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Stream: learning: questions

Topic: Really basic notation questions

Bobby T (Apr 03 2020 at 10:07):

You can't say I didn't warn you about laughably simple questions - here goes:
So I'm having trouble with a couple of concepts in intro textbooks, nameably Hom-functors and adjunction. Of course part of the problem is conceptual or otherwise comes from my difficulty finding an intuition for the ideas (and if anyone has any good ways of making these things "make sense", please do send them my way!). But I think one of the big issues for me is notational. I'm not familiar with this (-) notation, and every example I can find that tries to clarify Hom-functors or adjunction will always include it without explanation. I'm assuming that that means it's a common notation, even outside CT?
For example, the wikipedia article about Hom-functors describes Hom(A , - ) and Hom( - , B), and then sort of shows what results from this. The section in Simmons (2011) that dives into the finer parts of the definition of adjunctions (section 5.3 "Adjunctions uncoupled" for those following along at home) gives a commutative diagram like so:
image.png

Again, we've got these (-) symbols everywhere, but I don't actually know what they mean. As silly as this might sound, could someone just walk me through how they "work"? As in like a derivation, a substitution. I remember being absolutely baffled by the lambda calculus until someone just sat me down and said: "See that? The lambda is looking for an x. It'll eat whatever's next to it, and then turn into 'x + y', but replacing the x with whatever it ate."

I hope this isn't too basic to fit in the "basic questions" stream, but everybody's gotta start somewhere. Thanks in advance.

Matteo Capucci (he/him) (Apr 03 2020 at 10:15):

The $-$ is sort of standard notation for a placeholder. Writing $F(-)$ is like writing $\lambda x. Fx$.

Matteo Capucci (he/him) (Apr 03 2020 at 10:17):

To give an example from your picture: I assume $(A, GS)$ is an hom, then $- \circ k$ means "precompose with $k$". Indeed an 'argument' of that arrow is an arrow $\alpha : A \to GS$ and $k$ is an arrow $B \to A$, so
$(- \circ k)(\alpha) = \alpha \circ k = B \overset{k}\to A \overset{\alpha}\to GS$

Bobby T (Apr 03 2020 at 10:19):

Okay! So in the ordered pairs at the vertices of that commuting diagram, the first or second item is changing because it's the beginning or end of a path of morphisms composed together, right?

Matteo Capucci (he/him) (Apr 03 2020 at 10:20):

Precisely!

Matteo Capucci (he/him) (Apr 03 2020 at 10:20):

I think those things are super confusing to read, but really easy to write

Matteo Capucci (he/him) (Apr 03 2020 at 10:20):

Try to sit down and reproduce the diagram by yourself, by reasoning where morphisms go/start and writing down the 'stupid' details

Bobby T (Apr 03 2020 at 10:21):

Yeah, haha, my entire experience with CT has very much been reading a book written by someone who assumes I'm all caught up and they can cut a few corners for convenience haha

Bobby T (Apr 03 2020 at 10:21):

I'll go through the one on that page and check back if anything breaks haha. Again, thanks a million

Matteo Capucci (he/him) (Apr 03 2020 at 10:21):

Matteo Capucci said:

$(- \circ k)(\alpha) = \alpha \circ k = B \overset{k}\to A \overset{\alpha}\to GS$

Something that really helped be getting around that is denoting composition like this, since I can see what the arrows are doing

Matteo Capucci (he/him) (Apr 03 2020 at 10:23):

Another useful perspective I only learnt after, but which I think has a lot of untapped pedagogical power: you can always think 'pointwise' if you call arrows into an object 'elements' of the object. See 'generalized elements'

Bobby T (Apr 03 2020 at 10:24):

Yeah, I think with Hom-functors, I'm gonna need to take on a new perspective like that, because things get tricky quickly otherwise

Bobby T (Apr 03 2020 at 10:24):

Plus it kind of fits with the "diagrams are functors are diagrams" thing

Matteo Capucci (he/him) (Apr 03 2020 at 10:24):

Yeah basically $\rm{Hom}(-,X)$ is "the functor of elements of $X$"

Matteo Capucci (he/him) (Apr 03 2020 at 10:25):

Matteo Capucci said:

Another useful perspective I only learnt after, but which I think has a lot of untapped pedagogical power: you can always think 'pointwise' if you call arrows into an object 'elements' of the object. See 'generalized elements'

This is useful because it allows you to do element chase instead of diagram chase

Matteo Capucci (he/him) (Apr 03 2020 at 10:25):

And Yoneda guarantees it is consistent

Bobby T (Apr 03 2020 at 10:25):

And that would be related to the concept of the (either initial or) terminal object "picking out" the objects of the category?

Bobby T (Apr 03 2020 at 10:26):

I'm sure I'm explaining this poorly, but I think I'm on the same page as you

Bobby T (Apr 03 2020 at 10:27):

The distinction between objects and arrows is blurred because the object's identity is completely determined by its relationships, so the unique arrow pointing to an object is just as useful as the object itself. This is the intuition behind Yoneda, yeah?

