@TomBachmann This looks like this might be exactly what I need. Thanks!

@HarryGindi .

thanks

@CWcx The morphism W(F) → F is not pro-smooth, by the way. Pro-smooth morphisms are always flat.

@DenisNardin @HarryGindi For what it's worth Lurie uses the term "replete subcategory" in SAG 20.1.1 for the "correct" (up to equivalence) notion of subcategory. One can also say they're functors inducing monomorphisms on spaces of objects and spaces of arrows.

@MarcHoyois Isn't this asking for the subcategory to contain all equivalences between its objects? (this condition seems reasonable to me, and it kills off my pathological example, but it's not the "standard" one I think)

Indeed, I personally wouldn't call it is a subcategory otherwise.

Is there even an invariant notion of "subcategory" which is more general than this?

I noticed that Denis's example actually works for 1-categories with no ∞-component at all, so I think the issue is that the classical notion of subcategory is not really equivalence-invariant. Therefore, it makes sense to give a new name for such a notion.

I would rather say that the classical definition should be adjusted, now that we better understand what categories really are (as opposed to "strict categories").

Sure, but for better or for worse, the places where most people learn about category theory first are in books by deceased mathematicians (Lang, Mac Lane) that would be hard to reissue, so I think we're stuck with it (at least for the foreseeable future) by sheer inertia.

So I think Jacob probably made the right call by having his definitions line up on-the-nose with the terminology most people already know well, even if it's not the best definition.

I somewhat disagree. The classical notion of subcategory seems mostly useless (even more so in higher category theory), so I think at the very least we shouldn't use it in cases where it doesn't match the invariant notion.

@MarcHoyois If you asked me I'd say that what I want to call an "invariant subcategory" is a functor F:C→D that's faithful and conservative

I don't think I want the subcategory to contain the full automorphism group of every one of its objects

But you're right that "replete" is probably the better behaved notion

Isn't any map between sets both faithful and conservative?

@DenisNardin That is what we mean by "subspace" though.

@MarcHoyois Hmm... I guess. I suspect I'm just used to not consider a subcategory as a subobject, (while I do want subspaces to be subobjects), because the 1-categorical notion is not a subobject (up to equivalence)

Surely not every set should be a subcategory of the one-point set though?

@MarcHoyois That's a fair point

When somebody says that a map $f: X \to Y$ is "$\pi_1$-essential", do they mean that it is injective on $\pi_1$, or just that it is nonzero on $\pi_1$?

@BrunoStonek nice!

i would advocate for the convention that the prefix "sub" always refers to monomorphisms. so, (thinking invariantly) a "sub-$\infty$-category" should mean what marc said. if you are learning about classical category theory, this is still valid because you are studying the 1-category of strict categories, rather than its homotopy category / localization.

@MarcHoyois Ahh your right! thanks for pointing that out.