# January 2018

# Limits and Colimits, Part 2 (Definitions)

Welcome back to our mini-series on categorical limits and colimits! In Part 1 we gave an intuitive answer to the question, "What *are* limits and colimits?" As we saw then, there are *two* main ways that mathematicians construct *new* objects from a collection of given objects: 1) take a "sub-collection," contingent on some condition or 2) "glue" things together. The first construction is usually a limit, the second is usually a colimit. Of course, this might've left the reader wondering, "Okay... but what are we taking the (co)limit *of* ?" The answer? A *diagram*. And as we saw a couple of weeks ago, a diagram is really a functor.

# Brouwer's Fixed Point Theorem (Proof)

Today I'd like to talk about Brouwer's Fixed Point Theorem. *Literally!* It's the subject of this week's episode on PBS Infinite Series. Brouwer's Fixed Point Theorem is a result from topology that says no matter how you stretch, twist, morph, or deform a disc (so long as you don't tear it), there's always one point that ends up in its original location.

# 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.