Direct products/sums: when the associated morphisms are epi/mono

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…

Theorems about the direct product of two topological spaces

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…

Direct sum of Hausdorff topological spaces

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…

Theorems about the direct sum of two topological spaces

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…

Universal definition of the indiscrete topology

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…

Representation frameworks and representations

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….

One universal enveloping algebra

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…

No maverick elements in a free space

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…