In category theory, two categories C and D are isomorphic if there existfunctorsF : C → D and G : D → C which are mutually inverse to each other, i.e. FG = 1D and GF = 1C. This means that both the objects and the morphisms of C and Dstand 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 be equal to, but only naturally isomorphic to, and likewise that be naturally isomorphic to.
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 functorF : 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.
for every v in V and every element Σ agg 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, which describes a group homomorphism G → GL.. See also Representation theory of finite groups#Representations, modules and the convolution algebra.
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 algebraB, 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 a'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 is isomorphic to C. Dually, if t is a terminal object in C, the functor category is isomorphic to C. Similarly, if 1 is the category with one object and only its identity morphism, and C is any category, then the functor category C1, with objects functors c: 1' → C, selecting an object c∈Ob, and arrows natural transformationsf: c → d between these functors, selecting a morphismf: c → d in C, is again isomorphic to C''.
