Editing Isomorphism of categories

{{Use American English|date = January 2019}}
{{Short description|Relation of categories in category theory}}
In [[category theory]], two categories ''C'' and ''D'' are '''isomorphic''' if there exist [[functor]]s ''F'' : ''C'' → ''D'' and ''G'' : ''D'' → ''C'' that are mutually inverse to each other, i.e. ''FG'' = 1<sub>''D''</sub> (the identity functor on ''D'') and ''GF'' = 1<sub>''C''</sub>.<ref name="catswork">{{cite book |last=Mac Lane |first=Saunders |title=[[Categories for the Working Mathematician]] |publisher=Springer-Verlag |year=1998 |edition=2nd |series=Graduate Texts in Mathematics | volume=5 |author-link=Saunders Mac Lane |isbn=0-387-98403-8 | mr=1712872 | page=14}}</ref> This means that both the [[object (category theory)|object]]s and the [[morphism]]s of ''C'' and ''D'' stand in a [[one-to-one correspondence]] 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.

Isomorphism of categories is a very strong condition and rarely satisfied in practice. Much more important is the notion of [[equivalence of categories]]; roughly speaking, for an equivalence of categories we don't require that <math>FG</math> be ''equal'' to <math>1_D</math>, but only ''[[natural transformation|naturally isomorphic]]'' to <math>1_D</math>, and likewise that <math>GF</math> be naturally isomorphic to <math>1_C</math>.

==Properties==
As is true for any notion of [[isomorphism]], we have the following general properties formally similar to an [[equivalence relation]]:
* any category ''C'' is isomorphic to itself
* if ''C'' is isomorphic to ''D'', then ''D'' is isomorphic to ''C''
* if ''C'' is isomorphic to ''D'' and ''D'' is isomorphic to ''E'', then ''C'' is isomorphic to ''E''.

A functor ''F'' : ''C'' → ''D'' yields an isomorphism of categories if and only if it is [[bijective]] on objects and on [[Hom set|morphism sets]].<ref name="catswork"/> This criterion can be convenient as it avoids the need to construct the inverse functor ''G''.

==Examples==
* {{anchor|Category of representations}} Consider a finite [[group (mathematics)|group]] ''G'', a [[field (mathematics)|field]] ''k'' and the [[Group ring#Group algebra over a finite group|group algebra]] ''kG''. The category of ''k''-linear [[group representation]]s of ''G'' is isomorphic to the category of [[module (mathematics)|left module]]s over ''kG''. The isomorphism can be described as follows: given a group representation ρ : ''G'' → GL(''V''), where ''V'' is a [[vector space]] over ''k'', GL(''V'') is the group of its ''k''-linear [[automorphism]]s, and ρ is a [[group homomorphism]], we turn ''V'' into a left ''kG'' module by defining <math display="block">\left(\sum_{g\in G} a_g g\right) v = \sum_{g\in G} a_g \rho(g)(v)</math> for every ''v'' in ''V'' and every element Σ ''a<sub>g</sub>'' ''g'' in ''kG''. {{pb}} Conversely, given a left ''kG'' module ''M'', then ''M'' is a ''k'' vector space, and multiplication with an element ''g'' of ''G'' yields a ''k''-linear automorphism of ''M'' (since ''g'' is invertible in ''kG''), which describes a group homomorphism ''G'' → GL(''M''). (There are still several things to check: both these assignments are functors, i.e. they can be applied to maps between group representations resp. ''kG'' modules, and they are inverse to each other, both on objects and on morphisms.) See also {{slink|Representation theory of finite groups#Representations, modules and the convolution algebra}}.
* Every [[ring (mathematics)|ring]] can be viewed as a [[preadditive category]] with a single object. The [[functor category]] of all [[additive functor]]s from this category to the [[category of abelian groups]] is isomorphic to the category of left modules over the ring.
* Another isomorphism of categories arises in the theory of [[Boolean algebra (structure)|Boolean algebra]]s: the category of Boolean algebras is isomorphic to the category of [[Boolean ring]]s. Given a Boolean algebra ''B'', we turn ''B'' into a Boolean ring by using the [[symmetric difference]] as addition and the meet operation <math>\land</math> as multiplication. Conversely, given a Boolean ring ''R'', we define the join operation by ''a''<math>\lor</math>''b'' =  ''a'' + ''b'' + ''ab'', 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.
* If ''C'' is a category with an initial object s, then the [[slice category]] (''s''↓''C'') is isomorphic to ''C''. [[Dual (category theory)|Dually]], if ''t'' is a terminal object in ''C'', the functor category (''C''↓''t'') is isomorphic to ''C''. Similarly, if '''1''' is the category with one object and only its identity morphism (in fact, '''1''' is the [[terminal object|terminal category]]), and ''C'' is any category, then the functor category ''C''<sup>'''1'''</sup>, with objects functors ''c'': '''1''' → ''C'', selecting an object ''c''∈Ob(''C''), and arrows natural transformations ''f'': ''c'' → ''d'' between these functors, selecting a morphism ''f'': ''c'' → ''d'' in ''C'', is again isomorphic to ''C''.

== References ==
{{reflist}}

{{DEFAULTSORT:Isomorphism Of Categories}}
[[Category:Adjoint functors]]
[[Category:Equivalence (mathematics)]]