A number of people have very kindly sent corrections to the last version of Notes on Category Theory. There were some possibly confusing typos and also a downright wrong proof. Embarrassing. So here’s a “maintenance upgrade”, making some needed repairs.
[Added: a few more needed corrections are noted in the comments below.]
NB, as David Théret pointed out to me, on p. 123, in (2), repeatedly pairs $latex \langle X, 2\rangle$ should be products $latex X \times 2$ etc. And the counter example to products being preserved should involve finite $latex X, Y$.
More progress for me (thanks again)! Your notes keep being as good as I expected them to be. But also more typos:
Page 91 reads “[in a poset] the product of p and q must be their supremum”, but that should read the infimum/greater lower bound.
Page 95, proving Thm. 68, reads “u : Z -> O”, but “Z” should read “S”.
Page 111, definition 76: “A colimit for D is a terminal object in the category of cocoons under D.” Shouldn’t “terminal” read “initial” (since limits were terminal in the category of cones”, or is the category of cocones dual to what I expect? I don’t think, as you reveal when talking of initial objects: “So the category of cocones over the empty diagram is just the category C we started with, and a limit cocone is just an initial object in C !”
Moreover, Thm. 68 seems to have a much slicker proof at this point: we know universal elements are initial and we know products are terminal in the category of wedges, so we just need show that the category of wedges is isomorphic to (the dual of) the category of elements of F (which is just an exercise in checking the definitions); the dualization is only needed because F is contravariant.
I imagine you avoid this proof as too abstract, but it seems useful as an additional proof, also because it anticipates what we’re going to see later (with Theorem 79), and because if you discuss the categories of elements of F (an abstract thing) it can be helpful to see that it’s something more concrete already seen.
(Again, I’ve seen limits too many times by now, so I’m by far not the beginner reader).
Many thanks again for very useful corrections and comments — I’m very grateful!
A couple typos: on page 77, remark (3), “\hat{C} is just a relabeling… of [C, Set]” seems a typo for [C^{op}, Set].
Page 74 has “the the set”.
Finally, a genuine question: in 11.5 you write that the category of elements can be written down with the symbol of an integral. However, up to now in category theory I’ve always seen that symbol used for ends and coends, whatever they actually are (my only reference on ends is the nLab, which is not so helpful :-(). Is the terminology misleading, or is the category of elements indeed related to ends?
On the other hand, let me really thank you for Sec. 11.4. My previous meeting with universal elements was hosted by Awodey’s book, and I lost by knock-out.
Yet, that definition sounded awfully familiar (I’ve seen limits first). It took a bit of luck to find Theorem 79, confirming that limits are universal elements, so universal elements and universal properties are related. Then I’ve seen theorem 59! To be frank, I find theorem 59 much easier to remember than definition 44, which is scarily long and unmemorable.
Now, the next thing I expect to finally learn is the relation between limits (and universal properties) and adjunctions that MacLane keeps hinting at. I’ve gotten that e.g. products and sums can be defined by an adjunction (as in your Sec. 18.4). But I haven’t heard officially (yet) that when we say, for an arbitrary limit, for each Cone(D) there’s a unique arrow to the limit, we’re actually stating an isomorphism of homsets, hence (if the homsets have the right shape) an adjunction.
(I suspect what I’m looking for is just an explanation of nLab’s http://ncatlab.org/nlab/show/limit#global_definition_in_terms_of_adjoint_of_the_constant_diagram_functor, which seems within reach).
The link on https://logicmatters.net/categories/ to the notes is now broken – it’s just a 1 page PDF saying to check https://logicmatters.net/categories/ which is where I just came from! I think it should be to 6a not 6.
Oops, now fixed!