Okay, here's another kind of dumb $\infty$-category question. I feel mixed up about how to think of "commutative diagrams" in $\infty$-categories. In other words, back to my overcategory situation, how should I think of morphisms in a slice-$\infty$-category? Just 2-cells (with a fixed rightmost vertex of course), or something like... homotopy coherent diagrams of some kind?

In other words, the choice of filling 2-cell for a triangle may not be unique. So am I also choosing homotopies between all possible fillings when I say a diagram "commutes" in an $\infty$-category?

I guess the filling cell must be unique "up to a contractible space of choices."

@JonBeardsley Yeah, that's my impression. Giving such a 2-cell is precisely the same as giving a homotopy coherent diagram, and the weak Kan condition guarantees that the 2-cell and higher cells are determined up to contractible choice, so we don't have to worry.

No. There are many different choices in general, and each one determines a different morphism in the slice category.

Hm. Then I was under the wrong impression, apparently. I thought you could use the higher horn fillings to construct the needed 3-cells.

Oh, the 2-cell is not determined, of course.

@ZhenLin ok, so unlike in regular category theory, there are potentially many different ways a diagram can commute?

I mean, assuming I've fixed the boundary of the 2-cell

That seems to be what you're saying, but that seems dangerous to me somehow.

I guess to clarify, I'm saying you've handed me the information of $\partial\Delta^2$ and said "this diagram commutes in this $\infty$-category $C$." How am I to interpret this? And you're saying that there's non-contractible indeterminacy in this statement?

I'm not very fluent in \infty-categorical language, but I was very strongly under the impression that "this diagram commutes" is not a condition in that world.

You're right, it's a big glob of simplices.

Or at least, it should be.

I'm wondering HOW MANY simplices exactly, I guess.

@JonBeardsley yes: the set (or really space) of ways that it can commute is precisely the set of 2-cells extending the boundary!

i would say that "a diagram $\partial \Delta^2 \to C$" (for $C$ a quasicategory) is the data of two composable arrows $[f;g]$ along with another arrow $h$ from the first source to the second target. you can ask for the data of an equivalence between $h$ and "the" composite of $g$ after $f$, and that's the data of an extension over $\partial \Delta^2 \hookrightarrow \Delta^2$

@QiaochuYuan @ZhenLin okay right, thanks for clarifying (in re "total" categories). i thought this sounded a little fishy

@BenLim yes, i was there this past summer! send me an email and we can correspond about it privately

@JonBeardsley for an easy conceptual example, consider a composite $S^1 \to pt \to S^2$ and some arbitrary map $S^1 \to S^2$: certainly these are going to be homotopic (since $\pi_1(S^2) = 0$), but the space of homotopies is disconnected -- this comes from the fact that $\pi_2(S^2) \not= 0$

yes, indeed

incidentally, happy birthday @AaronMazel-Gee

Ok. Thanks very much @AaronMazel-Gee and @ZhenLin. So there are multiple ways that a diagram can commute. Weird.

And yeah, happy birthday!

The standard left-module/right-module notation confuses me. I.e. a right module is a module $M$ on which $R$ acts on the right, $M\otimes R\to M$. But I want it to be a module such that $M$ appears on the right side in the action diagram, i.e. $R\otimes M\to M$. What is the matter with me...

I think the train has sailed on that one, Jon.

I acknowledge and respect your use of mixed metaphors.

Or perhaps you really are referring to trains that sail:

Anyway, no time to talk, playing some serious solitaire right now.

the thing that annoys me the most about leftness / rightness is that sometimes they swap when you change your convention for the order of composition and sometimes they don't. like right now with my conventions the meaning of left / right adjoint doesn't change if i switch to diagrammatic composition, but the meaning of left / right module does

which is really weird and annoying when you want to make statements of the form "a (left/right) adjoint giving rise to a monad is a (left/right) module over that monad"

Is there a flavor of HoTT that uses directed spaces / (infty,1)-categories rather than spaces / (infty,1)-groupoids

(the notion of a "directed identity type" sounds a bit ridiculous, i will admit)

@Justaskin_ I think that Mike Shulman is working on "directed homotopy theory." I know close to zero details about it, but searching his name might be a place to start.

For instance, there's this: golem.ph.utexas.edu/category/2012/06/…

funnily enough, the first thing you have to give up is the identity type

@ZhenLin you don't have to give it up entirely, surely? You would just have to drop the construction rule from $a =_A b$ to $b =_A a$

that can be derived from the induction principle

it's funny because, at first glance, the induction principle looks directed

unrelatedly are there any homotopy flavored talks i should go to at JMM