Corollaries of the Yoneda Lemma in Analysis?

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.

Thanks William for reaching out (and thanks David Roberts for the hat tip to my talk).

Let me give an intentionally fuzzy, high-level answer. Generally speaking, the Yoneda Lemma allows you to make an identification between an "object" and the "maps into (or out of) the object". Because of the structural depth of this result, just about every result in analysis that exploits such an identification could be proven using the Yoneda Lemma. This may or may not add value depending on the audience of such a work: analysts are mostly uninterested in categorical formulations of their models (unless the work proves a novel analytical result). I'm sure category theorists would be interested in an analytical survey using CT.

The basic application is of course Cayley's theorem: every group can be admitted as a group action on some space.

For a more modern example, I believe the Kolmogorov-Centsov theorem can be formulated in this fashion: every stochastic process can be characterized using its finite-dimensional distributions (uniqueness), and all consistent families of finite-dimensional distributions generate a stochastic process (existence). The uniqueness result should follow immediately using Yoneda, and existence should follow from some "existence of limits" axiom of the category. I suggest trying to write this up as an exercise.

Moving beyond that, every analytical or probability category should admit some Yoneda consequence, since the result holds for every category. In most categories, this will be uninteresting. In some categories, this will correspond to some interesting classical theorem. Those are the ones to latch onto, because now every other classical CT result might correspond to some interesting theorem. Many will be known (so analysts won't care), but you have a chance of proving new, cutting-edge results too.

My suspicion is that if you're very good at both analysis and CT, then this can lead to a profitable research career:

• Use analytical results (e.g. Central Limit Theorem) to motivate the construction of suitable concrete categories,
• Grab general CT theorems and apply them to those categories,
• Prove new analytical results by unwrapping the abstract formalism into concrete, quantitative statements.

Hope this helps!