Editing Isomorphism (section)

==Category theoretic view==
In [[category theory]], given a [[category (mathematics)|category]] ''C'', an isomorphism is a morphism <math>f : a \to b</math> that has an inverse morphism <math>g : b \to a,</math> that is, <math>f g = 1_b</math> and <math>g f = 1_a.</math> <!-- This is discussed below. Consider the [[equivalence relation]] that regards two objects as related if there is an isomorphism between them. The [[equivalence class]]es of this equivalence relation are called isomorphism classes. -->

Two categories {{mvar|C}} and {{mvar|D}} are [[Isomorphism of categories|isomorphic]] if there exist [[functor]]s <math>F : C \to D</math> and <math>G : D \to C</math> which are mutually inverse to each other, that is, <math>FG = 1_D</math> (the identity functor on {{mvar|D}}) and <math>GF = 1_C</math> (the identity functor on {{mvar|C}}).

===Isomorphism vs. bijective morphism===
In a [[concrete category]] (roughly, a category whose objects are sets (perhaps with extra structure) and whose morphisms are structure-preserving functions), such as the [[category of topological spaces]] or categories of algebraic objects (like the [[category of groups]], the [[category of rings]], and the [[category of modules]]), an isomorphism must be bijective on the [[underlying set]]s. In algebraic categories (specifically, categories of [[variety (universal algebra)|varieties in the sense of universal algebra]]), an isomorphism is the same as a homomorphism which is bijective on underlying sets. However, there are concrete categories in which bijective morphisms are not necessarily isomorphisms (such as the category of topological spaces).