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1) Fields have algebraic closures unique up to a non-unique isomorphism.

2) Nice spaces (without base point) have universal covering spaces unique up to a non-unique isomorphism.

3) Modules have injective hulls unique up to a non-unique isomorphism.

Such situations can lead to interesting groups - the absolute Galois group, the fundamental group, and the "Galois" groups of modules introduced by Sylvia Wiegand in Can. J. Math., Vol. XXIV, No. 4, 1972, pp. 573-579.

I'd appreciate any insight into the abstract features of situations which give rise to this type of phenomenon. And I'd appreciate as many examples from as many parts of mathematics as possible.

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It is possible to functorially define the universal covering space of a non-pointed space X: take the diagram $\Pi_1(X) \to Top$ sending a point $x\in X$ to the (pretty well unique) covering space one gets from the pointed space $(X,x)$, and a homotopy class of paths to the map between covering spaces this induces. The colimit of this diagram is the universal covering space of $X$, and this time this is functorial wrt maps $X\to Y$. I learned this from Todd Trimble, I think. – David Roberts Jan 30 '11 at 9:59
@ David: "the colimit of this diagram is the universal covering space of X", are you sure this is correct? What you described is essentially the groupoid of universal covering spaces and covering space maps inside Top and you are taking the colimit inside Top. This is the same as taking a single covering space and taking the quotient by all covering automorphisms, so you just get X back. – Chris Schommer-Pries Jan 30 '11 at 14:46
Doesn't this always happen when the construction has not-trivial automorphisms? – Nick S Jan 30 '11 at 18:36
@ David - this coequalizer still just gives B back. Try an explicit example like B = circle. The point is that there are many paths and so the different universal covers are identified with eachother in more than one way. This forces you to take a quotient of the universal covers which is too small (namely B itself) and no longer a universal cover. You can see that this has to be the case because the paths from b to b (up to homotopy) are just $pi_1$ and so this colimit factors through the quotient by the action of $pi_1$. Agreed? Where in the n-lab is this written? – Chris Schommer-Pries Jan 31 '11 at 18:57
There cannot be any such construction. The group of homeomorphisms from the circle to itself has no compatible action on the (or should I say "a") universal covering space. – Tom Goodwillie May 17 '11 at 4:10

12 Answers 12

The first two examples can be described more or less uniformly. Associated to a field $F$ is the category $C_F$ of algebraic field extensions of $F$ (whose objects are morphisms $F \to E$ and whose morphisms are commutative triangles). This category has a weak terminal object given by any algebraic closure $F \to \bar{F}$. The full subcategory on the algebraic closures is what one might call the absolute Galois groupoid of $F$ (which is a perfectly canonical construction), and choosing an object in this groupoid (which is not) gives the absolute Galois group.

Similarly, associated to a nice space $X$ is the category $C_X$ of connected covers of $X$ (whose objects are covering maps $Y \to X$ and whose morphisms are commutative triangles). This category has a weak initial object given by any universal cover $\bar{X} \to X$. The full subcategory on the universal covers is (equivalent to?) the fundamental groupoid of $X$ (again, a perfectly canonical construction), and choosing an object in this groupoid (which is not) gives the fundamental group.

So you will get this kind of behavior in any situation where you have a weak universal object instead of a universal one. (This partially covers the third example, since injectivity is also a weak universal property.) A general way to engineer a situation similar to the above two might be to look at something like the category of (epi?)morphisms into an object or (mono?)morphisms out of it in your favorite category and see what happens.

In any case, if you are only interested in these constructions because they produce interesting groups, then I think nowadays the modern thing to do is to produce interesting groups using Tannaka-Krein duality.

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Nice observation! In fact, the Galois group of modules alluded to in example 3) is also of this type: it's the automorphism group of the weak initial object given by the injective hull inside the category of embeddings (monomorphisms) $M \to I$ into an injective, so it's analogous to example 1) and dual to 2). – Theo Buehler Jan 30 '11 at 14:18
Not to mention it's also the weak terminal object in the category of essential extensions of M (with embeddings for the maps). – Harry Altman Jan 30 '11 at 14:27
"You will get this kind of behavior in any situation where you have a weak universal object instead of a universal one" --- this is so only when the weak universal objects are all isomorphic, no? – Steven Landsburg Jan 30 '11 at 15:00
@Steven: I guess that would give a groupoid which is not connected, but it's still possible to restrict attention to each of its connected components. – Qiaochu Yuan Jan 30 '11 at 15:17
I don't understand your last comment. It seems to me that this is due to the fact that after all I'm not so sure what the good definition of "weak initial/terminal" object is, anyway. – Theo Buehler Jan 30 '11 at 15:58

Any two injective resolutions (of an object in an abelian category) are homotopy equivalent, but this homotopy equivalence is not unique. This is of course because the lifting property in the definition of "injective" does not require any uniqueness.

