2-category - Revision history

Vipul: /* Raw definition */

2009-12-29T15:22:08Z

Raw definition

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	* '''Identity 2-morphism''': For any two objects <math>A,B \in \operatorname{Ob}\mathcal{C}</math> and any morphism <math>f \in \operatorname{Hom}(A,B)</math>, a 2-morphism <math>\operatorname{id}_f \in \operatorname{Hom}(f,f)</math>.		* '''Identity 2-morphism''': For any two objects <math>A,B \in \operatorname{Ob}\mathcal{C}</math> and any morphism <math>f \in \operatorname{Hom}(A,B)</math>, a 2-morphism <math>\operatorname{id}_f \in \operatorname{Hom}(f,f)</math>.
	* '''Composition of 2-morphisms''': For any two objects <math>A,B</math>, and any three morphisms <math>f,g,h \in \operatorname{Hom}(A,B)</math>, a map <math>\operatorname{Hom}(g,h) \times \operatorname{Hom}(f,g) \to \operatorname{Hom}(f,h)</math>.		* '''Composition of 2-morphisms''': For any two objects <math>A,B</math>, and any three morphisms <math>f,g,h \in \operatorname{Hom}(A,B)</math>, a map <math>\operatorname{Hom}(g,h) \times \operatorname{Hom}(f,g) \to \operatorname{Hom}(f,h)</math>.
	* '''Horizontal composition of 2-morphisms''': For any three objects <math>A,B,C</math>, with morphisms <math>f_1,f_2:A \to B</math> and morphisms <math>g_1,g_2:B \to C</math>, an operator <math>\operatorname{Hom}(f_1,f_2) \times operatorname{Hom}(g_1,g_2) \to \operatorname{Hom}(f_1 \circ g_1, f_2\ circ g_2)</math>.		* '''Horizontal composition of 2-morphisms''': For any three objects <math>A,B,C</math>, with morphisms <math>f_1,f_2:A \to B</math> and morphisms <math>g_1,g_2:B \to C</math>, an operator <math>\operatorname{Hom}(f_1,f_2) \times \operatorname{Hom}(g_1,g_2) \to \operatorname{Hom}(f_1 \circ g_1, f_2\ circ g_2)</math>.

	satisfying the following conditions:		satisfying the following conditions:

Vipul at 15:19, 29 December 2009

2009-12-29T15:19:11Z

← Older revision		Revision as of 15:19, 29 December 2009
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	===Definition as a category enriched over categories===		===Definition as a category enriched over categories===

	A 2-category is an category [[defining ingredient::enriched category\|enriched]] over the [[category of categories]] (~~in particular cases~~, ~~it is usually enough~~ to ~~enrich over~~ the [[category of locally small categories]]~~, which presents fewer foundational hassles~~).		A 2-category is an category [[defining ingredient::enriched category\|enriched]] over the monoidal category whose objects are categories, morphisms are functors, and whose monoidal operation is the product of categories. (To ease foundational issues, we can restrict attention to the monoidal category of [[locally small category\|locally small categories]] or [[small category\|small categories]]).

	===Raw definition===		===Raw definition===

Vipul at 15:00, 29 December 2009

2009-12-29T15:00:59Z

Vipul at 14:52, 29 December 2009

2009-12-29T14:52:16Z

← Older revision		Revision as of 14:52, 29 December 2009
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	==Definition==		==Definition==

			===Definition as a category enriched over categories===

			A 2-category is an category [[defining ingredient::enriched category\|enriched]] over the [[category of categories]] (in particular cases, it is usually enough to enrich over the [[category of locally small categories]], which presents fewer foundational hassles).

			===Raw definition===

	A '''2-category''' <math>\mathcal{C}</math> is the following data:		A '''2-category''' <math>\mathcal{C}</math> is the following data:

Vipul: Created page with '==Definition== A '''2-category''' $\mathcal{C}$ is the following data: * '''Objects''': A defining ingredient::collection $\operatorname{Ob}\mathcal{C}$
2009-12-29T14:50:39Z

Created page with '==Definition== A '''2-category''' <math>\mathcal{C}</math> is the following data: * '''Objects''': A defining ingredient::collection <math>\operatorname{Ob}\mathcal{C}</mat…'

New page
==Definition==

A '''2-category''' <math>\mathcal{C}</math> is the following data:

