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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]]).
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| == Definition ==
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| A 2-category '''C''' consists of:
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| * A class of ''0-cells'' (or ''objects'') {{mvar|A}}, {{mvar|B}}, ....
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| * 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''.
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| * 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}}.
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| * 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>.
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| 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:
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| * Vertical composition is associative and unital, the units being the identity 2-cells <math>id_f</math>.
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| * 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>.
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| * The [[interchange law]] holds; i.e. it is true that for composable 2-cells <math>\alpha,\beta,\gamma,\delta</math>
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| :<math>(\alpha\circ_0\beta)\circ_1(\gamma\circ_0\delta) = (\alpha\circ_1\gamma)\circ_0(\beta\circ_1\delta)</math> | |
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| 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:
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| {| cellpadding=2 align=center
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| |- valign=center
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| | [[Image:2-category horizontal composition upper.svg|240px]]
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| | rowspan=3 style="font-size:large" | =
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| | rowspan=3 |[[Image:2-category double composition.svg|240px]]
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| | rowspan=3 style="font-size:large" | =
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| | rowspan=3 |[[Image:2-category vertical composition.svg|132px]]
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| | rowspan=3 style="font-size:large" |∘<sub>0</sub>
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| | rowspan=3 |[[Image:2-category vertical composition.svg|132px]]
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| |-
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| | align=center style="font-size:large" |∘<sub>1</sub>
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| |-
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| | [[Image:2-category horizontal composition lower.svg|240px]]
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| |}
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| 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.
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| == Doctrines ==
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| 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]].
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| 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.''
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| 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.
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| 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.
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| 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.
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| Doctrines were invented by [[J. M. Beck]].
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| == See also ==
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| * [[n-category]]
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| == References ==
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| * ''Generalised algebraic models'', by Claudia Centazzo.
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| * {{nlab|id=2-category}}
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| [[Category:Higher category theory]]
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