~/~~math>~~
	~~such that the above three equations for extension operators are satisfied.~~

	~~==Kleisli adjunction==~~

	~~Kleisli categories were originally defined in order to show that every monad arises from an adjunction~~. ~~That construction is as follows.~~

	~~Let〈''T'', η, μ〉be a monad over a category ''C'' and let ''C''<sub>''T''</sub> be the associated Kleisli category~~. ~~Define a functor ''F'' : ''C'' → ''C''<sub>''T''<~~/~~sub> by~~
	~~:<math>FX = X\;<~~/~~math>~~
	~~:<math>F(f : X \to Y) = \eta_Y \circ f<~~/~~math>~~
	~~and a functor ''G'' : ''C''<sub>''T''</sub> → ''C'' by~~
	~~:<math>GY = TY\;</math>~~
	~~:<math>G(f : X \to TY) = \mu_Y \circ Tf\;</math>~~
	~~One can show that ''F'' and ''G'' are indeed functors and that ''F'' is left adjoint to ''G''. The counit of the adjunction is given by~~
	~~:<math>\varepsilon_Y = \mathrm{id}_{TY}.</math>~~
	~~Finally, one can show that ''T'' = ''GF'' and μ = ''G''ε''F'' so that 〈''T'', η, μ〉is the monad associated to the adjunction 〈''F'', ''G'', η, ε〉.~~

	~~==External links==~~

	* {{nlab\|id=Kleisli+category\|title=Kleisli category}}

	~~==References==~~
	~~{{reflist}}~~
	* {{cite book \| last=Mac Lane \| first=Saunders \| authorlink=Saunders Mac Lane \| title=[[Categories for the Working Mathematician]] \| edition=2nd \| series=[[Graduate Texts in Mathematics]] \| volume=5 \| location=New York, NY \| publisher=[[Springer-Verlag]] \| year=1998 \| isbn=0-387-98403-8 \| zbl=0906.18001 }}
	* {{cite book \| editor1-last=Pedicchio \| editor1-first=Maria Cristina \| editor2-last=Tholen \| editor2-first=Walter \| title=Categorical foundations. Special topics in order, topology, algebra, and sheaf theory \| series=Encyclopedia of Mathematics and Its Applications \| volume=97 \| location=Cambridge \| publisher=[[Cambridge University Press]] \| year=2004 \| isbn=0-521-83414-7 \| zbl=1034.18001 }}

	~~[[Category:Adjoint functors]~~]

en>EmausBot: Bot: Migrating 1 interwiki links, now provided by Wikidata on d:Q3269980

2013-04-17T17:54:18Z

Bot: Migrating 1 interwiki links, now provided by Wikidata on d:Q3269980

← Older revision		Revision as of 19:54, 17 April 2013
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	The ~~writer~~'~~s name~~ is ~~Christy Brookins~~. ~~Mississippi~~ is ~~exactly where his house~~ is~~. Since I was 18 I~~'~~ve been operating~~ as a ~~bookkeeper~~ but ~~quickly my wife~~ and ~~I will start our personal business~~. The ~~preferred hobby~~ for ~~him~~ and ~~his children~~ is ~~to perform lacross~~ and he'll be ~~starting some thing else along with it~~.<br><br>~~Look at my weblog~~; ~~accurate psychic readings~~ (~~[http~~://~~kpupf~~.~~com~~/xe/~~talk~~/~~735373 http~~://~~kpupf.com~~/xe/~~talk~~/~~735373~~])		In [[category theory]], a '''Kleisli category''' is a [[category (mathematics)\|category]] naturally associated to any [[monad (category theory)\|monad]] ''T''. It is equivalent to the [[Eilenberg–Moore category\|category of free ''T''-algebras]]. The Kleisli category is one of two extremal solutions to the question ''Does every monad arise from an [[Adjunction (category theory)\|adjunction]]?'' The other extremal solution is the [[Eilenberg–Moore category]]. Kleisli categories are named for the mathematician [[Heinrich Kleisli]].

			==Formal definition==

			Let〈''T'', η, μ〉be a [[monad (category theory)\|monad]] over a category ''C''. The '''Kleisli category''' of ''C'' is the category ''C''<sub>''T''</sub> whose objects and morphisms are given by
			:<math>\begin{align}\mathrm{Obj}({\mathcal{C}_T}) &= \mathrm{Obj}({\mathcal{C}}), \\
			\mathrm{Hom}_{\mathcal{C}_T}(X,Y) &= \mathrm{Hom}_{\mathcal{C}}(X,TY).\end{align}</math>
			That is, every morphism ''f: X → T Y'' in ''C'' (with codomain ''TY'') can also be regarded as a morphism in ''C''<sub>''T''</sub> (but with codomain ''Y''). Composition of morphisms in ''C''<sub>''T''</sub> is given by
			:<math>g\circ_T f = \mu_Z \circ Tg \circ f</math>
			where ''f: X → T Y'' and ''g: Y → T Z''. The identity morphism is given by the monad unit η:
			:<math>\mathrm{id}_X = \eta_X</math>.