* '''Objects''': A [[defining ingredient::collection]] <math>\operatorname{Ob}\mathcal{C}</math> of '''objects'''.
* '''Morphisms''': For any objects <math>A,B \in \operatorname{Ob}\mathcal{C}</math>, a collection <math>\mathcal{C}(A,B)</math> of '''morphisms'''. Every element in <math>\mathcal{C}_1(A,B)</math> is termed a ''morphism'' from <math>A</math> (i.e., with source or domain <math>A</math>) to <math>B</math> (i.e., with target or co-domain <math>B</math>). The morphism sets for different pairs of objects are disjoint. Note that <math>f \in \mathcal{C}(A,B)</math> is also written as <math>f:A \to B</math>. The collection <math>\mathcal{C}(A,B)</math> is sometimes also denoted <math>\operatorname{Hom}_{\mathcal{C}}(A,B)</math> or simply <math>\operatorname{Hom}(A,B)</math>.
* '''Identity morphism''': For every object <math>A \in \operatorname{Ob}\mathcal{C}</math>, a distinguished morphism <math>\operatorname{id}_A \in \mathcal{C}(A,A)</math>. This is called the identity morphism of <math>A</math>.
* '''Composition rule''': For <math>A,B,C \in \operatorname{Ob}\mathcal{C}</math>, a map, called ''composition of morphisms'', from <math>\mathcal{C}(B,C) \times \mathcal{C}(A,B)</math> to <math>\mathcal{C}(A,C)</math>. This map is denoted by <math>\circ</math>.
* '''2-morphisms''': For any two objects <math>A,B \in \operatorname{Ob}\mathcal{C}</math> and any two morphisms <math>f,g \in \mathcal{C}_1(A,B)</math>, a collection <math>\mathcal{C}_2(f,g)</math> of '''2-morphisms''' from <math>f</math> to <math>g</math>. If <math>\alpha</math> is such a morphism, we write <math>\alpha:f \implies g</math>. The collection of such 2-morphisms may be denoted <math>\operatorname{Hom}(f,g)</math>.
* '''Identity 2-morphism''': For any two objects <math>A,B \in \operatorname{Ob}\mathcal{C}</math> and any morphism <math>f \in \operatorname{Hom}(A,B)</math>, a 2-morphism <math>\operatorname{id}_f \in \operatorname{Hom}(f,f)</math>.
* '''Composition of 2-morphisms''': For any two objects <math>A,B</math>, and any three morphisms <math>f,g,h \in \operatorname{Hom}(A,B)</math>, a map <math>\operatorname{Hom}(g,h) \times \operatorname{Hom}(f,g) \to \operatorname{Hom}(f,h)</math>.
* '''Horizontal composition of 2-morphisms''': For any three objects <math>A,B,C</math>, with morphisms <math>f_1,f_2:A \to B</math> and morphisms <math>g_1,g_2:B \to C</math>, an operator <math>\operatorname{Hom}(f_1,f_2) \times operatorname{Hom}(g_1,g_2) \to \operatorname{Hom}(f_1 \circ g_1, f_2\ circ g_2)</math>.

satisfying the following conditions:

* '''Associativity of composition of morphisms''': For <math>A,B,C,D \in \operatorname{Ob}\mathcal{C}</math>, with <math>f \in \mathcal{C}(A,B), g \in \mathcal{C}(B,C), h \in \mathcal{C}(C,D)</math>, we have <math>h \circ (g \circ f) = (h \circ g) \circ f</math>.
* '''Identity morphism behaves as an identity''': For <math>A,B \in \operatorname{Ob}\mathcal{C}</math>, with <math>f \in \mathcal{C}(A,B)</math>, we have <math>f \circ \operatorname{id}_A = \operatorname{id}_B \circ f = f</math>.
* '''Associativity of composition of 2-morphisms'''
* '''Identity 2-morphism behaves as an identity'''
* '''Associativity of horizontal composition'''

===Definition in terms of category definition===

A 2-category <math>\mathcal{C}</math> is a [[defining ingredient::category]] (that we'll also call <math>\mathcal{C}</math>), along with, for every <math>A,B \in \mathcal{C}</math>, a category whose collection of objects is the collection of morphisms from <math>A</math> to <math>B</math>, along with a horizontal composition...{{fillin}}

← Older revision		Revision as of 15:00, 29 December 2009
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	==Definition==		==Definition==

			The notion discussed here is that of a '''strict 2-category'''. For the somewhat more general notion that is also loosely referred to by the same name, refer [[bicategory]]. The main difference is the loosening of the requirement for strict composition of the 1-morphisms.
	===Definition as a category enriched over categories===		===Definition as a category enriched over categories===

