nLab isomorphism



Category theory

Equality and Equivalence



The concept of isomorphism generalizes the concept of bijection from the category Set of sets to general categories.

An isomorphism is an invertible morphism, hence a morphism with an inverse morphism.

Two objects of a category are said to be isomorphic if there exists an isomorphism between them. This means that they “are the same for all practical purposes” as long as one does not violate the principle of equivalence.

But beware that two objects may be isomorphic by more than one isomorphism. In particular a single object may be isomorphic to itself by nontrivial isomorphisms other than the identity morphism. Frequently the particular choice of isomorphism matters.

Every isomorphism is in particular an epimorphism and a monomorphism, but the converse need not hold.

Common jargon includes “is a mono” or “is monic” for “is a monomorphism”, and “is an epi” or “is epic” for “is an epimorphism”, and “is an iso” for “is an isomorphism”.


An isomorphism, or iso for short, is an invertible morphism, i.e. a morphism with a 2-sided inverse.

A morphism could be called isic (following the more common ‘monic’ and ‘epic’) if it is an isomorphism, but it's more common to simply call it invertible. Two objects xx and yy are isomorphic if there exists an isomorphism from xx to yy (or equivalently, from yy to xx). An automorphism is an isomorphism from one object to itself.


It is immediate that isomorphisms satisfy the two-out-of-three property. But they also satisfy two-out-of-six property satisfied by the weak equivalences in any homotopical category.

Note that the inverse morphism of an isomorphism is an isomorphism, as is any identity morphism or composite of isomorphisms. Thus, being isomorphic is an equivalence relation on objects. The equivalence classes form the fundamental 0-groupoid of the category in question.

Every isomorphism is both a split monomorphism (and thus about any other kind of monomorphism) and a split epimorphism (and thus about any other kind of epimorphism). In a balanced category, every morphism that is both a monomorphism and an epimorphism is invertible, but this does not hold in general. However, any monic regular epimorphism (or dually, any epic regular monomorphism) must be an isomorphism.

A groupoid is precisely a category in which every morphism is an isomorphism. More generally, the core of any category CC is the subcategory consisting of all objects but only the isomorphisms; it is a kind of underlying groupoid of CC. In a similar way, the automorphisms of any given object xx form a group, the automorphism group of xx.

In higher categories, isomorphisms generalise to equivalences, which we expect to have only weak inverses.

In the context of homotopy type theory, for every morphism f:hom A(a,b)f : hom_A(a,b) the type “f is an isomorphism” is a proposition. Therefore, for any a,b:Aa,b:A the type aba \cong b is a set.


Suppose given g:hom A(b,a)g:hom_A(b,a) and η:1 a=gf\eta:1_a = g \circ f and ϵ:fg=1 b\epsilon : f \circ g = 1_b, and similarly g,ηg',\eta', and ϵ\epsilon'. We must show (g,η,ϵ)=(g,η,ϵ)(g,\eta,\epsilon)=(g',\eta', \epsilon'). But since all hom-sets are sets, their identity types are mere propositions, so it suffices to show g=gg=g'. For this we have g=1 ag=(gf)g=g(fg)=g1 b=g g' = 1_{a} \circ g'= (g \circ f) \circ g' = g \circ (f \circ g') = g \circ 1_{b}= g


Last revised on June 7, 2022 at 16:01:41. See the history of this page for a list of all contributions to it.