Balance spring - Revision history

14.169.55.173 at 08:04, 21 November 2014

2014-11-21T08:04:39Z

← Older revision		Revision as of 10:04, 21 November 2014
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	~~In [[category theory]], a '''2-category''' is a [[category (mathematics)\|category]] with "morphisms between morphisms"; that is, where each [[hom-set]] itself carries the structure of a category~~. ~~It can be formally defined as a category [[enriched category\|enriched]] over '''Cat''' (~~the ~~[[Category of small categories\|category of categories and functors]]~~, ~~with the [[monoidal category\|monoidal]] structure given by [[product category\|product of categories]])~~.		Hi there. Let me begin by introducing the writer, her title is Sophia. My spouse and I live in Kentucky. Distributing manufacturing is exactly where her primary earnings arrives from. One of the issues she enjoys most is canoeing and she's been doing it for quite a whilst.<br><br>Visit my homepage :: [http://chungmuroresidence.com/xe/reservation_branch2/152663 email psychic readings]

	~~== Definition ==~~
	~~A 2-category '''C''' consists of:~~
	* A class of ''0-cells'' (or ''objects'') {{mvar\|A}}, {{mvar\|B}}, ....

	* For all objects {{mvar\|A}} and {{mvar\|B}}, a category <math>\mathbf{C}(A,B)</math>. The objects <math>f,g:A\to B</math> of this category are called ''1-cells'' and ~~its morphisms <math>\alpha:f\Rightarrow g</math> are called ''2-cells''; the composition~~ in ~~this category is usually written <math>\circ</math> or <math>\circ_1</math> and called ''vertical composition'' or ''composition along a 1-cell''~~.

	* For any object {{mvar\|A}} there is ~~a [[functor]]~~ from the [[terminal object\|terminal]] [[category (mathematics)\|category]] (with one object and one arrow) to <math>\mathbf{C}(A,A)</math>, that picks out the [[Identity morphism\|identity]] 1-cell {{math\|id<sub>''A''</sub>}} on {{mvar\|A}} and its identity 2-cell {{math\|id<sub>id<sub>''A''</sub></sub>}}. ~~In practice these two are often denoted simply by {{mvar\|A}}.~~

	* For all objects {{mvar\|A}}, {{mvar\|B}} and {{mvar\|C}}, there is a functor <math>\circ_0 : \mathbf{C}(B,C)\times\mathbf{C}(A,B)\to\mathbf{C}(A,C)</math>, called ''horizontal composition'' or ''composition along a 0-cell'', which is associative and admits the ~~identity 1 and 2-cells of {{math\|id<sub>''A''</sub>}} as identities. The composition symbol <math>\circ_0</math>~~ is ~~often omitted, the horizontal composite of 2-cells <math>\alpha:f\Rightarrow g:A\to B</math>~~ and ~~<math>\beta:f~~'~~\Rightarrow g':B\to C</math> being written simply as <math>\beta\alpha:f'f\Rightarrow g'g:A\to C</math>.~~

	The notion of 2-category differs from the more general notion of a [[bicategory]] in that composition of 1-cells (horizontal composition) is required to be strictly associative, whereas in a bicategory it ~~need only be associative up to a 2-isomorphism. The axioms of~~ a ~~2-category are consequences of their definition as '''Cat'''-enriched categories:~~

	* Vertical composition is associative and unital, the units being the identity 2-cells <math>id_f</math>.
	* Horizontal composition is also (strictly) associative and unital, the units being the identity 2-cells <~~math~~>~~A=id_{id_A}~~<~~/math~~> ~~on the identity 1-cells <math>id_A</math>.~~
	* The [[~~interchange law]] holds; i.e. it is true that for composable 2-cells <math>\alpha,\beta,\gamma,\delta</math>~~
	:~~<math>(\alpha\circ_0\beta)\circ_1(\gamma\circ_0\delta) = (\alpha\circ_1\gamma)\circ_0(\beta\circ_1\delta)<~~/~~math>~~

	~~The interchange law follows from the fact that <math>\circ_0<~~/~~math> is a functor between hom categories. It can be drawn as a [[pasting diagram]] as follows:~~
	~~{\| cellpadding=2 align=center~~
	~~\|- valign=center~~
	~~\| [[Image:2-category horizontal composition upper.svg\|240px]]~~
	~~\| rowspan=3 style="font-size:large" \| = ~~
	~~\| rowspan=3 \|[[Image:2-category double composition.svg\|240px]]~~
	~~\| rowspan=3 style="font-size:large" \| = ~~
	~~\| rowspan=3 \|[[Image:2-category vertical composition~~.~~svg\|132px]]~~
	~~\| rowspan=3 style="font-size:large" \|∘<sub>0<~~/~~sub>~~
	~~\| rowspan=3 \|[[Image:2-category vertical composition.svg\|132px]]~~
	\|-
	~~\| align=center style="font-size:large" \|∘<sub>1<~~/~~sub>~~
	\|-
	~~\| [[Image:2-category horizontal composition lower.svg\|240px]]~~
	\|}

	Here the left-hand diagram denotes the vertical composition of horizontal composites, the right-hand diagram denotes the horizontal composition of vertical composites, and the diagram in the centre is the customary representation of both.

