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to each other. Two isomorphic categories share all properties that are defined solely in terms of category theory; for all practical purposes, they are identical and differ only in the notation of their objects and morphisms.
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750:, and the meet operation as multiplication. Again, both of these assignments can be extended to morphisms to yield functors, and these functors are inverse to each other.
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Isomorphism of categories is a very strong condition and rarely satisfied in practice. Much more important is the notion of
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Representation theory of finite groups ยง Representations, modules and the convolution algebra
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modules, and they are inverse to each other, both on objects and on morphisms). See also
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411:. The isomorphism can be described as follows: given a group representation ฯ :
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557:{\displaystyle \left(\sum _{g\in G}a_{g}g\right)v=\sum _{g\in G}a_{g}\rho (g)(v)}
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127:; roughly speaking, for an equivalence of categories we don't require that
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is the category with one object and only its identity morphism (in fact,
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672:: the category of Boolean algebras is isomorphic to the category of
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is isomorphic to the category of left modules over the ring.
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Another isomorphism of categories arises in the theory of
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yields an isomorphism of categories if and only if it is
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as multiplication. Conversely, given a
Boolean ring
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684:into a Boolean ring by using the
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676:. Given a Boolean algebra
663:category of abelian groups
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952:Equivalence (mathematics)
880:Equivalence of categories
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125:equivalence of categories
117:one-to-one correspondence
85:(the identity functor on
817:, with objects functors
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701:{\displaystyle \land }
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18:Isomorphic categories
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143:{\displaystyle FG}
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941:Categories
886:References
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564:for every
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337:A functor
268:Properties
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723:∨
696:∧
610:yields a
534:ρ
516:∈
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476:∈
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351:bijective
105:morphisms
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874:See also
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395:-linear
384:and the
365:Examples
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103:and the
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927:1712872
809:), and
805:is the
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618:(since
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630:โ GL(
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921:.
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857:โ
853::
845:โ
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825:โ
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