The connected sum of oriented manifolds is unique up to homeomorphism, but this homeomorphism is not unique.

A bit silly, but: In a short exact sequence $0 \to A \to B \to C \to 0$ in a semisimple abelian category $B$ is unique up to isomorphism (namely, $B \cong A \oplus C$), but the isomorphism is not unique.

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But the equivalence of injective resolutions is unique in the correct higher sense, i.e. the space of choices is contractible. – Thomas Nikolaus May 17 '11 at 7:31
@Thomas, is that indeed true? I've never thought of it but it looks like an interesting question. What do you mean by 'space of choices' in this case? – Fernando Muro Feb 10 '12 at 12:22

For a field $k$ and a natural number $n$, the vector space of dimension $n$ over $k$ is unique up to a non-unique isomorphism, though this somehow feels "less unique" to me than your other examples. I thought at first that this might be due to its not fitting into the class of examples described by Qiaochu, but I suppose you can force it into that class by considering the category of $n$-dimensional vector spaces over $k$. But that in turn feels considerably more ad hoc (at least to me) than considering the category of algebraic field extensions.

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Except when the field is $\mathbf{F}_2$ and the dimensions is $1$:) – Chandan Singh Dalawat Jan 30 '11 at 15:52
Or any field and the dimension is 0 ;-)) – Johannes Hahn Jan 31 '11 at 0:15
I attended a talk about some problem in the classification of—well, of something to do with finite fields, I don't remember—at which a colleague was moved to ask “What happens here if $\lambda \ne \mu$?” “Oh, yes,” said the speaker, as if discounting a trivial special case, “the results don't work if there are more than 2 non-$0$ scalars in the field.” – L Spice May 17 '11 at 5:03

My favourite: the mapping cone of a morphism in a triangulated category is unique up to non-unique isomorphism. This fact has originated a lot of research in this topic, and it still does.

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In recent work in set theory the concept of "canonical structure" has emerged, in connection with combinatorial work on pcf theory. The idea is that there are many constructions that depend on the axiom of choice but, once realized, are actually independent of the specific choices made. Usually, this involves two steps: You construct an object, which is not quite canonical (say, a collection of subsets of a cardinal $\kappa$), but then you recognize that there is a natural ideal (say, the non-stationary ideal on $\kappa$) and the corresponding equivalence classes are canonical. Of course, by switching to a new model of set theory, the "canonical structure" may change, so sometimes one thinks of it as a sort of invariant of the models.

The first papers that explicitly mentioned the name "canonical structure" are by Cummings, Foreman, and Magidor, "Canonical structures in the universe of set theory", Parts I and II, Annals of Pure and Applied Logic 129 (2004), 211-243, and 142 (2006), 55-75.

The following quote is from the beginning of the introduction to Part I:

It is a distinguishing feature of modern set theory that many of the most interesting questions are not decided by ZFC, the theory in which we profess to work; to put it another way, ZFC admits a large variety of models. A natural response to this is to identify invariants which may take different values in different models, and which codify a large amount of information about a model.

Of particular interest are invariants which are canonical, in the sense that the Axiom of Choice is needed to show that they exist, but once shown to exist they are independent of the choices made. For example the uncountable regular cardinals are canonical in this sense.

Shelah discovered a large class of canonical invariants, the study of which he labeled PCF theory. These invariants include two which are central in this paper; Shelah [24, 26] (under some mild cardinal arithmetic assumptions on the singular cardinal $\mu$) defined two stationary subsets of $\mu^+$, the sets of good and approachable points. The definitions of these sets appear to depend on certain arbitrary choices, but (modulo the club filter) are in fact independent of these choices. Other canonical structures we study in this paper include the stationary sets of tight and internally approachable structures, and the collection of good points on a scale.

The two references cited in the quote are S. Shelah, "On successors of singular cardinals", in M. Boffa, D. van Dalen, and K. McAloon, editors, Logic Colloquium ’78, pages 357–380, Amsterdam, 1979. North-Holland; and S. Shelah, "Cardinal Arithmetic". Oxford University Press, Oxford, 1994.

