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Concept
Isomorphism of categories
Summary
In , 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 s 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 be equal to , but only naturally isomorphic to , and likewise that be naturally isomorphic to .
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.
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).
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