			An alternative way of writing this, which clarifies the category in which each object lives, is used by Mac Lane.<ref>Mac Lane(1998) p.147</ref> We use very slightly different notation for this presentation. Given the same monad and category <math>C</math> as above, we associate with each object <math>X</math> in <math>C</math> a new object <math>X_T</math>, and for each morphism <math>f:X\to TY</math> in <math>C</math> a morphism <math>f^*:X_T\to Y_T</math>. Together, these objects and morphisms form our category <math>C_T</math>, where we define
			:<math>g^\circ_T f^ = (\mu_Z \circ Tg \circ f)^*.</math>
			Then the identity morphism in <math>C_T</math> is
			:<math>\mathrm{id}_{X_T} = (\eta_X)^*.</math>

			==Extension operators and Kleisli triples==

			Composition of Kleisli arrows can be expressed succinctly by means of the ''extension operator'' (-)* : Hom(''X'', ''TY'') → Hom(''TX'', ''TY''). Given a monad 〈''T'', η, μ〉over a category ''C'' and a morphism ''f'' : ''X'' → ''TY'' let
			:<math>f^* = \mu_Y\circ Tf.</math>
			Composition in the Kleisli category ''C''<sub>''T''</sub> can then be written
			:<math>g\circ_T f = g^* \circ f.</math>
			The extension operator satisfies the identities:
			:<math>\begin{align}\eta_X^* &= \mathrm{id}_{TX}\\
			f^*\circ\eta_X &= f\\
			(g^\circ f)^ &= g^* \circ f^*\end{align}</math>
			where ''f'' : ''X'' → ''TY'' and ''g'' : ''Y'' → ''TZ''. It follows trivially from these properties that Kleisli composition is associative and that η<sub>''X''</sub> is the identity.

			In fact, to give a monad is to give a ''Kleisli triple'', i.e.
			* A function <math>T:\mathrm{ob}(C)\to \mathrm{ob}(C)</math>;
			* For each object <math>A</math> in <math>C</math>, a morphism <math>\eta_A:A\to T(A)</math>;
			* For each morphism <math>f:A\to T(B)</math> in <math>C</math>, a morphism <math>f^*:T(A)\to T(B)</math>
			such that the above three equations for extension operators are satisfied.

			==Kleisli adjunction==

			Kleisli categories were originally defined in order to show that every monad arises from an adjunction. That construction is as follows.

			Let〈''T'', η, μ〉be a monad over a category ''C'' and let ''C''<sub>''T''</sub> be the associated Kleisli category. Define a functor ''F'' : ''C'' → ''C''<sub>''T''</sub> by
			:<math>FX = X\;</math>
			:<math>F(f : X \to Y) = \eta_Y \circ f</math>
			and a functor ''G'' : ''C''<sub>''T''</sub> → ''C'' by
			:<math>GY = TY\;</math>
			:<math>G(f : X \to TY) = \mu_Y \circ Tf\;</math>
			One can show that ''F'' and ''G'' are indeed functors and that ''F'' is left adjoint to ''G''. The counit of the adjunction is given by
			:<math>\varepsilon_Y = \mathrm{id}_{TY}.</math>
			Finally, one can show that ''T'' = ''GF'' and μ = ''G''ε''F'' so that 〈''T'', η, μ〉is the monad associated to the adjunction 〈''F'', ''G'', η, ε〉.

			==External links==

			* {{nlab\|id=Kleisli+category\|title=Kleisli category}}

			==References==
			{{reflist}}
			* {{cite book \| last=Mac Lane \| first=Saunders \| authorlink=Saunders Mac Lane \| title=[[Categories for the Working Mathematician]] \| edition=2nd \| series=[[Graduate Texts in Mathematics]] \| volume=5 \| location=New York, NY \| publisher=[[Springer-Verlag]] \| year=1998 \| isbn=0-387-98403-8 \| zbl=0906.18001 }}
			* {{cite book \| editor1-last=Pedicchio \| editor1-first=Maria Cristina \| editor2-last=Tholen \| editor2-first=Walter \| title=Categorical foundations. Special topics in order, topology, algebra, and sheaf theory \| series=Encyclopedia of Mathematics and Its Applications \| volume=97 \| location=Cambridge \| publisher=[[Cambridge University Press]] \| year=2004 \| isbn=0-521-83414-7 \| zbl=1034.18001 }}

			[[Category:Adjoint functors]]

en>Incnis Mrsi: I like Sierpiński space but it is offtopical (Wikipedia talk:WikiProject Logic/Archive 2#False (logic), Boolean domain and Truth value). also,{{merge|Two-element Boolean algebra}}

2012-02-26T09:22:32Z

I like Sierpiński space but it is offtopical (Wikipedia talk:WikiProject Logic/Archive 2#False (logic), Boolean domain and Truth value). also,{{merge|Two-element Boolean algebra}}

New page

The writer's name is Christy Brookins. Mississippi is exactly where his house is. Since I was 18 I've been operating as a bookkeeper but quickly my wife and I will start our personal business. The preferred hobby for him and his children is to perform lacross and he'll be starting some thing else along with it.<br><br>Look at my weblog; accurate psychic readings ([http://kpupf.com/xe/talk/735373 http://kpupf.com/xe/talk/735373])