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 = 1D (the identity functor on D) and GF = 1C. 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.
- 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.
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
for every v in V and every element Σ ag 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 as multiplication. Conversely, given a Boolean ring R, we define the join operation by ab = 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.
Finally, the category of pointed sets (whose objects are pairs (S,s) with S a set and s a distinguished element of S, and whose morphisms are functions which map distinguished elements to distinguished elements) is isomorphic to the category of all sets, with partial functions as morphisms (a partial function is a function defined on some subset). The isomorphism takes a pointed set and throws away the distinguished element; its action on morphisms is interesting: the "new map" is defined only on those elements which the old map did not send to the distinguished element. On those elements, the action of the new map is identical to the action of the old map.