Exercise 7 (Chapter 2, “Categories”): In the category of sets, the two morphisms ( and ) in a direct product are monomorphisms and the two morphisms in a direct sum are epimorphisms. Is this true in every category? There are two mistakes in this wording. The author has exchanged “monomorphisms” and “epimorphisms”. In a direct…
A lot of results concerning the direct product of two topological spaces and are trivial if you consider the usual particular construction of these entities. I find it more satisfying to derive them directly from the universal definitions. See also: the equivalent page on direct sums. In the following, we will always have and topological…
Exercise 185 (Chapter 28, “The category of topological spaces”): Prove that both the direct product and the direct sum of two Hausdorff topological spaces is Hausdorff. We will consider Hausdorff topological spaces and and their direct sum (in the first part) or product (in the second part) . We will proceed without reference to the…
A lot of results concerning the direct sum of two topological spaces and are trivial if you consider the usual particular construction of these entities. I find it more satisfying to derive them directly from the universal definitions. See also: the equivalent page on direct products. In the following, we will always have and topological…
Exercise 182 (Chapter 29, “The category of topological spaces”): Give a universal definition which leads to the introduction of the indiscrete topology on a set. On a set , the indiscrete topology is the unique topology such that for any set , any mapping is continuous. This isn’t really a universal definition. We have seen…
Chapter 22 on representations is, I feel, the worst written I have come upon up to now. I haven’t yet come to wrap my head around it. I certainly appreciate the desire to give a general definition of representations in category theory terms. This is how the author puts it: Let and be two categories….
In my search for less abstract or trivial examples of a universal enveloping algebra, I have come upon this one. Let be a vector space and a free associative algebra on . The mapping is a vector space morphism . can also be viewed as a Lie algebra. is a part of ; let be…
This keeps cropping up, and every time it takes me some effort to rediscover the proof. It happened first with free groups, then with free vector spaces and again with free Lie algebras. I’ll formulate the issue with groups, but it is easy to carry it over to other cases. A free group on set…
Chapter 12 (Functors) gives as an example of a functor the “free group functor” from the category of sets () to the category of groups (). We will call this functor , and its construction, as described, is the following: For any set , the object is the free group on . Let be a…
(Unfinished) In Chapter 12, R.G. describes three possible transformations “from real to complex vector spaces and back”. Inspired by Chapter 17 (Functors), I have examined these transformations from the functor point of view. We can distinguish not two, but three categories here, the first and third of which are equivalent: : Complex vector spaces with…