Besides the ongoing work by Cummings-Foreman-Magidor and Shelah, these ideas have been extended by others; Krueger and Ishiu come to mind.

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Sounds like a case for 'the': – David Roberts Jan 30 '11 at 20:47
Ok, @David, I'm curious. What do you mean? – Andrés E. Caicedo Jan 31 '11 at 0:47
@Andres: What David is referring to is the fact that universal constructions in category theory have this property, and that this is only a new feature of set theory. For instance, suppose we want to make the product with an object $S$ a functor (i.e. the functor $(-)\times S$. However, while the product is unique up to unique isomorphism, we have to choose a representative of each isomorphism class as well as the connecting morphisms between them. This requires choice or global choice, but after we choose a specific representative of the functor usign choice, it is unique up to unique iso. – Harry Gindi Jan 31 '11 at 9:16
So when I said above "the functor $(-)\times S$", I was abusing language, since any given construction of "the" functor is only "a functor $(-)\times S$". This might seem like we're being overly cautious, but when we move from isomorphism of objects to equivalence of objects (in a bicategory), this makes some difference. This can be resolved by using Mac Lane's coherence theorem for bicategories, but, when we move up to tricategories, such a coherence theorem is proven not to exist. – Harry Gindi Jan 31 '11 at 9:21
I see. Thanks, @Harry! – Andrés E. Caicedo Jan 31 '11 at 14:11

The homology of a differential graded algebra has an $A_\infty$-algebra structure which is unique up to non-unique isomorphism.

See Keller's nice expository paper, for instance. In particular, he states this result in Section 3.3 (as a theorem due to Kadeishvili, among others). It is stated there as a result about the homology of an $A_\infty$-algebra, but any differential graded algebra may be viewed as an $A_\infty$-algebra.

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Hi John, Thanks, I wish you would say more people or give a reference at least. – David Feldman Jan 30 '11 at 23:22
@David: I added a reference. – John Palmieri Jan 30 '11 at 23:36

Here are some examples that are less of an algebraic nature (but all seem to be subsumed by Qiaochu's observation in that they are "weakly initial" or "weakly terminal" objects in appropriate categories):

Consider the categories of metric spaces or complete metric spaces and $1$-Lipschitz maps. Isbell has shown that in these categories there are injective hulls, unique up to non-unique isomorphism. A metric space $I$ is injective if for every isometric embedding $A \to B$ and every $1$-Lipschitz map $A \to I$ there exists a $1$-Lipschitz extension $B \to I$. The automorphism groups of the injective hull of a space seems exceedingly hard to determine (even for finite spaces) but there's one case I find interesting. If $M$ happens to be a (real) Banach space and $I(M)$ is its injective hull then $I(M)$ is a Banach space, uniquely determined up to unique linear isometry, and it is of the form $C(K)$ where $K$ is an extremally disconnected Hausdorff space. H. Elton Lacey and co-authors have given a complete (finite!) list of possible injective hulls of separable Banach spaces.

Closely related are projective covers in the category of compact Hausdorff spaces and continuous maps. There, the projectives are precisely the extremally disconnected spaces (Gleason).

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I should add that there is an abstract version of injective hulls/projective covers considered in Adamek-Rosicky, Locally presentable and accessible categories, that seems to subsume all the examples given so far (except maybe the example involving bases). – Theo Buehler Jan 30 '11 at 14:48

A compact connected semisimple Lie Group $G$ has an essentially unique maximal torus $T$, a maximal abelian subgroup of maximum dimension (the rank of $G$ is the dimension of this torus). Although $G$ has lots of such torii (in fact any element of $G$ is contained in at least one), any two are conjugate to one another by some element of $G$.

In a similar vein, one can break a given maximal torus $T$ up into congruent pieces (the images of Weyl chambers under the exponential map applied to the Lie algebra of $T$), any two of which are equivalent to one another by an element of the Weyl Group of $G$. The value of any class function on $G$ is then completely determined on all of $G$ by its values on a single one of these pieces.

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The countable dense linear order is unique up to a non-unique isomorphism. The countable atomless Boolean algebra is unique up to a non-unique isomorphism. The random graph is unique up to a non-unique isomorphism. (Algebraically closed fields of a given characteristic and transcendence degree, and vector spaces of given dimension over a given field, are other examples already mentioned above.)

In general, if $T$ is a $\kappa$-categorical first-order theory (in a countable language), then the model $M$ of $T$ of cardinality $\kappa$ is unique up to a non-unique isomorphism.

Even more generally: if $T$ is any complete theory and $\kappa$ an infinite cardinal, then the saturated model $M$ of $T$ of cardinality $\kappa$—if it exists at all—is unique up to a non-unique isomorphism. ($M$ is unique by a standard back-and-forth argument. Non-uniqueness of the isomorphism amounts to saying that $\operatorname{Aut}(M)$ is nontrivial. By homogeneity, it suffices to exhibit two elements of $M$ with the same type. If there exists a nonprincipal parameter-free $1$-type, we can easily find two elements that realize it. If all $1$-types are principal, there are only finitely many, hence two elements of $M$ have to realize the same type by the pigeonhole principle.)

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Let me mention Sullivan's minimal models.

Every commutative differential graded $\mathbb{Q}$-algebra (cdga) $A^*$ concentrated in non-negative degrees and such that $H^0(A^*)=\mathbb{Q}$ admits a minimal Sullivan model $i:M^*\to A^*$ where $M^*$ is a free commutative graded algebra obtained from $\mathbb{Q}$ by adding generators of non-negative degrees so that the differential of each generator is a $\mathbb{Q}$-linear combination of products of length $\geq 2$ of the previous generators, and $i$ is a map of cgda's that induces a cohomology isomorphism (i.e., a quasi-isomorphism).

The minimal model is unique up to a non-unique isomorphism. More generally, if $f:A^*\to B^*$ is a map of cdga's and $j:N^*\to B^*$ is a minimal model of $B^*$, then there is a cdga map $g:M^*\to N^*$, defined up to cdga homotopy, such that $fi=gj$ up to cdga homotopy; moreover, if $f$ is a quasi-isomorphism, then $g$ is an isomorphism.

This reduces the classification of non-negative cdga's up to quasi-isomorphism (and as a consequence, the classification of simply connected topological spaces up to rational homotopy) to the classification of algebras of a certain kind up to isomorphism.

Of course, this example is similar to some mentioned before (in a sense it is the commutative analog of the answer of John Palmieri).

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Vector spaces have a basis that is unique up to a non-unique isomorphism.
Hilbert spaces have an orthonormal basis that is unique up to a non-unique unitary.
(At least, if you accept Zorn's lemma, i.e. the axiom of choice)

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Wait, how do vector spaces have a unique basis? If $V = \langle x, y\rangle$, then $x + y, y$ is also a basis of $V$, but it is not the same basis. – Simon Rose Jan 30 '11 at 16:04
@Simon: but there is an linear isomorphism that maps the former basis to the latter. So indeed a basis is not unique, but unique up to an isomorphism. Finally, this isomorphism itself is not unique, as it can for example be composed with any isomorphism that leaves the latter basis invariant. – Chris Heunen Jan 30 '11 at 16:14
@Chris: We might be splitting hairs, but I think that this misses the spirit of the question. There is some notion of canonicality which defines an algebraic closure, for example, but no such notion that defines a basis of a vector space. – Simon Rose Jan 30 '11 at 16:23
@Simon: I agree this might only answer the letter of the question and perhaps not its spirit. Neverthelesss, OP asked for as many examples from as many parts of mathematics as possible, and this is one. Moreover, these isomorphisms certainly form interesting groups, namely $SL(n)$ and $U(n)$. Anyway, if one is after insight into the abstract features of such situations, isn't it important to also consider examples that fall outside one's initial intuition? – Chris Heunen Jan 30 '11 at 17:29
I do think an example lurks here, but you haven't nailed it. You don't want a basis, you want only that for which you thought you wanted a basis, namely rigidification. So for one given field and a given dimension fix a model vector space $M$ of that dimension. Define a rigidification of any given vectors space (over the same field, with the same dimension) as in isomorphism $i:V\rightarrow M$. An isomorphism of rigidified vector spaces is a map $j:M\rightarrow M$ that makes the triangle commute. Now you get $GL(n)$ (and other groups if you force more structure on $M$.) – David Feldman Jan 31 '11 at 5:00

One example that springs to mind is when you are secretly working with the objects of a higher category, and so the choice is not unique up to a unique isomorphism, but the choice of isomorphism is also subject to higher coherence data. In a 2-category this would mean the isomorphisms are unique up to a unique invertible 2-arrow and so on. In $\omega$-categories, you may have such coherence all the way to infinity, and so end up with no uniqueness after all. One place where this emerges is when your $\omega$-category has all duals - is then an $\omega$-groupoid, because all the duals make everything weakly invertible!

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