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	* '''Objects''': A [[defining ingredient::collection]] <math>\operatorname{Ob}\mathcal{C}</math> of '''objects'''.		* '''Objects''': A [[defining ingredient::collection]] <math>\operatorname{Ob}\mathcal{C}</math> of '''objects'''.
	* '''~~Morphisms~~''': For any objects <math>A,B \in \operatorname{Ob}\mathcal{C}</math>, a collection <math>\mathcal{C}(A,B)</math> of '''morphisms'''. Every element in <math>\mathcal{C}_1(A,B)</math> is termed a ''morphism'' from <math>A</math> (i.e., with source or domain <math>A</math>) to <math>B</math> (i.e., with target or co-domain <math>B</math>). The morphism sets for different pairs of objects are disjoint. Note that <math>f \in \mathcal{C}(A,B)</math> is also written as <math>f:A \to B</math>. The collection <math>\mathcal{C}(A,B)</math> is sometimes also denoted <math>\operatorname{Hom}_{\mathcal{C}}(A,B)</math> or simply <math>\operatorname{Hom}(A,B)</math>.		* '''1-morphisms''': For any objects <math>A,B \in \operatorname{Ob}\mathcal{C}</math>, a collection <math>\mathcal{C}(A,B)</math> of '''morphisms'''. Every element in <math>\mathcal{C}_1(A,B)</math> is termed a ''morphism'' from <math>A</math> (i.e., with source or domain <math>A</math>) to <math>B</math> (i.e., with target or co-domain <math>B</math>). The morphism sets for different pairs of objects are disjoint. Note that <math>f \in \mathcal{C}(A,B)</math> is also written as <math>f:A \to B</math>. The collection <math>\mathcal{C}(A,B)</math> is sometimes also denoted <math>\operatorname{Hom}_{\mathcal{C}}(A,B)</math> or simply <math>\operatorname{Hom}(A,B)</math>.
	* '''Identity morphism''': For every object <math>A \in \operatorname{Ob}\mathcal{C}</math>, a distinguished morphism <math>\operatorname{id}_A \in \mathcal{C}(A,A)</math>. This is called the identity morphism of <math>A</math>.		* '''Identity 1-morphism''': For every object <math>A \in \operatorname{Ob}\mathcal{C}</math>, a distinguished morphism <math>\operatorname{id}_A \in \mathcal{C}(A,A)</math>. This is called the identity morphism of <math>A</math>.
	* '''Composition rule''': For <math>A,B,C \in \operatorname{Ob}\mathcal{C}</math>, a map, called ''composition of morphisms'', from <math>\mathcal{C}(B,C) \times \mathcal{C}(A,B)</math> to <math>\mathcal{C}(A,C)</math>. This map is denoted by <math>\circ</math>.		* '''Composition rule''': For <math>A,B,C \in \operatorname{Ob}\mathcal{C}</math>, a map, called ''composition of morphisms'', from <math>\mathcal{C}(B,C) \times \mathcal{C}(A,B)</math> to <math>\mathcal{C}(A,C)</math>. This map is denoted by <math>\circ</math>.
	* '''2-morphisms''': For any two objects <math>A,B \in \operatorname{Ob}\mathcal{C}</math> and any two morphisms <math>f,g \in \mathcal{C}_1(A,B)</math>, a collection <math>\mathcal{C}_2(f,g)</math> of '''2-morphisms''' from <math>f</math> to <math>g</math>. If <math>\alpha</math> is such a morphism, we write <math>\alpha:f \implies g</math>. The collection of such 2-morphisms may be denoted <math>\operatorname{Hom}(f,g)</math>.		* '''2-morphisms''': For any two objects <math>A,B \in \operatorname{Ob}\mathcal{C}</math> and any two morphisms <math>f,g \in \mathcal{C}_1(A,B)</math>, a collection <math>\mathcal{C}_2(f,g)</math> of '''2-morphisms''' from <math>f</math> to <math>g</math>. If <math>\alpha</math> is such a morphism, we write <math>\alpha:f \implies g</math>. The collection of such 2-morphisms may be denoted <math>\operatorname{Hom}(f,g)</math>.
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	satisfying the following conditions:		satisfying the following conditions:

	* '''Associativity of composition of morphisms''': For <math>A,B,C,D \in \operatorname{Ob}\mathcal{C}</math>, with <math>f \in \mathcal{C}(A,B), g \in \mathcal{C}(B,C), h \in \mathcal{C}(C,D)</math>, we have <math>h \circ (g \circ f) = (h \circ g) \circ f</math>.		* '''Associativity of composition of 1-morphisms''': For <math>A,B,C,D \in \operatorname{Ob}\mathcal{C}</math>, with <math>f \in \mathcal{C}(A,B), g \in \mathcal{C}(B,C), h \in \mathcal{C}(C,D)</math>, we have <math>h \circ (g \circ f) = (h \circ g) \circ f</math>.
	* '''Identity morphism behaves as an identity''': For <math>A,B \in \operatorname{Ob}\mathcal{C}</math>, with <math>f \in \mathcal{C}(A,B)</math>, we have <math>f \circ \operatorname{id}_A = \operatorname{id}_B \circ f = f</math>.		* '''Identity 1-morphism behaves as an identity''': For <math>A,B \in \operatorname{Ob}\mathcal{C}</math>, with <math>f \in \mathcal{C}(A,B)</math>, we have <math>f \circ \operatorname{id}_A = \operatorname{id}_B \circ f = f</math>.
	* '''Associativity of composition of 2-morphisms'''		* '''Associativity of composition of 2-morphisms'''
	* '''Identity 2-morphism behaves as an identity'''		* '''Identity 2-morphism behaves as an identity'''