We discuss the notion of representable functors in Category Theory. Then, we present Yoneda's lemma and apply it to the situation of group objects in categories and their relation to functors into the category of groups, resulting in a surprising result that these two concepts are essentially the same. Most proofs are included, as well as lots of commutative diagrams.
Posts for August 2009.
In Linear Algebra, one often has the problem that one wants to talk about complex eigenvalues of objects defined over the reals. If the object is a matrix, it is clear what that means. But what if the object is an endomorphism of a non-canonical real vectorspace? This question is strongly related an important use of tensor products, namely base changes.
We explain the aims of functional calculus and specialize to polynomials evaluated at endomorphisms. We reconsider the Jordan decomposition and prove it with more generality. Then, we discuss Taylor expansion in the nilpotent part for endomorphisms with separable minimal polynomials, and prove Cayley-Hamilton again for arbitrary fields.
In real and complex analysis, the Taylor series expansion is a very important tool. For polynomials over arbitrary unitary rings, it is possible to define a derivative which behaves similar to the usual derivative; unfortunately, the Identity Theorem and Taylor's formula do not transfer to this new situation. Fortunately, there exists a different definition of derivatives for these cases, namely the Hasse derivative. Not only does it gives a Identity Theorem and Taylor's formula back, but also allows to write other identities in a simpler way.