For a bit I mistook the category of "Set" for the category of "Set and Injections". Which begs the question, how many functors are there "Set and Injections" -> 3.
In "Set and Injections" you can have the singleton sets serve as a sort of dual t…
Yup, you got it! My intuition is that splitting Obj_C into any desired two (empty or non-empty) sets picks a functor F -> I, but I'm wondering if there is some structure I am not seeing.
In b: am confused by what it means by "the constant functor" aren't there two constant functors?
Edit: I was confused because we went from a constant functor from Set_Bool -> Set_Bool to a constant functor from Type -> Type. I get now tha…
Hello everyone, sorry again for the late notice, but I'm hanging out in this virtual room working on PS2, if anyone wants to join and work along side me! https://zoom.us/j/4839633293
Hmm, I suppose you could look at it like I'm only defining one operation which is "absolute value of x - y" so all we have is |1 - 2| = 1, not the concept of -1. But if there is something wrong with this logic, I would like to know!
a:
Unit holds because \(0 + m = m \) and \(m + 0 = m \),
associativity holds because \((m_1 + m_2) + m_3 = m_1 + (m_2 + m_3) \)
b:
Unit holds because \([] ++ m = m \) and \( m ++ [] = m \)
associativity holds because \((m_1 ++ m_2) ++ m_3 = m_1…
Hi everyone, I'm hosting a discussion tomorrow at 10AM Central for a couple of friends. The plan is to review the first problem set and talk about any questions that have come up during the lecture. All are welcome to drop in, link below:
Jake Gill…
Can anyone think of an "almost-category" that satisfies everything but the right identity law?
(for any \(f : c \to d \), the equation \(id_c ; f = f \) holds but \(f ; id_d = f \) doesn't)
How about an "almost-category" satisfying everything but t…
@kenwebb Here is a hint: look at 2. in the definition of category above. For every pair of objects, we must have a set of morphisms (the set may be empty). There must be four pairs of objects in our category. Which pairs do you see?
I'll kick this off with my solution because this is a fun one to see a bunch of different answers
a) A single object category whose morphisms are the Natural numbers, composition given by \(x ; y = x * y \), and the identity morphism on the sole ob…
Thanks for setting things up! I like the idea of Blog post type discussions. It would be great if we could convince someone / collaboratively create "lecture" posts like Baez did for the "Applied Category Theory" Course.
@IssacDeFrain Yup, feel free to add your solutions or questions to the exercises on the exercise-specific thread (or create new exercise threads for exercises that you come up with or unposted exercises). Feel free to also contribute to the discussi…
Created a wiki-page to make navigation of this discussion page easier, similar to https://www.azimuthproject.org/azimuth/show/Applied+Category+Theory+Course:
https://www.azimuthproject.org/azimuth/show/Programming+With+Categories+Course
Hello! Attending from Boston, but virtually. Glad to see you all here :)
For the "7 sketches" course, each question (not each problem set) had its own post. Something similar might work well here.