Substrate

With given category $\mathcal{C}$ and its objects $A$ and $B$, a hom-set $\hom_\mathcal{C}(A, B)$ is the collection of all morphisms from $A$ to $B$. There is also a related notion of hom-functor ...

Thinking about the relationship between pieces was an exercise in frustration, a continual feeling that the solution was just out of reach, as concentrating on one part would push some other critical piece of knowledge out of my head. ​ Why do we leave material out of classes and then fail students who can't figure out that material for themselves? Why do we make the first couple years of an engineering major some kind of hazing ritual, instead of simply teaching people what they need to know to be good engineers? For all the high-level talk about how we need to plug the leaks in our STEM education pipeline, not only are we not plugging the holes, we're proud of how fast the pipeline is leaking.

When you import a module into Swift code, you expect the result to be entirely additive. But as we’ll see, this isn’t always the case.

That was a day I felt our friendship leveled-up, because I knew I could trust her to give me honest feedback on any subject. "Truth is Kindness" ​ all forms of lying --including while lies meant to spare feelings-- are associated with less satisfying relationships ​ I slide into dishonesty more often than I’d like. It’s easy and it’s comfortable.

I love a good secret. More to the point, I love a good surprise. Seeing the excitement and joy that people feel when you give them good news? The best…

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.

Happy the people whose annals are tiresome

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?

The bicycle, as we know it today, was not invented until the late 1800s. Here are some theories about why

This quarter’s experiment has helped me see that every class we attend, every word we write, every article we read is where we are going. We are already there. ​ I do not want that experience to feel like some unrelenting ultra-marathon. I want it to feel alive and loving, nourishing and compelling. I want to feel hungry and then full and then hungry again. ​ May reading, like all things we do, become an invitation to experience the miracle that we are alive — still, and in the first place. And may we use the very act of reading itself to challenge the idea that life is about collecting the most knowledge or arriving at some finish line or final page. ​ I felt a greater sense of agency because I got to decide what to read each time I read. Choosing intuitively meant I looked forward to making a choice about what to read. ​ This quarter’s experiment has taught me that I must do both to become the scholar I want to be — a person who can hold uncertainty as well as she can hold knowledge, who can be slow and discerning, and insatiably curious and eager at the same time.

Why do people care so much about a piece of — no offense — punctuation?

Thursdays are wily: Unlike most days, there are no expectations for Thursday, and it deftly plays that lack of promise into a wealth of possibility. Thursday is that guy at work who you never talk to, the one that seems nice enough, always shows up on time, doesn’t raise a fuss, and quietly does a quality job every time. Thursday is humble, understated excellence. It will never make you feel ashamed of not “doing enough with your Thursday.” You’re welcome. (Thursday would never say it sarcastically like that, which is why I’m saying it on behalf of Thursday.)

Want to learn how to cook a steak perfectly each time? Look no further. We’re focusing on both ribeye and skirt steak in this episode of Basics with Babish. Watch the rebroadcast of the Twitch livestream for this episode here: https://youtu.be/HpzbyjyUf1k Recipe: https://basicswithbabish.co/basicsepisodes/2017/10/23/sauces-9w5tm Grocery List: Tomahawk ribeye Skirt steak Vegetable oil Butter Garlic Fresh sprig rosemary Kosher salt Freshly cracked pepper Special equipment: Stainless steel pan OR cast iron pan Instant read thermometer My first cookbook, Eat What You Watch, is available now in stores and online! Amazon: http://a.co/bv3rGzr Barnes & Noble: http://bit.ly/2uf65LX Theme song: "Stay Tuned" by Wuh Oh https://open.spotify.com/track/5lbQ6nKPgzkfFigheb467z Music: “Feel Good“ and “Add And” by Broke for Free https://soundcloud.com/broke-for-free http://www.bingingwithbabish.com/podcast Binging With Babish Website: http://bit.ly/BingingBabishWebsite Basics With Babish Website: http://bit.ly/BasicsWithBabishWebsite Patreon: http://bit.ly/BingingPatreon Instagram: http://bit.ly/BabishInstagram Facebook: http://bit.ly/BabishFacebook Twitter: http://bit.ly/BabishTwitter Twitch: http://bit.ly/BabishTwitch

(See PDF in reMarkable for annotations and notes).

