@JoeBerner it's an important thing to learn and ultimately part of the job. earlier is easier than later

not that you don't know, but just to say it again

Ive been told that "pointed spaces" is a presentable $\infty$-category. I want to believe this but I can't see in concrete terms what could be the generating set for this category.

Also there's this fact that classical brown representability doesn't work (to my knowledge) for the homotopy category of pointed (not nessasarily connected) spaces.

@SaalHardali the generating object is S^0. To have Brown representability you need your generating objects to be cogroup objects in the homotopy category, that's why it doesn't work there

I see. In what generality is it true that if I have a set of objects in my category whose left orthogonal is empty then they generate the category?

I think you need the cstegory to be stable cocomplete and the "generating" objects to be compact

Aha, thanks for the help!

@SaalHardali you probably know this, but presentable categories are stable under various constructions; see e.g. HTT section 5.5.3. In particular, if C is presentable then so is C_{c/} for any object c in C.