# 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.

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.)

There are also many other coordinate-free representations of objects in analysis, and all seem to be characterized by the interactions of the objects with all possible coordinate systems.

Is this related to the Yoneda Lemma at all? It sounds very similar to the particle physics/probing analogy given frequently. If it is, I think this improves my understanding of such objects tremendously. But I might be leading myself along a false path.

Motivation/Context: For example, in Qiachu 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).

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.

• @JuanFran: You underestiate the power of inertia. – user14972 Jun 21 '16 at 19:04
• @Hurkyl You too hehe – Juan Fran Jun 29 '16 at 12:33

The underlying idea is extremely important in analysis; a simple example is that you can embed any inner product space into the space of linear functionals on its conjugate by the linear transformation $v \mapsto \langle -, v\rangle$.
• @William: That is one example, yes. Anywhere double duals appear is another type of example, giving $V \subseteq V^{**}$ by a similar trasnformation. – user14972 Jun 22 '16 at 18:13