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
@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.