nLab natural isomorphism

Definition

A natural isomorphism $\eta\colon F \Rightarrow G$ between two functors $F$ and $G$

$$\array{ & \nearrow \searrow^{F} \\ C &{}^{\simeq}\Downarrow^\eta& D \\ & \searrow \nearrow_{G} }$$

is equivalently

• a natural transformation with a two-sided inverse;

• a natural transformation each of whose components $\eta_c : F(c) \to G(c)$ for all $c \in Obj(C)$ is an isomorphism in $D$;

• an isomorphism in the functor category $[C,D]$.

In this case, we say that $F$ and $G$ are naturally isomorphic. (Synonym: $F$ and $G$ are isomorphic functors; the naturality is understood when one says that two functors are isomorphic.)

Notably, especially in expositions and lectures, despite the third bullet point above, one does not need to define the concept of a functor category in order to define isomorphic functors. Authors sometimes make cautionary remarks about the category of all functors from $C$ to $D$ (cf. e.g. Auslander (1971, p.9))

If you want to speak of ‘the’ functor satisfying certain conditions, then it should be unique up to unique natural isomorphism.

A natural isomorphism from a functor to itself is also called a natural automorphism.

Some basic uses of isomorphic functors

Defining the concept of equivalence of categories

A fundamental use of the concept of isomorphic functors is the usual definition of equivalent categories which involves functors isomorphic to identity functors.

Re-defining isomorphism of objects in terms of isomorphism of functors

The Yoneda lemma implies that in any category $\mathcal{C}$, and for any objects $O$ and $O'$ of $\mathcal{C}$, the following are equivalent:

• $O$ and $O'$ are isomorphic in $\mathcal{C}$,

• the representable presheaves $\mathcal{C}(-,O)$ and $\mathcal{C}(-,O')$ are isomorphic functors,

• the representable copresheaves $\mathcal{C}(O,-)$ and $\mathcal{C}(O',-)$ are isomorphic functors.

Related entries

• functor category

• limit

References

• M. Auslander?, The representation dimension of artin algebras.

  Queen Mary College Mathematics Notes (1971) Republished in: Selected works of Maurice Auslander. American Mathematical Society (1999)

Related concepts

• natural equivalence