Commutative diagram on slide #15
kputnam opened this issue · comments
I'm really enjoying your slides so far. Perhaps I've misunderstood something, but the diagram of function composition on slide 15 was confusing: should the lower-left corner be labeled C instead of B? The diagram on the previous slide shows g o f : A -> C. If so, then should C in the lower-right be relabeled D?
Sorry for the late reply, I wasn't notified.
So, the diagram contains all necessary informations. From the diagram:
- f: A -> B
- g: B -> C
- h: C -> D
Then the diagram state that, each time you have such three morphisms, you must have:
- g.f: A -> C
- h.g: B -> D
by definition of composition, and h.(g.f) must be equal to (h.g).f.
By definition of composition we have both, h.(g.f):A -> D and (h.g).f: A->D
But their equality is necessary between all morphisms inside a Category.
So the diagram is in fact correct. The confusing part (for me at least) is the reverse order of function applications with the composition. h.g.f, means, apply f, then g, then h.
Thank you for your reply. I realized just now the slide I'm referring, "Commutative Diagrams" to is actually labeled "16" in the corner of the slide (my confusion was because the URL ends with #slide-15). I understand your answer completely if you're talking about the slide titled "Category laws: Associativity".
On the next slide, "Commutative Diagrams", if I am interpreting the diagram correctly, it says h.g: B -> C and also g.f: A -> B. Perhaps I'm still misunderstanding the diagram though.
Oh, yes, good catch, thanks. I fixed it now :).