Thanks So8res!
Yeah the page is still under development and I’m planning to add more negative examples as I go along. It was just copy-pasted from the Isomorphism: Intro (Math 0) page to have something here to start.
Do you feel the isomorphism page should have more negative examples for bijections? Or that it’s long enough already?
Could be nice to add a concrete “real-life” (non-math) example, say like the following:
You are a defense lawyer. Your client is accused of stealing the cookie from the cookie jar. You want to prove her innocence. Lets say you have evidence that the jar is still sealed. Reason as follows:
Assume she stole the cookie from the cookie jar.
Then she would have had to open the jar.
The jar is still sealed.
For the jar to be sealed and for her to have opened it is a contradiction.
Hence the assumption in 1 is false (given the deductions below it are true).
Hence she did not steal the cookie from the cookie jar.
(Yes, I’m sure you can still figure out a way in which she stole the cookie. You’re very clever. This is just an example to illustrate the method.)
The urls are displaying as: https://arbital.com/learn/?path=$bayes_rule_details,$bayes_update_details,$bayes_guide_end
I.e., $-signs are being interpreted as math mode.
Eric Rogstad But… but… poset office was a pun, not a typo.
Yeah, I think keeping it as it is now is probably the best way of following the “one idea per page” methodology. The page on Products (mathematics) can have this page as child.
Eric Bruylant Whether Product (mathematics) is appropriate really depends if you’re asking a category theorist (who would say yes) or not . ;-)
In seriousness, specific kinds of products include cartesian products, products of algebraic structures, products of topological spaces and the most well known: product of numbers. All of these are special cases of the categorical product (if you pick your category right), but I can imagine someone wanting to look up ‘product’ as in multiplication and getting hit with category theory.
I don’t know. It’s a matter of taste I suppose. I get the idea that category theory is not yet quite widely-known enough for this to be considered “the” definition by most mathematicians, but if other contributors feel it should be given that status I certainly won’t complain. I just thought this was the safer approach.
See, for example product on Wikipedia.
Does the first question seem a bit much of a ‘gotcha’? I was slightly annoyed I got it wrong despite being quite capable of working with posets. XD
I suppose the way it is asked is completely valid, and will drive home the fact that relations have to be reflexive and illustrates very well what would be necessary to get a valid order.
What are other people’s thoughts?
(Maybe it will be better once it isn’t the only question, nor the first?).
Eric Rogstad Elmo comes to visit. Does that seem fine you think?
Orthonormal I think that’s a fair assumption for the moment. Later as Arbital grows the requisites can be refined to Calculus 0, 1, 2, 3; abstract algebra 0, 1, 2, 3 etc. Or else pages can just make use of specific pages as requisites (e.g. do you know what a ‘group homomorphism’ is).
Patrick Stevens I agree completely. Along with some other pictures. However, due tomy current circumstances I can’t make any pictures at thr moment.
If someone else is willing to, I would be very grateful. Otherwise I could probably do it in about a month? Month and a half?
Eric Rogstad The post has been updated with an isomorphic version of what you suggested. Thanks!
Joke stolen shamelessly from the latest post on slatestarcodex.com
I’d like to add some pictures to this page at some point, but due to current circumstances I can’t for now. If anyone wants to add pics (say different station maps with the same connections, two ‘boxes’ with random items) please feel welcome.
I also think I’ll change the names of the stations from a, b etc. to funny made up station names.
The majority of this page will probably end up in the least technical lens.
Eric Bruylant Thank you very much! just to be clear, are you talking about the ‘clickbait’, the intro paragraph in the text itself, or both?
Feel free to suggest / make your own changes if you have anything specific in mind by the way.
Patrick Stevens Yeah I’ve been wondering about the convention of things like this. I’ve been calling my pages things like category_mathematics.
Jaime Sevilla Molina I’ve submitted an edit to the page on morphisms and wrote an intuitive guide to isomorphisms. I hope it’s suitable for now.
My plan is also eventually to write an intuitive guide to categories with lots of concrete examples which hopefully tie in to some real-world ideas and are not as reliant on abstract mathematics.
However, I’d like to get some of the basic concepts down for the sake of the main lens. Also I’m formally trained in pure math so these are the setting and examples with which I’m more familar.
How can I get permission to edit this page, please?
I don’t understand? A morphism is just an abstract element of a category. Its behaviour is completely characterized by the axioms of a category. It would be like formally defining an element of a set.
This is still very much a work in progress. Anyone is welcome to submit more info or edit. I’ll add more details later. The current page probably won’t remain the main lens.
I was thinking of starting here, then splitting into lenses once the structure is more certain. Do you think I should do that earlier?