A natural isomorphism is the “correct” notion of isomorphism between functors. Informally, one sometimes speaks of two objects being “naturally isomorphic” if they are the images of two functors that are naturally isomorphic. However, sometimes one encounters functors that are “pointwise isomorphic” but not naturally. There is not much to say mathematically about such “unnatural isomorphisms”, but some discussion of them can help the intuitive understanding of natural isomorphisms and their importance.
On this page we will adhere to the convention that an “unnatural isomorphism” means an isomorphism that is not necessarily natural (but might be), similarly to how a “noncommutative ring” might happen to be commutative. The word “unnatural” is not to be taken perjoratively or as suggesting that unnatural isomorphisms are rare; in cardinality terms there are many more unnatural isomorphisms than natural ones, although in mathematical practice surprisingly many isomorphisms are indeed natural. Other possible terms with the same meaning are “pointwise isomorphic” or “objectwise isomorphic”.
The term “natural” is used for traditional reasons, cf. EilenbergMac Lane45, and is an example of the common phenomenon that concepts branded natural, normal or regular tend to be rather non-generic or non-random. In fact, Eilenberg and MacLane comment (cf. p. 233, first paragraph in loc. cit) on the term, associating it with an intuition of simultaneity, rather than genericity. This intuition is in tune with conceiving of naturality as generalized commutatitivity.
Let $F,G:C\to D$ be functors. An unnatural isomorphism between $F$ and $G$ consists of, for each object $c\in C$, an isomorphism $F(c) \cong G(c)$, satisfying no further conditions.
In other words, it is an unnatural transformation whose components are isomorphisms, or equivalently that is an isomorphism in the category of unnatural transformations.
Note that if each component of a natural transformation is an isomorphism, then it is necessarily a natural isomorphism. (Depending on the definition of “natural isomorphism”, this could be a tautology; but the non-tautologous statement is that if each component of a natural transformation is an isomorphism then it is an isomorphism in the functor category. This statement, though non-tautologous, is easy to prove for 1-categories, but more difficult for higher categories.)
More generally, an unnatural isomorphism (or transformation) might be natural with respect to only some morphisms in the domain, such as for instance the isomorphisms. Some people have proposed calling transformations that are natural only with respect to isomorphisms “canonical”.
If one says that two objects $A$ and $B$ are “unnaturally isomorphic”, it might mean that they are the images of two functors that are unnaturally isomorphic, but it might also mean only that there is no obvious way even to make them the images of two functors. For instance, this seems to be the meaning in Section 1.1 of Strickland.
Somewhat relatedly, the complex numbers $\mathbb{C}$ and the p-adic complex numbers $\mathbf{C}_p$ are “unnaturally isomorphic”. More precisely, there is no way of making constructions such as “algebraic closure of a field” or “injective hull of an $R$-module” functorial, if the desired embeddings into these constructions are to be natural, and this precludes speaking of any two such constructions as “naturally isomorphic”. We touch on this again below, but see here.
Intuitively, there are several reasons that an isomorphism might fail to be natural.
Two functors of opposite variances (one covariant and one contravariant) cannot be naturally isomorphic in the usual sense. However, they can be dinaturally isomorphic, so this is not really a “reason for a failure of naturality”.
In defining an isomorphism, it may be necessary to make certain arbitrary choices. If making these choices differently would yield a different isomorphism, then usually the isomorphism will not be natural.
Unnatural isomorphisms are not very well-behaved mathematically, so if we have an unnatural isomorphism we may want to “make it natural”. There are several ways to do this:
Use a different definition that includes more structure, such as derivators instead of triangulated categories. See for instance this Wikipedia discussion on axioms for triangulated categories.
Relatedly, pass to a higher-categorical context in which a non-commuting square can commute up to a higher transformation; see pseudonatural transformation and lax natural transformation.
Cut down to a smaller class of morphisms in the domain category for which the naturality square does commute.
The use of adjectives like canonical, invariant, functorial, natural is sometimes satirized. For example, as quoted in MO2010, André Weil writes:
I can assure you, at any rate, that (…) my results are invariant, probably canonical, perhaps even functorial. (Oeuvres, vol. 2, page 558)
which may be subtly mocking.
These three terms can be distinguished in the following way:
In particular, the adjectives “functorial” and “natural” in general apply to different classes of things.
See non-canonical isomorphism for examples of natural isomorphisms that are not canonical.
