I just recently learned about the yoneda lemma and I have this question: If we look at a arbitrary functor F in [C,Set] and we see that F a (where a is an object in C) is not the empty set, does that imply that there is a natural transformation from the homefunctor C(a,_) to F? And does that mean that the functor F is representable?
When dealing with real numbers (and I believe complex numbers too), the exponential function can be defined as being the function which satisfies e0 = 1, ea+b = ea eb , and ( et )’| (t=0) =1.
Now if I’ve got my terminology right (which I might not), I think (but am not sure) that this can be described as an endomorphism which maps the one group structure of a set to another where those structures have a linear relationship, along with some other conditions I’m not sure how to generalise.
My question is whether a similar type of definition exists for matrices (or more generally linear operators)? I know the power series definition, but compared to the neat definition of the reals I described above, it’s deeply unsatisfying and non-insightful. I can appreciate aspects of it, especially for computational purposes, but I still want a deeper definition.
This might sound silly but I don’t know why stochastic analysis starts with Ito calculus/SDEs invariably. There’s so much to the field other than that.
Are there introductions to stochastic analysis other than SDEs/Ito calculus?