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Stream: learning: questions

Topic: Source and target of 0-morphisms

Jonathan Arnoult (Oct 10 2024 at 08:27):

It's a bit of a degenerate case, but I'm trying to get what should be a good definition for the source and target of a 0-morphism (so that I could have a general notion of composition of k-cells for all $k\ge0$). ncatlab says that "it may be useful to recognise that every ∞-category has a (−1)-morphism, which is the source and target of every object.". I get that there is this thing about the (-1)-simplex being empty, so there should be a unique (-1)-morphism.
But in that sense, I think that composition rules would tell you that any two objects (as 0-morphisms) are composable, since their (-1)-source and target meet, which is not the case in practice. What am I missing?

On the other hand, could we not just introduce two degenreate cells for each level below 0 (let's say $\star s_k$ and $\star t_k$ for all $k < 0$), that would definitionally not be equal, and such that the source of any k-morphism for $k \le 0$ is $\star s_{k-1}$ and their target is $\star t_{k-1}$?

Mike Shulman (Oct 10 2024 at 15:27):

I think that nLab remark should not be taken too seriously, and maybe even deleted, for the reasons you give. I would only say that an $\infty$-category has a unique $(-1)$-morphism if it is monoidal.

Mike Shulman (Oct 10 2024 at 15:27):

Your other description does make sense: you can always take the "directed suspension" of any higher category by making it the only nontrivial hom-category of something with two objects.

John Baez (Oct 10 2024 at 16:01):

It's indeed extremely unwise to say every n-category has a unique (-1)-morphism $\star$ such that all objects are really morphisms from $\star$ to itself. It reminds me a bit of how in an augmented simplicial set there are "(-1)-simplices", and each 0-simplex has a (-1)-simplex as face. But in this setup 0-simplex has just a single face, not a source and target! Only when we get to 1-simplexes (edges) do they have two faces that can be seen as a kind of 'source' and 'target'.

Jonathan Arnoult (Oct 10 2024 at 16:56):

Mike Shulman said:

Your other description does make sense: you can always take the "directed suspension" of any higher category by making it the only nontrivial hom-category of something with two objects.

I actually realized that in my description you don't have identity morphisms at all levels anymore, since there is no identity 0-morphism on $\star s_{-1}$ for instance.

Mike Shulman (Oct 10 2024 at 16:58):

Oh, well, you can just add in those identities formally.

Jonathan Arnoult (Oct 10 2024 at 17:00):

But then you have more 0-morphisms than the true objects of your category?

Mike Shulman (Oct 10 2024 at 17:39):

Well, it's kind of a violation of type discipline to talk about the set of all $k$-morphisms of any category. In nearly all cases what you want to talk about is the set (or category, or whatever) of $k$-morphisms with some fixed domain and codomain, and the $\infty$-category you started with is indeed precisely one of the hom-categories of the directed suspension.