If we want a name to fully describe the concept it points to, then it must be a very simple concept indeed. ​ But we still have to explain what a Mappable is. “What’s a Mappable? Well, it’s something you can map over!” is a terrible explanation! It’s literally just the grammatic expansion of the word. All it does is move the question one bit further ​ Functor is hard to learn. It is not hard to learn because it is named Functor. If you renamed it to anything else, you’d have just as hard of a time, and you’d be cutting off your student from all of the resources and information currently using the word “functor” to refer to that concept.

Keynote by Dr. Emily Riehl C◦mp◦se :: Conference http://www.composeconference.org/ May 18, 2017 Slides: http://www.math.jhu.edu/~eriehl/compose.pdf Monads have famously been used to model computational effects, although, curiously, the computer science literature presents them in a form that is scarcely recognizable to a category theorist — I’d say instead that a monad is just a monoid in the category of endofunctors, what’s the problem? ;) To a categorical eye, computational effects are modeled using the Kleisli category of a monad, a perspective which suggests another categorical tool that might be used to reason about computation. The Kleisli category is closely related to another device for categorical universal algebra called a Lawvere theory, which may be a more natural framework to model computation (an idea suggested by Gibbons, Hinze, Hyland, Plotkin, Power and certainly others). This talk will survey monads, Lawvere theories, and the relationships between them and illustrate the advantages and disadvantages of each framework through a variety of examples: lists, exceptions, side effects, input-output, probabilistic non-determinism, and continuations.

Facebook News Feed simultaneously increased the efficiency of distribution of new posts and pitted all such posts against each other in what was effectively a single giant attention arena, complete with live updating scoreboards on each post. It was as if the panopticon inverted itself overnight, as if a giant spotlight turned on and suddenly all of us performing on Facebook for approval realized we were all in the same auditorium, on one large, connected infinite stage, singing karaoke to the same audience at the same time. ​ As humans, we intuitively understand that some galling percentage of our happiness with our own status is relative. What matters is less our absolute status than how are we doing compared to those around us. By taking the scope of our status competitions virtual, we scaled them up in a way that we weren't entirely prepared for. Is it any surprise that seeing other people signaling so hard about how wonderful their lives are decreases our happiness? ​ Facebook, with its explicit attachment to the real world graph and its enforcement of a single public identity, is just a poor structural fit for the more complex social capital requirements of the young. ​ Every network has some ceiling on its ultimate number of contributors, and it is often a direct function of its proof of work. ​ This season, the color of the moment might be saffron. Why? Because someone cooler than me said so. Tech tends to prioritize growth at all costs given the non-rival, zero marginal cost qualities of digital information. In a world of abundance, that makes sense. However, technology still has much to learn from industries like fashion about how to proactively manage scarcity, which is important when goods are rivalrous. Since many types of status are relative, it is, by definition, rivalrous. There is some equivalent of crop rotation theory which applies to social networks, but it's not part of the standard tech playbook yet. ​ but if I have anything to offer on that front, it’s this: if you want control of your own happiness, don’t tie it to someone else’s scoreboard.

Selective Applicative Functors: Declare Your Effects Statically, Select Which to Execute Dynamically - GitHub - snowleopard/selective: Selective Applicative Functors: Declare Your Effects Staticall...

Contribute to cohomolo-gy/cats-in-context development by creating an account on GitHub.

Slides for Scalaworld 2019. Contribute to cohomolo-gy/Type-Arithmetic-and-the-Yoneda-Perspective development by creating an account on GitHub.

Guessing Swift takes a similar approach in [SE-0185](https://github.com/apple/swift-evolution/blob/098152eafbfbd7faa74d81e1443231bd7caabc45/proposals/0185-synthesize-equatable-hashable.md).

Introduction Operator-fusion, one of the cutting-edge research topics in the reactive programming world, is the aim to have two of more su...

Course notes for Pure Mathematics Topic D 2019. Contribute to DavidMichaelRoberts/AlgebraicTopology2019 development by creating an account on GitHub.

From [Adrien Vakili](vaki.li) General Topology Munkres -- Topology Chapter 1, sections 1-6 Chapter 2 Chapter 3, sections 1-2 Algebraic Topology Hatcher -- Algebraic Topology Provides good geometric intuition but avoids categorical language. Chapters 0, 1, 4 Strom -- Modern Classical Homotopy Theory More axiomatic/categorical treatment. Fomenko -- Visual Geometry and Topology For pictures.

We present Type Classes: a series of focused video courses on Haskell, Nix, and related subjects.

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