I created Math3ma in early 2015 as an aid in transitioning from undergraduate-level to graduate-level mathematics.

I've been collaborating on an exciting project for quite some time now, and today I'm happy to share it with you. There is a new topology book on the market! Topology: A Categorical Approach is a graduate-level textbook that presents basic topology from the modern perspective of category theory. Coauthored with Tyler Bryson and John Terilla, Topology is published through MIT Press and will be released on August 18, 2020. But you can pre-order on Amazon now!

In last last week's episode of PBS Infinite Series, we talked about different flavors of multiplication (like associativity and commutativity) to think about when multiplying things that aren't numbers. My examples of multiplying non-numbers were vectors and matrices, which come from the land of algebra. Today I'd like to highlight another example: We can multiply shapes!

Have you heard the buzz? Applied category theory is gaining ground! But, you ask, what is applied category theory? Upon first seeing those words, I suspect many folks might think either one of two thoughts: 1. Applied category theory? Isn't that an oxymoron? or 2. Applied category theory? What's the hoopla? Hasn't category theory always been applied? (Visit the blog to read more!)

Viewers like you help make PBS (Thank you 😃) . Support your local PBS Member Station here: https://to.pbs.org/donateinfi What happens when you multiply shapes? This is part 2 of our episode on multiplying things that aren't numbers. You can check out part 1: The Multiplication Multiverse right here https://www.youtube.com/watch?v=H4I2C3Ts7_w Tweet at us! @pbsinfinite Facebook: facebook.com/pbsinfinite series Email us! pbsinfiniteseries [at] gmail [dot] com And discuss the episode further over on reddit at https://www.reddit.com/r/PBSInfiniteSeries/ Previous Episode The Multiplication Multiverse | Infinite Series https://www.youtube.com/watch?v=H4I2C3Ts7_w In our last episode, we talked about different properties of multiplication: associativity and commutativity are the most familiar, but they’re just two of many. We also saw it’s possible to multiply things that aren’t numbers, and in that case we may not have... associativity, for instance. But that’s not a bad thing. In fact, it’s a beautiful thing! References:: More on the associahedra: http://www.ams.org/samplings/feature-column/fcarc-associahedra http://www.claymath.org/library/academy/LectureNotes05/Lodaypaper.pdf https://arxiv.org/pdf/math/0212126.pdf More on multiplying non-numbers: http://www.math3ma.com/mathema/2017/11/24/multiplying-non-numbers An introduction to operads: http://www.math3ma.com/mathema/2017/10/23/what-is-an-operad-part-1 http://www.math3ma.com/mathema/2017/10/30/what-is-an-operad-part-2 Some applications in math and physics: https://arxiv.org/abs/1202.3245 http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.22.2871&rep=rep1&type=pdf http://bookstore.ams.org/conm-227 http://www.springer.com/us/book/9780817647346 Richard Stanley’s book on the Catalan Numbers: https://www.amazon.com/Catalan-Numbers-Richard-P-Stanley/dp/1107427746 Written and Hosted by Tai-Danae Bradley Produced by Rusty Ward Graphics by Ray Lux Assistant Editing and Sound Design by Meah Denee Barrington Made by Kornhaber Brown (www.kornhaberbrown.com) Thanks to Matthew O'Connor and Yana Chernobilsky who are supporting us on Patreon at the Identity level! And thanks to Nicholas Rose and Mauricio Pacheco who are supporting us at the Lemma level!

relaxing a notion of “sameness” gives us extra currency with which to explore new phenomena

@JadeMasterMath: There are lots of mathematical concepts which don’t have well written resources to learn about them. I think that explaining something in a clear way with a story arc can sometimes be enough. @jeremyjkun: Write about the topics that you learned, where there was a succinct phrase, picture, or idea that suddenly made it clear. Then arrange the whole blog post around getting the reader to that same understanding.

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.

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.

This post made the concept of “indexing,” used across mathematics, click for me.

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 existsa [yadda yadda] such thatthe following diagram commutes." and perhaps it left you wondering what it all meant.

Previously on the blog, we've discussed a recurring theme throughout mathematics: making new things from old things. Today, I'd like to focus on a particular way to build a new vector space from old vector spaces: the tensor product. This construction often come across as scary and mysterious, but I hope to shine a little light and dispel a little fear. In particular, we won't talk about axioms, universal properties, or commuting diagrams. Instead, we'll take an elementary, concrete look: Given two vectors $\mathbf{v}$ and $\mathbf{w}$, we can build a new vector, called the tensor product $\mathbf{v}\otimes \mathbf{w}$. But what is that vector, really? Likewise, given two vector spaces $V$ and $W$, we can build a new vector space, also called their tensor product $V\otimes W$. But what is that vector space, really?

an often fruitful way to discover properties of an object is not to investigate the object itself, but rather to study the collection of maps to or from the object.