# 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.

*W*hat'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.