Characterizing specific "concrete" mathematical objects by abstract general properties - MathOverflow

Characterizing specific "concrete" mathematical objects by abstract general properties

2011-12-18T00:52:32Z

In this note by Tom Leinster the Banach space $\mathrm{L}^1[0,1]$ is recovered by "abstract nonsense" as the initial object of a certain category of (decorated) Banach spaces. So a function space, that would habitually be defined through the machinery of Lebesgue measure and integration, is uniquely described (up to isomorphism) in terms of abstract functional analysis and a bit of category theory.

I would be curious to see more results, ideally in diverse areas of mathematics, in the spirit of the above one, in which a familiar and important "concrete" mathematical object is recovered by a universal property (in the technical categorical sense) or -more generally- by a characterizing property that is abstract and general or doesn't delve into the "concrete" habitual definition of that object.

Community wiki, so put one item per answer please.

Answer by Benjamin Steinberg for Characterizing specific "concrete" mathematical objects by abstract general properties

2011-12-18T01:04:24Z

The Stone-Cech compactification. Neither, Stone nor Cech was thinking about category theory at the time (since it didn't exist), but of course the Stone-Cech compactification is a left adjoint to the forgetful functor from compact Hausdorff spaces to completely regular Hausdorff spaces.

If the general construction is not specific enough, then restrict my answer to $\beta \mathbb N$ which is a key object in Ramsey theory.

Answer by Benjamin Steinberg for Characterizing specific "concrete" mathematical objects by abstract general properties

2011-12-18T01:05:37Z

Free groups. If I am not mistaken, they were first introduced by Dyck via the reduced words description. The modern universal property definition only came about later.

Answer by Yemon Choi for Characterizing specific "concrete" mathematical objects by abstract general properties

2011-12-18T01:17:35Z

The real line as "the' complete Archimedean ordered field, as opposed to a bunch of Dedekind cuts.

Answer by Johannes Ebert for Characterizing specific "concrete" mathematical objects by abstract general properties

2011-12-18T14:49:18Z

My impression is that most, if not all, ''natural objects'' in linear algebra, analysis or differential geometry, ..., are usefully characterized by some \emph{symmetry} property, for eaxmple

''The exterior derivative is, up to a constant multiple, the only linear operator from $k$-forms to $k+1$-forms such that for each open embedding $f:U \to M$ and each form $\omega \in \Omega^k (M)$, the idenity $f^{\ast} d \omega = d (f^{\ast}\omega)$ holds.''

Answer by Yemon Choi for Characterizing specific "concrete" mathematical objects by abstract general properties

2011-12-18T19:51:14Z

Slightly facetious one here: the 3-sphere is, up to diffeomorphism, the unique simply connected, closed, 3-manifold.

Answer by George C. Modoi for Characterizing specific "concrete" mathematical objects by abstract general properties

2011-12-19T11:08:04Z

Probably, Johannes Ebert is right: (almost) all natural mathematical objects may be characterized by a universal property. The question is now what we understand exactly by the the fact that universal property is delving in the concrete habitual definition.

More concrete, let consider the usual definition of a factor structure, let say a factor group (of $G$ modulo a normal subgroup $H$). There is also a universal one: A factor group is (up to an isomorphism) an epimorphism (i.e. a surjective group homomorphism) $G\to G'$. Does the second definition delve the first? I really don't know!

Another example: Having two $R$-modules, $M$ of the right and $N$ of the left, one may define the tensor product as a factor of the free abelian group with the basis the cartezian product $M\times N$ modulo the relations which emphasize the bilinearity. Secondly, we may define the tensor $M\otimes_R-$ as the right adjoint of the functor $Hom_R(M,-)$, definition which may be extended for $M$ in a cocomplete abelian category. This time the possibility to change the settings leading to a more general definition stands as an argument that the universal definition is not delving in the usual one.

Answer by Jan Weidner for Characterizing specific "concrete" mathematical objects by abstract general properties

2011-12-20T09:39:53Z

The natural numbers, maybe the oldest known mathematical obeject, have many universal properties in various categories. They are for example the free monoid on one generator, the initial rig, the free inductive set on one generator,...

Answer by jbc for Characterizing specific "concrete" mathematical objects by abstract general properties

2012-09-04T06:59:54Z

The space of Radon measures on the closed unit interval is the free topological vector space over the interval. It has universal property that evey continuous function on the interval has a unique extension to a continuous linear mapping. This has zillions of generalisations---Radon measures on compacta, bounded Radon measures on a completely regular space, uniform measures on a uniform space and, and ...

Answer by Victor Makarov for Characterizing specific "concrete" mathematical objects by abstract general properties

2012-09-13T11:19:28Z

The integers are the unique commutative ordered ring whose positive elements are well-ordered (thanks to Harry Altman).

Answer by Todd Trimble for Characterizing specific "concrete" mathematical objects by abstract general properties

2012-09-13T13:51:59Z

The category $Set$ of sets is, up to equivalence, the only locally small category $C$ whose Yoneda embedding $y: C \to Set^{C^{op}}$ admits a string of adjoint functors

$$u \dashv v \dashv w \dashv x \dashv y.$$

A precise treatment is given here.

Answer by Qfwfq for Characterizing specific "concrete" mathematical objects by abstract general properties

2012-11-19T14:10:15Z

I happen to have just read Manes' theorem in the n-category café:

Theorem The algebras for the ultrafilter monad are the compact Hausdorff spaces.

The "ultrafilter monad" $X\mapsto \mathrm{U}(X)$ maps a set $X$ to the set of ultrafilters on it. The abstractness of the characterization of compact Hausdorff spaces lies in the fact that $\mathrm{U}$ is defined in purely set-theoretical (or, rather, category-theoretical) terms: it appears to be the "codensity monad" (don't ask me the meaning of this because I don't know!) of the inclusion $\mathrm{FinSet}\to\mathrm{Set}$.