I totally fell down on the job last year (compared to my goal), but I did read a lot of additional material online instead and lot of what I did read, (but didn’t necessarily finish toward my goal) was of a highly dense/technical nature. We’ll do better this year.
This short introduction to category theory is for readers with relatively little mathematical background. At its heart is the concept of a universal property, important throughout mathematics. After a chapter introducing the basic definitions, separate chapters present three ways of expressing universal properties: via adjoint functors, representable functors, and limits. A final chapter ties the three together. For each new categorical concept, a generous supply of examples is provided, taken from different parts of mathematics. At points where the leap in abstraction is particularly great (such as the Yoneda lemma), the reader will find careful and extensive explanations.
h/t to John Carlos Baez for the notice:
My friend Tom Leinster has written a great introduction to that wonderful branch of math called category theory! It’s free:
https://arxiv.org/abs/1612.09375
It starts with the basics and it leads up to a trio of related concepts, which are all ways of talking about universal properties.
Huh? What’s a ‘universal property’?
In category theory, we try to describe things by saying what they do, not what they’re made of. The reason is that you can often make things out of different ingredients that still do the same thing! And then, even though they will not be strictly the same, they will be isomorphic: the same in what they do.
A universal property amounts to a precise description of what an object does.
Universal properties show up in three closely connected ways in category theory, and Tom’s book explains these in detail:
through representable functors (which are how you actually hand someone a universal property),
through limits (which are ways of building a new object out of a bunch of old ones),
through adjoint functors (which give ways to ‘freely’ build an object in one category starting from an object in another).
If you want to see this vague wordy mush here transformed into precise, crystalline beauty, read Tom’s book! It’s not easy to learn this stuff – but it’s good for your brain. It literally rewires your neurons.
Here’s what he wrote, over on the category theory mailing list:
…………………………………………………………………..
Dear all,
My introductory textbook “Basic Category Theory” was published by Cambridge University Press in 2014. By arrangement with them, it’s now also free online:
https://arxiv.org/abs/1612.09375
It’s also freely editable, under a Creative Commons licence. For instance, if you want to teach a class from it but some of the examples aren’t suitable, you can delete them or add your own. Or if you don’t like the notation (and when have two category theorists ever agreed on that?), you can easily change the Latex macros. Just go the arXiv, download, and edit to your heart’s content.
There are lots of good introductions to category theory out there. The particular features of this one are:
• It’s short.
• It doesn’t assume much.
• It sticks to the basics.
The "Meet the Press" host opens up about the surprising thing he shares with the president-elect.
Sooner or later, enough people I like are going to abandon the service, and the pain-to-pleasure ratio will tip unfavorably. I don't know how Twitter will survive 2017 without making some drastic changes to its service. Maybe it's already too late.
Tweet a positive 9-digit (or smaller) integer at @PrimesAsAService. It will reply via Twitter to tell you if the number prime or not.
Some of the usable commands one can tweet to the bot for answers follow. (Hint: Click on the buttons with the tweet text to auto-generate the relevant Tweet.)
If you ask about a prime number with a twin prime, it should provide the twin.
Pro tip: You should be able to drag and drop any of the buttons above to your bookmark bar for easy access/use in the future.
Happy prime tweeting!
Summits, elections, anniversaries and other momentous dates for Trump’s first year
A reporter reveals how he investigated Trump’s claims on his donations to charity.
AUROVILLE, India — In the Colours of Nature dye house, Vijayakumar Varathan is busy prepping a vat of indigo. At 51, he looks frail, with a tanned body made mostly of bones, but he runs to and fro, setting up an open fire where he’ll brew cauldrons of natural colorants made from plants.He’s worked here for 15 years. But until his early 30s, Varathan mixed chemicals in a conventional clothing factory in the same region of southern India. There he developed a disease that caused layers of his skin to peel off. Even today, it is discolored. “It was pretty bad,” he says, in his fragmented English. “But I didn’t have a choice.”
Neuroengineering is faced with unique challenges in repairing or replacing complex neural systems that are composed of many interacting parts. These interactions form intricate patterns over large spatiotemporal scales, and produce emergent behaviors that are difficult to predict from individual elements. Network science provides a particularly appropriate framework in which to study and intervene in such systems, by treating neural elements (cells, volumes) as nodes in a graph and neural interactions (synapses, white matter tracts) as edges in that graph. Here, we review the emerging discipline of network neuroscience, which uses and develops tools from graph theory to better understand and manipulate neural systems, from micro- to macroscales. We present examples of how human brain imaging data is being modeled with network analysis and underscore potential pitfalls. We then highlight current computational and theoretical frontiers, and emphasize their utility in informing diagnosis and monitoring, brain-machine interfaces, and brain stimulation. A flexible and rapidly evolving enterprise, network neuroscience provides a set of powerful approaches and fundamental insights critical to the neuroengineer's toolkit.
It is argued that if the non-unitary measurement transition, as codified by Von Neumann, is a real physical process, then the "probability assumption" needed to derive the Second Law of Thermodynamics naturally enters at that point. The existence of a real, indeterministic physical process underlying the measurement transition would therefore provide an ontological basis for Boltzmann's Stosszahlansatz and thereby explain the unidirectional increase of entropy against a backdrop of otherwise time-reversible laws. It is noted that the Transactional Interpretation (TI) of quantum mechanics provides such a physical account of the non-unitary measurement transition, and TI is brought to bear in finding a physically complete, non-ad hoc grounding for the Second Law.
The Web grows every day. Tools, approaches, and styles change constantly, and keeping up is a challenge. We've compiled the best insights from subject matter experts for you in one place, so you can dive deep into the latest of what's happening in web development.
100 years after Smoluchowski introduces his approach to stochastic processes, they are now at the basis of mathematical and physical modeling in cellular biology: they are used for example to analyse and to extract features from large number (tens of thousands) of single molecular trajectories or to study the diffusive motion of molecules, proteins or receptors. Stochastic modeling is a new step in large data analysis that serves extracting cell biology concepts. We review here the Smoluchowski's approach to stochastic processes and provide several applications for coarse-graining diffusion, studying polymer models for understanding nuclear organization and finally, we discuss the stochastic jump dynamics of telomeres across cell division and stochastic gene regulation.
This morning I launched the Kickstarter project for Micro.blog. Really happy with the response. Thank you, everyone!
As a fellow IndieWeb proponent, and since I know how much work such an undertaking can be, I’m happy to help you with the e-book and physical book portions of your project on a voluntary basis if you’d like. I’ve got a small publishing company set up to handle the machinery of such an effort as well as being able to provide services that go above and beyond the usual low-level services most self-publishing services might provide. Let me know if/how I can help.
KTLA’s live-stream of the 128th Rose Parade Presented by Honda in Pasadena occurred Monday, Jan. 2, 2017. It marked our 70th consecutive broadcast of the parade, which this year had the theme “Echoes of Success.” KTLA's live-stream of the 128th Rose Parade Presented by Honda in Pasadena occurred Monday, Jan. 2, 2017. It marked our 70th consecutive broadcast of the parade, which this year had the theme "Echoes of Success." Our "band cam," a raw feed of the parade’s bands, presented by Jack in the Box, is below:
Somehow I overslept and missed the B2 Bomber flying over the house on the way to kick off the parade.