In the middle of my sketching hours, I don’t want to be worrying about whether I’ll be ready for my classroom prototype next month. Within a given day, action-oriented “butt-in-chair”-style advice does help; meta-thought is just distracting. But go too long without error correction, and you’ll misspend hours in the chair. ​ The rest of the day’s work becomes roughly deontological. I give myself permission to be satisfied with the day if I spent three focused hours sketching like I’d planned. ​ From time to time, I flip back into execution mode. It feels like an old friend. We say hello, dance for a while, and part ways smiling, just as it always was. Open-ended mode is more enigmatic, reserved—yet occasionally it sparks some moment so singular it lights up the whole year. Those moments don’t happen without the days spent together between those moments. I’m slowly learning to make the most of our quiet strolls.

instead of interesting thoughts and aperçus, I find instead a lot of self-promotion, water carrying, awkward efforts to impress people, attempts to @ my way into conversations I didn’t belong in, and lots of stray opinions that would have been better off kept to myself in any circumstances. It’s like I had no concept of a “lane” to stay in. I no longer feel like I need to narrate my entire day’s reading to the site as if it were a surrogate listener. But reading them over as I eliminate them from the public record, I see that there was nothing there, nothing that can redeem for me now the time I spent on the platform in the past. It’s more an illustration of the time I wasted while never trying to write something that might have had the remote chance of actually being lastingly useful. No one on social media is speaking to the future.

but the thought experiment’s contrast with how shortages and surpluses are resolved in real life is the reason I think about it a lot. As an example, a chronological Twitter feed becomes chaotic once it comprises thousands of followed accounts, but Twitter restores order algorithmically, making that feed less crowded by regulating which messages actually appear in it.

(referenced in https://kneelingbus.substack.com/p/97-the-coroner-at-dreamgate-frontier)

But the person with Chill is crucially missing these last ingredients because they are too far removed from anything that looks like intensity to have passions. Because Chill is the opposite of something else too: warmth. And kindness, and earnestness, and vulnerability. And we need just enough of those things to occasionally do something so remarkably unchill as fall in love.

Ultralearning, in my opinion, often works well because it compresses the frustration barrier to a shorter period of time. ​ Whenever we, as human beings, sense a comparative disadvantage, it’s as if our brain immediately tries to avoid practicing the skill. ​ One-on-one tutoring immediately removes the “I’m the worst in the class” feeling. It also removes the “I’m the best in the class” laziness that can afflict high-performing students. Woah, hadn’t thought of it this way. À la the more “single-player games” tendency I’ve been leaning into.

My writing pace accelerates whenever I find myself in a learning rich environment, which is why I wrote so much in my first two years out of school and over the past three years at Stripe. ​ but even more important for me is that Stripe Press is a bit unusual: they typically buy completed manuscripts, rather than proposals. This gave me an extraordinary amount of latitude in my approach to writing, the book’s format and marketing the book. ​ I am truly amazed by folks who are able to write when raising young children, caring for their parents, or otherwise committed: it takes a great deal of privilege to write a book. ​ and I’m deeply grateful that I’ve gotten to do it. So far, I think the hardest bit will be a small sense of loss after it all quiets down, e.g. the return to normalcy. ​ If I wrote another book, I would spend more time outline in detail to build the small pieces more intentionally over the course of the book. _whispers: “composition”_ Honest feedback is very hard to find when writing a book, since you have to find (a) someone who will give hard feedback, and (b) someone who is willing to read your book. That’s a small intersection.

A paradigm is sneaking in when you aren’t paying attention. Pay attention. There’s a chance here to influence development practice for decades ​ We really can have languages that give the benefits of tomorrow without losing all the working components of today. I think Swift can be one of those languages. ​ Much of that, I believe, is education.

Yotto b2b Jeremy Olander, live in London at Anjunadeep Open Air. Brought to you by Beatport.