Limits and Colimits, Part 1 (Introduction)
I'd like to embark on yet another mini-series here on the blog. The topic this time? Limits and colimits in category theory! But even if you're not familiar with category theory, I do hope you'll keep reading. Today's post is just an informal, non-technical introduction. And regardless of your categorical background, you've certainly come across many examples of limits and colimits, perhaps without knowing it! They appear everywhere - in topology, set theory, group theory, ring theory, linear algebra, differential geometry, number theory, algebraic geometry. The list goes on. But before diving in, I'd like to start off by answering a few basic questions.
What ARE limits and colimits?
They are a (and in some sense, the most efficient) way to construct a new object from an existing collection of objects. For example, suppose we were given a bunch of sets. Plain, old, vanilla-flavored sets. Can we use these to construct new sets? Definitely. We can isolate a single element and form a one-point set. We can isolate no elements and form the empty set. We can form a quotient set. We can take a few sets and form their Cartesian product or their fibered product, or we can form their disjoint union. We can also look at their intersection or their (not-necessarily disjoint) union. And if we have a function from one set to another, we can look at the preimage of an element in its range.
Each of the bold-faced words is an example of either a limit or a colimit.
What's the difference between a limit and a colimit?
You'll notice that constructions above seem to come in two flavors.
flavor #1: taking sub-things
A single element, an intersection, a preimage, a product. These are all formed by picking out a sub-collection of elements from given sets, contingent on some condition. (Let's think of the product as a subset of itself. Bear with me.) These are examples of limits.
flavor #2: gluing things together
A set with no elements, disjoint unions, not-necessarily-disjoint unions, and quotients are all formed by assembling or 'gluing' things together. (Let's think of forming the empty set by 'gluing' no things together. Bear with me.) These are examples of colimits.
In practice, limits tend to have a "sub-thing" feel to them, whereas colimits tend to have a "glue-y" feel to them. That's terribly imprecise, but it's an intuitive feel for how mathematicians construct things. More formally, the defining property of a limit is characterized by maps whose domain is the limit. On the other hand, the defining property of a colimit is characterized by maps whose codomain is the colimit. In this sense, limits and colimits are dual to each other.
You'll notice I used the word "maps" instead of "functions." That's because set theorists aren't the only people who like to construct new things from existing things. As I mentioned in the opening paragraph, limits and colimits appear all across the mathematical landscape. A direct sum of abelian groups, the kernel of a group homomorphism, the direct product of vector spaces, the free product of groups, the least upper bound of a poset, and a CW complex are all examples of limits and colimits.
What are some prerequisites for learning more about (co)limits?
Although category theory wasn't a prerequisite for today's post, it is necessary for the remaining posts in this series. I'll assume familiarity with categories and functors and natural transformations. For a bird's eye view of the subject, check out What is Category Theory, Anyway? At that link, you'll also find a list of suggested resources for further reading.
By the way, in every discussion of limits and colimits there is a heavy emphasis on maps - homomorphisms, functions, linear transformations, continuous functions, smooth functions, you name it. In fact, limits and colimits are defined in terms of a universal property. This emphasis on maps, or relationships, is a trademark of category theory and has roots in what I like to call the Yoneda perspective, which is the categorical maxim that objects are completely determined by their relationships to other objects.I recommend taking a look at that post to get a better feel for how categorically-minded mathematicians think. With that said, you'll also want to be familiar with commutative diagrams.
Are (co)limits related to limits of sequences in topology and analysis?
No, not that I'm aware of. But there is one similarity. In analysis, we ask for the limit OF a sequence. In category theory, we also ask for the (co)limit OF something. But that "something" is not a sequence of points.
So, what is it?
That's where we'll pick up next time.
In this series: