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"). The confusing part (also for me) is that the only node could be identified with anything. Perhaps this could help: math.stackexchange.com/a/1332693
My gut instinct is that your correct. Your absolutely right in spring that the bottom of what the nodes and arrows represent does not need to match how is thought of in the algebraic world. ncatlab.org/nlab/show/category+theory "The classical examples of categories are concrete categories whose objects are sets with extra structure and whose morphisms are structure preserving functions of sets, [...] But it is far from the case that all categories are of this type. Categories are much more versatile than these classical examples suggest. After all, a category is just a quiver (a directed graph) with a notion of composition of its edges."
This is incorrect. In bartosz example, you can either say that a set monoid is always a category by first defining the set monoid, ie integer addition, and then partially applying addition to get a morphism which you can then easily compose and prove is a category. The other way without a set category is to look at the hom set and set that every small category has a hom set and a monoid small category only has C(m,m) which consists of all the morphisms from m to m. The binary operator for this set is obviously compose, thus every one element small category is a monoid. The idea being that a small category can be transformed whenever you want into a hom set, basically making them isomorphic
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!
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
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.
Not quite; groups have one more item of structure. In a monoid you don't need to have inverse elements, but in a group, every element has an inverse element such that the two combine to the identity element.
I am never going to be able to look at Bartosz again without thinking of Inigo Montoya!
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").
The confusing part (also for me) is that the only node could be identified with anything. Perhaps this could help: math.stackexchange.com/a/1332693
My gut instinct is that your correct. Your absolutely right in spring that the bottom of what the nodes and arrows represent does not need to match how is thought of in the algebraic world.
ncatlab.org/nlab/show/category+theory
"The classical examples of categories are concrete categories whose objects are sets with extra structure and whose morphisms are structure preserving functions of sets, [...] But it is far from the case that all categories are of this type. Categories are much more versatile than these classical examples suggest. After all, a category is just a quiver (a directed graph) with a notion of composition of its edges."
This is incorrect. In bartosz example, you can either say that a set monoid is always a category by first defining the set monoid, ie integer addition, and then partially applying addition to get a morphism which you can then easily compose and prove is a category. The other way without a set category is to look at the hom set and set that every small category has a hom set and a monoid small category only has C(m,m) which consists of all the morphisms from m to m. The binary operator for this set is obviously compose, thus every one element small category is a monoid. The idea being that a small category can be transformed whenever you want into a hom set, basically making them isomorphic
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!
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
He's not correct though, look at my reply
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.
hi, anyone has a link to this article of Polymorphism mentioned? Original link is not accessible anymore
5:30 Godammit. Ben Deane made a latin joke.
Is a monoid the same thing as a group?
Not quite; groups have one more item of structure. In a monoid you don't need to have inverse elements, but in a group, every element has an inverse element such that the two combine to the identity element.
Nice video
Good