This post will venture further into abstract mathematics than most of my posts. If this isn’t what you’re looking for, you might try browsing here for more concrete articles.
Incidentally, although I’m an applied mathematician, I also appreciate pure math. I imagine most applied mathematicians do as well. But what I do not appreciate is pseudo-applied math, pure math that pretends to be more useful than it is. Pure math is elegant. Applied math is useful. The best math is elegant and useful. Pseudo-applied math is the worst because it is neither elegant nor useful [1].
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A common theme in pure mathematics, and especially the teaching of pure mathematics, is to strip items of interest down to their most basic properties, then add back properties gradually. One motivation for this is to prove theorems assuming no more structure than necessary.
Choosing a level of abstraction
For example, we can think of the Euclidean plane as the vector space ℝ², but we can think of it as having less structure or more structure. If we just think about adding and subtracting vectors, and forget about scalar multiplication for a moment, then ℝ² is an Abelian group. We could ignore the fact that addition is commutative and think of it simply as a group. We could continue to ignore properties and go down to monoids, semigroups, and magmas.
Going the other direction, there is more to the plane than it’s algebraic structure. We can think of ℝ² as a topological space, in fact a Hausdorff space, and in fact a metric space. We could think of the plane as topological vector space, a Banach space, and more specifically a Hilbert space.
In short, there are many ways to classify the plane as a mathematical object, and we can pick the one best suited for a particular purpose, one with enough structure to get done what we want to get done, but one without additional structure that could be distracting or make our results less general.
Topological groups
A topological group is a set with a topological structure, and a group structure. Furthermore, the two structures must play nicely together, i.e. we require the group operations to be continuous.
Unsurprisingly, an Abelian topological group is a topological group whose group structure is Abelian.
Not everything about Abelian topological groups is unsurprising. The motivation for this post is a surprise that we’ll get to shortly.
Category theory
A category is a collection of objects and structure-preserving maps between those objects. The meaning of “structure-preserving” varies with context.
In the context of vector spaces, maps are linear transformations. In the context of groups, the maps are homomorphisms. In the context of topological spaces, the maps are continuous functions.
In the previous section I mentioned structures playing nicely together. Category theory makes this idea of playing together nicely explicit by requiring maps to have the right structure-preserving properties. So while the category of groups has homomorphisms and the category of topological spaces has continuous functions, the category of topological groups has continuous homomorphisms.
Abelian categories
The category of Abelian groups is much nicer than the category of groups. This takes a while to appreciate. Abelian groups are groups after all, so isn’t the category of Abelian groups just a part of the category of groups? No, it’s more subtle than that. Here’s a post that goes into the distinction between the categories of groups and Abelian groups.
The category of Abelian groups is so nice that the term Abelian category was coined to describe categories that are as nice as the category of Abelian groups. To put it another way, the category of Abelian groups is the archetypical Abelian category.
Now here’s the surprise I promised above: the category of topological Abelian groups is not an Abelian category. More on that at nLab.
If you naively think of an Abelian category as a category containing Abelian things, then this makes no sense. If a grocery cart is a cart containing groceries, then you’d think a gluten-free grocery cart is a grocery cart containing gluten-free groceries.
A category is not merely a container, like a shopping cart, but a working context. What makes an Abelian category special is that it has several nice properties as a working context. If that idea is new to you, I’d recommend carefully reading this post.
Related posts
[1] This reminds me of the quote by William Morris: “Have nothing in your houses that you do not know to be useful or believe to be beautiful.”