Isomorphism of categories

In category theory, two categories "C" and "D" are isomorphic if there exist functors "F" : "C" → "D" and "G" : "D" → "C" which are mutually inverse to each other, i.e. "FG" = 1_"D" (the identity functor on "D") and "GF" = 1_"C". This means that both the objects and the morphisms 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 "FG"("x") be "equal" to "x", but only "isomorphic" to "x" in the category "D", and likewise that "GF"("y") be isomorphic to "y" in "C".

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 morphism sets. This criterion can be convenient as it avoids the need to construct the inverse functor "G".

Examples

Consider a finite group "G", a field "k" and the group algebra "kG". The category of "k"-linear group representations of "G" is isomorphic to the category of left modules 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 automorphisms, and ρ is a group homomorphism, we turn "V" into a left "kG" module by defining:$(sum\_\{gin\; G\}\; a\_g\; g)\; v\; =\; sum\_\{gin\; G\}\; a\_g\; ho(g)(v)$for every "v" in "V" and every element Σ "a_g" "g" in "kG".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).

Every ring can be viewed as a preadditive category with a single object. The functor category of all additive functors 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 algebras: the category of Boolean algebras is isomorphic to the category of Boolean rings. Given a Boolean algebra "B", we turn "B" into a Boolean ring by using the symmetric difference as addition and the meet operation $land$ as multiplication. Conversely, given a Boolean ring "R", we define the join operation by "a"$lor$"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.