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Can anyone give an example of an unnatural isomorphism? Or, maybe, somebody can explain why unnatural isomorphisms do not exist.

Consider two functors $F,G: {\mathcal C} \rightarrow {\mathcal D}$. We say that they are unnaturally isomorphic if $F(x)\cong G(x)$ for every object $x$ of ${\mathcal C}$ but there exists no natural isomorphism between $F$ and $G$. Any examples?

Just to clarify the air, $V$ and $V^\ast$ for finite dimensional vector spaces ain't no gud: one functor is covariant, another contravariant, so they are not even functors between the same categories. A functor should mean a covariant functor here.

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A better terminology would be "pointwise isomorphic". –  Martin Brandenburg Aug 14 '13 at 8:57
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maybe this question should be CW since there is no unique correct answer. –  HenrikRüping Aug 14 '13 at 9:03
    
read "Are there better axioms?" in en.wikipedia.org/wiki/Triangulated_category –  Buschi Sergio Aug 14 '13 at 15:10
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10 Answers

up vote 31 down vote accepted

For a simpler, but arguably more artificial, example than Mark's, take $\mathcal{C}$ to be the category with one object and two morphisms. Then the identity functor $\mathcal{C}\to\mathcal{C}$ is "unnaturally isomorphic" to the functor that sends both morphisms to the identity map.

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What you've written doesn't uniquely define a category. There are two such categories: either the non-identity morphism $f$ satisfies $f^2 = \text{id}$ or it satisfies $f^2 = f$. –  Qiaochu Yuan Aug 14 '13 at 8:35
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True. Either will do. –  Jeremy Rickard Aug 14 '13 at 8:38
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That is very cool! –  Bugs Bunny Aug 14 '13 at 10:43
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If $F,G : C \to D$ are functors such that $F(x) \cong G(x)$ for every $x \in C$, I would call $F,G$ "pointwise isomorphic". You ask for examples of non-isomorphic functors which are pointwise isomorphic. There are plenty natural examples.

  1. Consider the interval category $I=\{0 \to 1\}$. The category of functors $I \to C$ is isomorphic to the category of morphisms in $C$. Of course for most $C$ there are non-isomorphic morphisms in $C$ whose domain and codomain are isomorphic or even equal. For example take the identity and a constant map on a nontrivial set or space.

  2. Let $C$ be the category of finite sets with bijections as morphisms. Then we have the functor $\mathrm{Sym} : C \to C$ which maps every set to its set of permutations, and the functor $\mathrm{Ord} : C \to C$ which maps every set to its set of total orderings; the action on morphisms is "conjugation". These functors are pointwise isomorphic, but not isomorphic (in fact between these functors there is no natural transformation at all). Actually this example (when restricted to sets of a given size) can be seen as a special case of the next one.

  3. Let $G$ be a group (or monoid), considered as a category with one object $\star$. Then a functor $G \to \mathsf{Set}$ is the same as a $G$-set. In fact, the category of $G$-sets is isomorphic to the category of functors $G \to \mathsf{Set}$. The value at $\star$ is the underlying set. Of course for $G \neq 1$ there are non-isomorphic $G$-sets whose underlying sets are isomorphic (for example the underlying set of $G$ with the regular action and with the trivial action of $G$).

  4. If $C$ denotes the category of finite abelian groups, then $\mathrm{Tor}_1^{\mathbb{Z}}$ and $\otimes_{\mathbb{Z}} : C \times C \to C$ are pointwise isomorphic (since $\mathrm{Tor}_1(\mathbb{Z}/n,\mathbb{Z}/m) \cong \mathbb{Z}/\mathrm{gcd}(n,m) \cong \mathbb{Z}/n \otimes_{\mathbb{Z}} \mathbb{Z}/m$), but they are not isomorphic (for example since $\mathrm{Tor}_1^{\mathbb{Z}}$ is not right exact in the second or first variable).

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I think that this answer is severely underappreciated. Personally, I consider the first example here to be the most illuminating among all the examples mentioned in the answers to this question. It's a very useful exercise to explicitly write down both functors for some small sets and analyze them. The non-existence of a natural isomorphism between these functors really helps me appreciate what naturality is all about. –  Karol Szumiło Aug 14 '13 at 12:00
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By the way, example 1 is well-known in combinatorics, there people say "the species of permutations and the species of total orders have (1) the same exponential generating functions, but (2) different isomorphism type generating functions". (1) says that $\mathrm{Sym}(X) \cong \mathrm{Ord}(X)$ for all $X$, (2) shows that the functors can't be isomorphic. –  Omar Antolín-Camarena Aug 14 '13 at 15:15
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@OmarAntolín-Camarena: Indeed, if $G$ is a group and $X$ is a principal homogeneous space under $G$, then $G\times X$ is naturally isomorphic to $X\times X$. –  ACL Aug 14 '13 at 17:39
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The first example is perhaps more transparent if you consider it as a special case of the second example, where $G=S_n$ and you let it act on itself either by translation (to get Ord) or conjugation (to get Sym). –  Eric Wofsey Aug 14 '13 at 18:39
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Omar, I don't think you said what you meant to say. If what you said was true, we could compose your natural iso with the first-projection functor $C \times C \to C$ to obtain a natural iso between Ord and Sym - and that's impossible. I guess you actually meant to say that there's a natural iso between the two endofunctors of $C$ defined by $X \mapsto \text{Ord}(X) \times \text{Sym}(X)$ and $X \mapsto \text{Ord}(X) \times \text{Ord}(X)$. –  Tom Leinster Aug 16 '13 at 0:29
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The Universal Coefficient Theorem for, say, singular cohomology should give examples. For any abelian group $G$ and $n> 0$, the functors from spaces to abelian groups given by $$X\mapsto H^n(X;G),\qquad X\mapsto \mathrm{Ext}(H_{n-1}(X),G)\oplus\mathrm{Hom}(H_n(X),G)$$ are isomorphic, but not naturally so. See Hatcher's "Algebraic Topology", Chapter 3.1 (in particular Exercise 11 at the end of that section).