	~~== Doctrines ==~~
	In mathematics, a '''doctrine''' is simply a 2-category which is heuristically regarded as a system of theories. For example, [[algebraic theory\|algebraic theories]], as invented by [[Lawvere]], is an example of a doctrine, as are [[multi-sorted theory\|multi-sorted theories]], [[operad]]s, [[category (mathematics)\|categories]], and [[topos (mathematics)\|toposes]].

	~~The objects of the 2-category are called ''theories'', the 1-morphisms <math>f\colon A\rightarrow B<~~/~~math> are called ''models'' of the {{mvar\|A}} in {{mvar\|B}}, and the 2-morphisms are called ''morphisms between models.''~~

	The distinction between a 2-category and a doctrine is really only heuristic: one does not typically consider a 2-category to be populated by theories as objects and models as morphisms. It is this vocabulary that makes the theory of doctrines worth while.

	For example, the 2-category '''Cat''' of categories, functors, and natural transformations is a doctrine. One sees immediately that all [[presheaf category\|presheaf categories]] are categories of models.

	As another example, one may take the subcategory of '''Cat''' consisting only of categories with finite products as objects and product-preserving functors as 1-morphisms. This is the doctrine of multi-sorted algebraic theories. If one only wanted 1-sorted algebraic theories, one would restrict the objects to only those categories that are generated under products by a single object.

	~~Doctrines were invented by [[J. M. Beck]].~~

	~~== See also ==~~
	* [[n-category]]

	~~== References ==~~
	* ''Generalised algebraic models'', by Claudia Centazzo.
	* {{nlab\|id=2-category}}

	~~[[Category:Higher category theory]~~]

en>David.Boettcher: Some minor edits of clarification

2014-01-30T10:44:52Z

Some minor edits of clarification

← Older revision		Revision as of 12:44, 30 January 2014
Line 1:		Line 1:
	~~Greetings! I am Marvella~~ and ~~I really feel comfortable when individuals use~~ the ~~complete name~~. ~~Years~~ in the ~~past we moved~~ to ~~North Dakota~~ and ~~I love each working day residing right here~~. ~~To collect cash~~ is a ~~factor~~ that I'~~m completely addicted to~~. ~~Hiring~~ is ~~her day occupation now~~ and ~~she will not alter~~ it ~~anytime soon~~.<br><br>~~Also visit my page~~ [~~http~~://xn--~~bp2bm9e86j~~.~~com/index~~.~~php?document_srl~~=~~2281&mid~~=~~review at home std testing~~]		In [[category theory]], a '''2-category''' is a [[category (mathematics)\|category]] with "morphisms between morphisms"; that is, where each [[hom-set]] itself carries the structure of a category. It can be formally defined as a category [[enriched category\|enriched]] over '''Cat''' (the [[Category of small categories\|category of categories and functors]], with the [[monoidal category\|monoidal]] structure given by [[product category\|product of categories]]).

			== Definition ==
			A 2-category '''C''' consists of:
			* A class of ''0-cells'' (or ''objects'') {{mvar\|A}}, {{mvar\|B}}, ....

			* For all objects {{mvar\|A}} and {{mvar\|B}}, a category <math>\mathbf{C}(A,B)</math>. The objects <math>f,g:A\to B</math> of this category are called ''1-cells'' and its morphisms <math>\alpha:f\Rightarrow g</math> are called ''2-cells''; the composition in this category is usually written <math>\circ</math> or <math>\circ_1</math> and called ''vertical composition'' or ''composition along a 1-cell''.

			* For any object {{mvar\|A}} there is a [[functor]] from the [[terminal object\|terminal]] [[category (mathematics)\|category]] (with one object and one arrow) to <math>\mathbf{C}(A,A)</math>, that picks out the [[Identity morphism\|identity]] 1-cell {{math\|id<sub>''A''</sub>}} on {{mvar\|A}} and its identity 2-cell {{math\|id<sub>id<sub>''A''</sub></sub>}}. In practice these two are often denoted simply by {{mvar\|A}}.

