This is a cross-post of my ~2 weeks (canonically) unanswered question on Math.SE: https://math.stackexchange.com/questions/1830287/corollaries-of-the-yoneda-lemma-in-analysis.
I am looking for some simple examples of how the Yoneda Lemma can be applied in analysis and probability theory and related fields.
One candidate example is the Fourier transform. See this question on Math.SE: https://math.stackexchange.com/questions/1667473/is-the-fourier-transform-a-special-case-of-this-version-of-the-yoneda-lemma.
Another candidate example would be the Riesz Representation Theorem (see this unanswered question on Math Overflow Can the similarity between the Riesz representation theorem and the Yoneda embedding lemma be given a formal undergirding? or one of the responses to my original question on Math.SE).
There are also many coordinate-free representations of objects in analysis, and all seem to be characterized by the interactions of the objects with all possible coordinate systems. (Although this seems to be the principle of equivalence rather than the Yoneda Lemma per se, see the comment by David Roberts below.)
Motivation/Context: For example, in Qiaochu Yuan's answer to a similar question, the Yoneda Lemma shows us that every element in a poset can be determined by either the set of all elements it is greater than or equal to, or the set of all elements which is less than or equal to, essentially each element is equivalent to a Dedekind cut. If we consider the partial order of equivalence classes of Cauchy convergent sequences of rational numbers, then this reasoning seems to justify the construction of the real numbers from Dedekind cuts (although I am not entirely sure).
A simple candidate example that I can think of and somewhat understand is any linear transformation on a finite-dimensional vector space. These can be represented by a matrix, which determines the action of the transformation on a certain basis of the vector space. But seemingly we can characterize the "inherent/intrinsic" linear transformation in a coordinate-free way by specifying its action on every possible set of bases. (This is basically the idea that tensors are actually coordinate-free objects.) Why do I think that the Yoneda Lemma might be related to coordinate-free representations of vectors? Because the automorphism group on a vector space (which consists of changes of basis and which is a monoidal category with one object), can be lifted to the category whose objects are the various representations of a single vector in different coordinate systems, using the group action. Whether or not this lifting constitutes a functor, I don't know. In any case, this lifting from a category with one object to the category of all possible coordinate representations of a vector suggests intuitively that we can consider all of those coordinate representations to be a single object. My question is whether or not the Yoneda lemma formalizes that intuition.
In another answer on MathOverflow, which was mentioned towards the end of Tom LaGatta's talk about category theory at the NYC Lisp meetup, an analogy to particle physics was made. Basically the intuition behind Yoneda's Lemma is supposed to be that one can characterize an object (up to equivalence I guess) by probing it via its interactions (i.e. morphisms) with all other objects. In the above example, we would be smashing a vector against all possible changes of basis in order to understand it completely.
Despite being an analyst/probabilist, Tom LaGatta did not go further into examples besides this particle physics metaphor. (He did say something along the lines of "You will find it meaningful in your own context, I guarantee it" around 1:34:00. However, I am curious, because this analogy suggests to me coordinate-free representations of objects.
EDIT: Related post on MO Is there an introduction to probability theory from a structuralist/categorical perspective?
Best Answer
Yoneda's lemma can indeed be used to construct the real numbers. Starting with the rational numbers Q, considered as a posetal category, we can write Yoneda embedding as $Q \to 2^{Q^{op}}$, where $2$ is the category with two objects $0$ and $1$ and one arrow from $0$ to $1$. Then the posetal category of real numbers, thought of as Dedekind cuts, is precisely the full subcategory $R$ of non-constant cocontinuous functors in $2^{Q^{op}}$.
The free cocompletion property of $2^{Q^{op}}$ can be used for example to defined addition of real numbers, provided such operation is defined for rational numbers. One proceeds in two steps; first note that summing a fixed rational number $r$ provides an endofunctor in $Q$ which lifts to a unique cocontinuous endofunctor in $2^{Q^{op}}$. The restriction $S_r$ to $R$ of this latter has in turn image in $R$ and has the effect of summing $r$. To define the endofunctor $T_a$ that sums an arbitrary real number $a$ one first restricts to $Q$ using the previous definition, which provides a functor $S_a: Q \to R$ and lifts this to a cocontinuous functor $2^{Q^{op}} \to R$. The restriction of this latter to $R$ is the desired $T_a$.
All this is of course rather trivial, but it gets more interesting if one is working within the axiomatization of the category of categories as a foundation, as worked out, for example, by McLarty ("Axiomatizing a category of categories", Journal of Symbolic Logic 56, 1991, no. 4, 1243–1260). Then working entirely in that axiomatization one can construct the posetal category of real numbers as explained above and prove that it is a complete ordered field. The completeness of the real numbers, for example, is expressed by the fact that $R$ is a cocomplete category, which follows in turn from the cocompleteness of $2$ and the fact that a colimit of cocontinuous functors is cocontinuous.
Another interesting viewpoint is that one can do the above construction inside a topos, and then one would get a real number object which might be different from the Dedekind reals or the Cauchy reals.