nLab isomorphism

Idea

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”.

Henri Poincaré wrote in Poincaré 1908 about isomorphisms:

“It is scarcely credible, as Mach said, how much a well-chosen word can economize thought. I do not know whether or not I have said somewhere that mathematics is the art of giving the same name to different things. We must so understand it. It is appropriate that things different in substance, but alike in form, should be put into the same mold, so to speak. When our language is well chosen, it is astonishing to see how all the demonstrations made upon some known fact immediately become applicable to many new facts. Nothing has to be changed, not even the words, since the names are the same in the new cases. There is an example, which comes at once to my mind; it is quaternions, upon which, however, I will not dwell.

A word well chosen very often causes the disappearance of exceptions to rules as announced in their former forms; it was for this purpose that the terms ‘negative quantities’, ‘imaginary quantities’, ‘infinite points’, have been invented. And let us not forget that these exceptions are pernicious, for they conceal laws. Very well then, one of those marks by which we recognize the pregnancy of a result is in that it permits a happy innovation in our language. The mere fact is oftentimes without interest; it has been noted many times, but has rendered no service to science; it becomes of value only on that day when some happily advised thinker perceives a relationship, which he indicates and symbolizes by a word.

The physicists also do it just the same way. They invented the term ‘energy’, a word of very great fertility, because through the elimination of exceptions it established a law; because it gave the same name to things different in substance, but alike in form.

Among the words which have had this happy result, I will mention the ‘group’ and the ‘invariant’. They make us perceive the gist of many mathematical demonstrations; they make us realize how often mathematicians of the past must have run across groups without recognizing them and how, believing these groups to be isolated things, they have found them to be in close relationship without knowing why. Today we would say that they were looking right in the face of isomorphic groups. We feel now that in a group the substance interests us but very little; it is the form alone which matters, and so, when we once know well a single group, then we know through it all the isomorphic groups; thanks to the words ‘groups’ and ‘isomorphism’, which sum in a few syllables this subtle law and make it at once familiar to us all, we take our step at once and in so doing economize all effort of thought.“

Definitions

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 $x$ and $y$ are isomorphic if there exists an isomorphism from $x$ to $y$ (or equivalently, from $y$ to $x$). An automorphism is an isomorphism from one object to itself.

Properties

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 $C$ is the subcategory consisting of all objects but only the isomorphisms; it is a kind of underlying groupoid of $C$. In a similar way, the automorphisms of any given object $x$ form a group, the automorphism group of $x$.

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 $f : hom_A(a,b)$ the type “f is an isomorphism” is a proposition. Therefore, for any $a,b:A$ the type $a \cong b$ is a set.

Proof

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

Examples

• A bijection is an isomorphism in Set.

• A homeomorphism is an isomorphism in Top.

• A diffeomorphism is an isomorphism in Diff.

• Every morphism in a groupoid is an isomorphism. By definition of groupoid.

Related concepts

• equality

• isomorphism

  • isomorphism class

  • exceptional isomorphism

• equivalence

• weak equivalence

• homotopy equivalence, weak homotopy equivalence

• equivalence in an (∞,1)-category

• equivalence of (∞,1)-categories

• definitional isomorphism

• EI-category

• gaunt category, where all isomorphisms are identities

• skeletal category, where the only non-trivial isomorphisms are automorphisms

• groupoid, where all morphisms are isomorphisms

 References

• Henri Poincaré, The future of mathematics, Revue generale des Sciences pures et appliquees, Paris, 19th year, No. 23, December, 1908 (pdf)

• Francis Borceux, Section 1.9 in: Handbook of Categorical Algebra Vol. 1: Basic Category Theory, Encyclopedia of Mathematics and its Applications 50, Cambridge University Press (1994) [doi:10.1017/CBO9780511525858]