			* For all objects {{mvar\|A}}, {{mvar\|B}} and {{mvar\|C}}, there is a functor <math>\circ_0 : \mathbf{C}(B,C)\times\mathbf{C}(A,B)\to\mathbf{C}(A,C)</math>, called ''horizontal composition'' or ''composition along a 0-cell'', which is associative and admits the identity 1 and 2-cells of {{math\|id<sub>''A''</sub>}} as identities. The composition symbol <math>\circ_0</math> is often omitted, the horizontal composite of 2-cells <math>\alpha:f\Rightarrow g:A\to B</math> and <math>\beta:f'\Rightarrow g':B\to C</math> being written simply as <math>\beta\alpha:f'f\Rightarrow g'g:A\to C</math>.

			The notion of 2-category differs from the more general notion of a [[bicategory]] in that composition of 1-cells (horizontal composition) is required to be strictly associative, whereas in a bicategory it need only be associative up to a 2-isomorphism. The axioms of a 2-category are consequences of their definition as '''Cat'''-enriched categories:

			* Vertical composition is associative and unital, the units being the identity 2-cells <math>id_f</math>.
			* Horizontal composition is also (strictly) associative and unital, the units being the identity 2-cells <math>A=id_{id_A}</math> on the identity 1-cells <math>id_A</math>.
			* The [[interchange law]] holds; i.e. it is true that for composable 2-cells <math>\alpha,\beta,\gamma,\delta</math>
			:<math>(\alpha\circ_0\beta)\circ_1(\gamma\circ_0\delta) = (\alpha\circ_1\gamma)\circ_0(\beta\circ_1\delta)</math>

			The interchange law follows from the fact that <math>\circ_0</math> is a functor between hom categories. It can be drawn as a [[pasting diagram]] as follows:
			{\| cellpadding=2 align=center
			\|- valign=center
			\| [[Image:2-category horizontal composition upper.svg\|240px]]
			\| rowspan=3 style="font-size:large" \| =
			\| rowspan=3 \|[[Image:2-category double composition.svg\|240px]]
			\| rowspan=3 style="font-size:large" \| =
			\| rowspan=3 \|[[Image:2-category vertical composition.svg\|132px]]
			\| rowspan=3 style="font-size:large" \|∘<sub>0</sub>
			\| rowspan=3 \|[[Image:2-category vertical composition.svg\|132px]]
			\|-
			\| align=center style="font-size:large" \|∘<sub>1</sub>
			\|-
			\| [[Image:2-category horizontal composition lower.svg\|240px]]
			\|}

			Here the left-hand diagram denotes the vertical composition of horizontal composites, the right-hand diagram denotes the horizontal composition of vertical composites, and the diagram in the centre is the customary representation of both.

			== Doctrines ==
			In mathematics, a '''doctrine''' is simply a 2-category which is heuristically regarded as a system of theories. For example, [[algebraic theory\|algebraic theories]], as invented by [[Lawvere]], is an example of a doctrine, as are [[multi-sorted theory\|multi-sorted theories]], [[operad]]s, [[category (mathematics)\|categories]], and [[topos (mathematics)\|toposes]].

			The objects of the 2-category are called ''theories'', the 1-morphisms <math>f\colon A\rightarrow B</math> are called ''models'' of the {{mvar\|A}} in {{mvar\|B}}, and the 2-morphisms are called ''morphisms between models.''

			The distinction between a 2-category and a doctrine is really only heuristic: one does not typically consider a 2-category to be populated by theories as objects and models as morphisms. It is this vocabulary that makes the theory of doctrines worth while.

			For example, the 2-category '''Cat''' of categories, functors, and natural transformations is a doctrine. One sees immediately that all [[presheaf category\|presheaf categories]] are categories of models.

			As another example, one may take the subcategory of '''Cat''' consisting only of categories with finite products as objects and product-preserving functors as 1-morphisms. This is the doctrine of multi-sorted algebraic theories. If one only wanted 1-sorted algebraic theories, one would restrict the objects to only those categories that are generated under products by a single object.

			Doctrines were invented by [[J. M. Beck]].

			== See also ==
			* [[n-category]]

			== References ==
			* ''Generalised algebraic models'', by Claudia Centazzo.
			* {{nlab\|id=2-category}}

			[[Category:Higher category theory]]

en>Chetvorno: Copyedited introduction to add more history and notability

2012-08-10T17:17:06Z

Copyedited introduction to add more history and notability

New page

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