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.
A great little introduction to category theory! Can’t wait to see what the future installments bring.
Interestingly I came across this on Instagram. It may be one of the first times I’ve seen math at this level explained in pictorial form via Instagram.
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Given a bunch of sets, what are some ways to construct a new set? Some options include: intersections, unions, Cartesian products, preimages, and quotients. And these are all examples of “limits and colimits” in #categorytheory! Notice how the examples come in two flavors? An intersection, a preimage, a product are all formed by picking out a sub-collection of elements from given sets, contingent on some condition. These are examples of limits. On the other hand, unions and quotients are formed by assembling or 'gluing' things together. 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. And these constructions are two of the most frequent ways that mathematicians build things, so they appear ALL over mathematics. But what are (co)limits, exactly? I’ve just posted a non-technical introduction on my blog. It’s Part 1 of the latest mini-series on Math3ma. Link in profile!