In category theory, two categories C and D are isomorphic if there exist functors F : C → D and G : D → C that are mutually inverse to each other, i.e. FG = 1D (the identity functor on D) and GF = 1C.[1] 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 be equal to , but only naturally isomorphic to , and likewise that be naturally isomorphic to .
Properties
editAs 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.[1] This criterion can be convenient as it avoids the need to construct the inverse functor G.
Examples
edit- 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
- 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 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 (s↓C) is isomorphic to C. 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 category), and C is any category, then the functor category C1, 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.
See also
editReferences
edit- ^ a b Mac Lane, Saunders (1998). Categories for the Working Mathematician. Graduate Texts in Mathematics. Vol. 5 (2nd ed.). Springer-Verlag. p. 14. ISBN 0-387-98403-8. MR 1712872.