Matteo Capucci (he/him) (Apr 03 2020 at 10:27):

Bobby T said:

And that would be related to the concept of the (either initial or) terminal object "picking out" the objects of the category?

Sort of. If $1$ is terminal, arrows $1 \to X$ are called "global elements". They are somewhat special, and in $\bf{Sets}$ they are the only elements you care about, because $1$ is a generator.

Matteo Capucci (he/him) (Apr 03 2020 at 10:29):

The first chapter of this explains really well how much you can pretend to have elements in a general category, and how much $\bf{Sets}$ is peculiar in the way it has elements:

• https://arxiv.org/abs/1012.5647

Bobby T (Apr 03 2020 at 10:29):

Alright, related basic question: how do you do the quote thing? I know it'll take some practice to get my latex down, but I figure the quoting in Zulip is something relatively simple

Matteo Capucci (he/him) (Apr 03 2020 at 10:32):

Bobby T said:

The distinction between objects and arrows is blurred because the object's identity is completely determined by its relationships, so the unique arrow pointing to an object is just as useful as the object itself. This is the intuition behind Yoneda, yeah?

See my previous message. Yoneda tells you $X$ is the "set" whose "elements" are the arrows it receives. But you really need arrows from every other object in the category to characterize $X$, unless your category is somewhat special and has special objects, called generators, whose arrows into $X$ suffice to characterize $X$ completely.

Matteo Capucci (he/him) (Apr 03 2020 at 10:32):

Bobby T said:

Alright, related basic question: how do you do the quote thing? I know it'll take some practice to get my latex down, but I figure the quoting in Zulip is something relatively simple

In the browser: hover on a message in the top right corner, click on the chevron, select "Quote and reply"

Matteo Capucci (he/him) (Apr 03 2020 at 10:34):

On mobile I don't know how to do it :(

Bobby T (Apr 03 2020 at 10:39):

Matteo Capucci said:

See my previous message. Yoneda tells you $X$ is the "set" whose "elements" are the arrows it receives. But you really need arrows from every other object in the category to characterize $X$, unless your category is somewhat special and has special objects, called generators, whose arrows into $X$ suffice to characterize $X$ completely.

Right, I'm overgeneralizing from the Set example. I'll go over Yoneda stuff again to make sure I'm catching the more formal aspects there as well. I've got the Leinster paper, and then I believe sarahzrf sent out a pdf about sheaves and Yoneda as well.

Again, thanks so much for taking the time here - this has been a huge help!

Bobby T (Apr 03 2020 at 10:39):

Oh damn

Matteo Capucci (he/him) (Apr 03 2020 at 10:40):

Use only three backticks, not 4

Matteo Capucci (he/him) (Apr 03 2020 at 10:40):

3 is quote, 4 is code I guess

Matteo Capucci (he/him) (Apr 03 2020 at 10:41):

Bobby T said:

Matteo Capucci said:

See my previous message. Yoneda tells you $X$ is the "set" whose "elements" are the arrows it receives. But you really need arrows from every other object in the category to characterize $X$, unless your category is somewhat special and has special objects, called generators, whose arrows into $X$ suffice to characterize $X$ completely.

Right, I'm overgeneralizing from the Set example. I'll go over Yoneda stuff again to make sure I'm catching the more formal aspects there as well. I've got the Leinster paper, and then I believe sarahzrf sent out a pdf about sheaves and Yoneda as well.

Again, thanks so much for taking the time here - this has been a huge help!

:) great! Take your time: it took me quite a lot of time as well!

Bobby T (Apr 03 2020 at 10:42):

I've tried 4, 3, 2, and 1 back tick haha. It's hopeless

Matteo Capucci (he/him) (Apr 03 2020 at 10:43):

Matteo Capucci said:

Use only three backticks, not 4

Nevermind, it has nothing to do with the number of backticks: you need to write "quote" after the opening

Matteo Capucci (he/him) (Apr 03 2020 at 10:43):

Like "quote .... "

Matteo Capucci (he/him) (Apr 03 2020 at 10:43):

Well, of course...

Bobby T (Apr 03 2020 at 10:43):

Yep, just realized the same. I think I love Zulip

Matteo Capucci (he/him) (Apr 03 2020 at 10:44):

Mmmh I like it but it's not really a very intuitive UX

Bobby T (Apr 03 2020 at 10:45):

Oh not at all, but it's good practice with all this kind of inline formatting, like the kind of practice I need in LaTeX :wink: :point_right:

Nathaniel Virgo (Apr 03 2020 at 15:46):

I don't know if this helps or not, but I found hom-functors a bit mysterious until I stopped thinking of them as mappings objects into arbitrary sets and started thinking of them as mapping into sets of morphisms instead. So if you have a category like this
image.png

then Hom(X,Y) is literally the set of arrows, drawn on the page, from X to Y.

(Of course, category theory doesn't care what's in the sets - it only cares how many members they have and how they're related by functions - but that means you're free to think of the elements as whatever you like, and I found that image made it much easier to understand.)