Category Theory for Programmers: Chapter 3 - Categories Great and Small

[amydebuitleir]

I am never going to be able to look at Bartosz again without thinking of Inigo Montoya!

[tenthlegionstudios]

Really nice to see the actual category table for the bools. He mentions these in his talks but I had a hard time grasping what a category table would look like. Thanks for the example!

[japedr]

Standard disclaimer: IANAM (I am not a mathematician).

9:00 I think that part of the confusion arises from the natural urge to identify the morphisms (or arrows/edges) as functions and the objects (nodes) as sets, which is the correct way if we use the standard definition of monoid as an algebraic structure; but this doesn't have to be the case, remember that objects and arrows only describe a graph and the only rule is that of arrow composition. If I understood Bartosz's explanation correctly, in this category the arrows are the elements of the monoid (e.g. integers) and the function composition is identified with the binary operation (e.g. sum). Of course, your interpretation is consistent because in "algebraic" monoids, elements can be represented as (i.e. are "isomorphic to") the "unary" operation of addition to a fixed value (this is similar in vector spaces to the concept of "linear forms" and "dual space").

[jiraguha]

For 9:00, I 100% agree with @jp48! Bartosz lectures, start by explaining that objects and arrows are primitives of the category theory. The definition of those things is what you give up when you reach that level of abstraction. IMO the simplest definition of Monoid is “a category of one object”… The definitions given by Den and Dave are consequences of category axioms applied to set theory.

- Identity
- composability
- associativity

[retagainez]

These challenges really tripped me up because it gets a little vague as to what Bartosz is actually looking for in the form of an answer. Well, if you just stick to drawing the objects and their morphisms it's not all too bad. What about discussing associativity and composition for question 5? Wasn't too sure how to show associativity.

[pmcgee]

5:30 Godammit. Ben Deane made a latin joke.

[RobotProctor]

Is a monoid the same thing as a group?

[raviskolli]

hi, anyone has a link to this article of Polymorphism mentioned? Original link is not accessible anymore