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I "almost knew" this one but I did not know how to show unnaturality. –  Bugs Bunny Aug 14 '13 at 10:45
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Beautiful example. –  Piotr Pstrągowski Aug 14 '13 at 13:04
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Your non-example of vector spaces and their duals can be souped up to a real example.

Let $C$ be the groupoid of finite-dimensional vector spaces and linear isomorphisms. Then there are two obvious functors $C \to C^{op}$: the linear dual, and the natural isomorphism $C \stackrel \sim \to C^{op}$ that one has for any groupoid. These functors are unnaturally isomorphic.

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Specifically, the second functor takes any map to its inverse. –  Eric Wofsey Aug 14 '13 at 18:34
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Take $C = BG$ for some group $G$ and take $D = \text{Set}$. A functor $BG \to \text{Set}$ is a $G$-set. Two $G$-sets are unnaturally isomorphic iff they have the same cardinality, and it's easy to find two $G$-sets of the same cardinality which are not isomorphic as $G$-sets, e.g. find a group with two non-conjugate subgroups of the same index.

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Perhaps a better construction of nonisomorphic G-sets of the same cardinality, that works for any nontrivial group G, is to take G acting on itself by multiplication and G acting on itself trivially. –  James Cranch Aug 14 '13 at 9:21
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The geometric realization of a simplicial set and the geoemetric realization of its barycentric subdivision are always homeomorphic. However there cannot be a natural isomorphism between these two functors. (Look at the diagram of simplicial sets $\Delta^1 \leftrightarrow \Delta^2\rightarrow \Delta^1$. The maps are induces by $1,2,3 \mapsto 1,1,2$ and $1,2,3\mapsto 1,2,2$).

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Here is an example of unnaturally isomorphic functors for which there does not exist any non-trivial natural transformation between them.

Let $\mathcal{C} = \mathbb{N}^{\mathrm{op}}$, $\mathcal{D} = \mathcal{Ab}$, and consider $F, G : \mathcal{C} \rightarrow \mathcal{D}$ defined by $$F(n) = G(n) = \mathbb{Z} \quad\text{for all } n \in \mathbb{N},$$ $$F(m \le n)(x) = 2^{n-m} x,\quad G(m \le n)(x) = 3^{n-m} x \quad\text{for all } x \in \mathbb{Z}.$$

Suppose $\eta = \{ \eta_n : F(n) \rightarrow G(n) \}_{n \in \mathbb{N}}$ is a natural transformation. Then for any $n$ we have that $\eta_0 (2^n x) = 3^n \eta_n (x)$, so $2^n \eta_0 (x) = 3^n \eta_n(x)$. But then $3^n \mid \eta_0 (x)$ for all $n$, which implies that $\eta_0(x) = 0$, and so $\eta = 0$.

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I do not think $3^n \mid \eta_0 (x)$ for all $n$ is clearly absurd: take $\eta_0 = 0$. In fact, there is always the zero natural transformation between two functors into abelian groups. –  Ricardo Andrade Aug 14 '13 at 9:20
    
You're right, I forgot to say non-zero, I'll edit that. –  Luca Bressan Aug 14 '13 at 9:31
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I'm pretty sure one can also categorify the fact that for ordinary complex representations of finite groups, number of irreducible representations = number of conjugacy classes. As in this closely related question, one has a bijection (which categorifies to a pointwise isomorphism) but not a natural one.

(The two functors I'm thinking of here are contravariant functors from the category of finite groups to the category of $k$-linear categories. The first is $F_1(G) = \mathrm{rep}_{\mathbb{C}}(G)$. The second, $F_2$, takes a finite group $G$ to the $k$-linear category freely generated by the conjugacy classes of $G$. I'd have to think about how to define $F_2$ on morphisms, but I don't think there's any choice of definition of $F_2$ on morphisms that will make $F_1$ and $F_2$ naturally isomorphic, despite the fact that they are pointwise isomorphic.)

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There's no ambiguity on how to define $F_2$ for isomorphisms of groups, and already in that case $F_1$ and $F_2$ cannot be naturally isomorphic; see the answers to mathoverflow.net/questions/21606/…. –  Eric Wofsey Aug 14 '13 at 19:29
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I gave a more elaborate example to the Universal Coefficient splitting being non natural in my paper ``Cohomology with chains as coefficients'', Proc. London Math. Soc. (3) 14 (1964), 545-565, available here. It is proved there that for chain complexes $K,L$ which are free and are zero below dimension $0$, there is an isomorphism for any abelian group $G$

$$H^*( K \otimes L, G) \cong H^*(K, H^*(L,G))$$

which can be chosen to be natural with respect to maps of $K$ but not with regard to maps of $L$, nor in Example 3.2 maps of $G$. The naturality with respect to maps of $K$ is useful to recover R. Thom's determination of the weak homotopy type of the function space $K(G,n)^Y$ and further to determine $k^Y$ where $k$ is a cohomology operation (see the paper ``On Kunneth suspensions'', Proc. Camb. Phil. Soc. 60 (1964) 713-720, available here.

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Although there are already too many answers, let me just add the observation that one of the real motivations for The General Theory of Natural equivlances, was to understand the distinction between the fact that a finite dimensional vector space is isomorphic to its dual space, but naturally isomorphic to its second dual.

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Which is undoubtedly the reason why this example is mentioned in the question itself. –  Emil Jeřábek Oct 14 '13 at 19:30
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