Let $U: Vect\to Aff$ be the functor taking the underlying affine space of a vector space, and $D:Aff\to Vect$ the functor constructing the vector space of displacements of an affine space. Then there is a natural isomorphism $D \circ U \cong Id_{Vect}$, but only an unnatural isomorphism “$U \circ D \cong Id_{Aff}$”. In particular, these functors do not form an equivalence of categories. A similar phenomenon occurs with groups and heaps, and with torsors of groups.
As a particular example of the last phenomenon: there is a bijective correspondence between linear orderings of a set $S$ and permutations of $S$. Indeed, the set $Lin(S)$ is a torsor over $Perm(S)$: there is a group action $\alpha: Perm(S) \times Lin(S) \to Lin(S)$ such that for any choice of linear ordering $L$, the map $\alpha(-, L): Perm(S) \to Lin(S)$ is a bijection. But, there is no canonical choice of ordering $L$.
Species are functors for which unnatural isomorphisms are sometimes discussed. For example, the species of linear orders is unnaturally isomorphic to the species of permutations; for a related example, see the commentary under this MathOverflow answer MO2014. Yorgey2014 explicitly uses the term: his definition 3.3.4 introduces an “equipotence between species (…) as an ”unnatural“ isomorphism between [the two species in question]”.
The term unnatural isomorphism is often used in homological algebra, in particular in discussions of splittings of exact sequences. An example is Hilton-Stammbach, Theorem 4.3, which explicitly uses the term to describe an isomorphism between Hom- and Ext-functors. More generally, unnatural transformations also often occur in homological algebra.
It is often said that while there is an isomorphism between the field of complex numbers $\mathbb{C}$ and the completion $\mathbf{C}_p$ (with respect to the canonical absolute value) of the algebraic closure of the p-adic numbers $\mathbb{Q}_p$, this isomorphism is not natural. This statement, while somewhat loose since there are no functors specified, can be made formally precise along the lines of Adámek, Herrlich, Rosický, and Tholen. The abstract reads: “In a category with injective hulls and a cogenerator, the embeddings into injective hulls can never form a natural transformation, unless all objects are injective. In particular, assigning to a field its algebraic closure, to a poset or Boolean algebra its MacNeille completion, and to an $R$-module its injective envelope is not functorial, if one wants the respective embeddings to form a natural transformation.”
If $C$ and $D$ have only one object (or slightly more generally, one isomorphism class of objects), then any two functors $F,G:C\to D$ are unnaturally isomorphic. For example, for any ring $R$, any two endofunctors of the category of rank-one free $R$-modules are unnaturally isomorphic.
See MathOverflow2013.
Ji?í Adámek, Horst Herrlich, Ji?í Rosický, and Walter Tholen, Injective Hulls are not Natural, algebra universalis
Volume 48, Issue 4 (December 2002), 379–388. (Citeseer link)
S. Eilenberg, S. Mac Lane, General Theory of Natural Equivalences, Transactions of the American Mathematical Society Vol. 58, No. 2 (Sep., 1945), pp. 231-294 (JSTOR)
P. Hilton, U. Stammbach, A course in homological algebra, Springer-Verlag, New York, 1971, Graduate Texts in Mathematics, Vol. 4.
Konrad Waldorf (https://mathoverflow.net/users/3473/konrad-waldorf), What is the definition of “canonical”?, URL (version: 2013-11-13): https://mathoverflow.net/q/19644
Bugs Bunny (https://mathoverflow.net/users/5301/bugs-bunny), Example of an unnatural isomorphism, URL (version: 2013-08-14): https://mathoverflow.net/q/139388
Almeo Maus (https://mathoverflow.net/users/29853/almeo-maus), Unicity of Yoneda isomorphism, URL (version: 2013-08-24): https://mathoverflow.net/q/140303
Todd Trimble (https://mathoverflow.net/users/2926/todd-trimble), What are some examples of interesting uses of the theory of combinatorial species?, URL (version: 2011-03-23): https://mathoverflow.net/q/59259
Charles Rezk, Stuff about quasicategories, Lecture Notes for course at University of Illinois at Urbana-Champaign, 2016, version May 2017,(pdf)
N. P. Strickland, Morava E-Theory, Topology, Vol. 37, No. 5 (1998), 757–779
B. A. Yorgey, Combinatorial Species and Labelled Structures, Dissertation, University of Pennsylvania, 2014