Commutative Diagrams Explained

Commutative Diagrams Explained

Have you ever come across the words "commutative diagram" before? Perhaps you've read or heard someone utter a sentence that went something like

"For every [bla bla] there exists
a [yadda yadda] such that
the following diagram commutes."

and perhaps it left you wondering what it all meant.

Read More

Group Elements, Categorically

Group Elements, Categorically

On Monday we concluded our mini-series on basic category theory with a discussion on natural transformations and functors. This led us to make the simple observation that the elements of any set are really just functions from the single-point set {✳︎} to that set. But what if we replace "set" by "group"? Can we view group elements categorically as well?

The answer to that question is the topic for today's post, written by guest-author Arthur Parzygnat.

Read More

What is a Natural Transformation? Definition and Examples

What is a Natural Transformation? Definition and Examples

I hope you have enjoyed our little series on basic category theory. (I know I have!) This week we'll close out by chatting about natural transformations which are, in short, a nice way of moving from one functor to another. If you're new to this mini-series, be sure to check out the very first post, What is Category Theory Anyway? as well as What is a Category? and last week's What is a Functor?

Read More

What is a Functor? Definition and Examples, Part 1

What is a Functor? Definition and Examples, Part 1

Next up in our mini series on basic category theory: functors! We began this series by asking What is category theory, anyway? and last week walked through the precise definition of a category along with some examples. As we saw in example #3 in that post, a functor can be viewed an arrow/morphism between two categories.

Read More

What is a Category? Definition and Examples

What is a Category? Definition and Examples

As promised, here is the first in our triad of posts on basic category theory definitions: categories, functors, and natural transformations. If you're just now tuning in and are wondering what is category theory, anyway? be sure to follow the link to find out!

A category 𝖢 consists of some data that satisfy certain properties...

Read More

A Quotient of the General Linear Group, Intuitively

A Quotient of the General Linear Group, Intuitively

Over the past few weeks, we've been chatting about quotient groups in hopes of answering the question, "What's a quotient group, really?" In short, we noted that the quotient of a group G by a normal subgroup N is a means of organizing the group elements according to how they fail---or don't fail---to satisfy the property required to belong to N. The key point was that there's only one way to belong to N, but generally there may be several ways to fail to belong.

Read More

The First Isomorphism Theorem, Intuitively

The First Isomorphism Theorem, Intuitively

Welcome back to our little discussion on quotient groups! (If you're just now tuning in, be sure to check out "What's a Quotient Group, Really?" Part 1 and Part 2!) We're wrapping up this mini series by looking at a few examples. I'd like to take my time emphasizing intuition, so I've decided to give each example its own post. Today we'll take an intuitive look at the quotient given in the First Isomorphism Theorem.

Read More

What's a Quotient Group, Really? Part 2

What's a Quotient Group, Really? Part 2

Today we're resuming our informal chat on quotient groups. Previously we said that belonging to a (normal, say) subgroup N of a group G just means you satisfy some property. For example,  5ℤ ⊂  means "You belong to 5 if and only if you're divisible by 5". And the process of "taking the quotient" is the simple observation that every element in G either....

Read More

What's a Quotient Group, Really? Part 1

What's a Quotient Group, Really? Part 1

I realize that most of my posts for the past, er, few months have been about some pretty hefty duty topics. Today, I'd like to dial it back a bit and chat about some basic group theory! So let me ask you a question: When you hear the words "quotient group," what do you think of? In case you'd like a little refresher, here's the definition....

Read More

The Most Obvious Secret in Mathematics

The Most Obvious Secret in Mathematics

Yes, I agree. The title for this post is a little pretentious. It's certainly possible that there are other mathematical secrets that are more obvious than this one, but hey, I got your attention, right? Good. Because I'd like to tell you about an overarching theme in mathematics - a mathematical mantra, if you will. A technique that mathematicians use all the time to, well, do math. 

Read More

Resources for Intro-Level Graduate Courses

Resources for Intro-Level Graduate Courses

In recent months, several of you have asked me to recommend resources for various subjects in mathematics. Well, folks, here it is! I've finally rounded up a collection of books, PDFs, videos, and websites that I found helpful while studying for my intro-level graduate courses.

Read More