# 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}.[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

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.[1] This criterion can be convenient as it avoids the need to construct the inverse functor *G*. (We use "bijection" informally here because, if a category is not concrete, we don't have such a notion.)

## 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

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). See also Representation theory of finite groups#Representations, modules and the convolution algebra.

- 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*C*^{1}, 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

## References

- Mac Lane, Saunders (1998).
*Categories for the Working Mathematician*. Graduate Texts in Mathematics.**5**(2nd ed.). Springer-Verlag. p. 14. ISBN 0-387-98403-8. MR 1712872.