So I just started learning about quasicategories... Alright, that's an understatement: I just listened to Julie Bergner talk about quasicategories, and then started reading Moritz Groth's short course about them. I did a quick google search for this question, but there's a high probability that I just wasn't using the right terminology, or that I'm being silly. In any case, here is my question:
Suppose you have yourself a quasicategory (or $\infty$-category, or weak kan complex, whatever you call it). Is it always possible to embed this into a finitely bicomplete, pointed quasicategory? (I purposefully didn't ask for a way to make it a stable $\infty$-category, though that may be equally possible). An existence statement would be nice, a construction or reference to an instruction would be even nicer.
A bit more down to earth: Can this even be done with small categories? If you have a small category, can you always add objects to it in some way to make it finitely bicomplete? I suppose if I wanna sound fancy then I might want to be asking for an adjoint to the inclusion functor from complete categories into categories... or something like that.
There's a good reason I'd like to know all of this... but it's Level 1 Classified by the NSA.
(In other words... I'd like to think about it a bit more to make sure I don't embarrass myself playing with machinery I don't understand yet.)
NB: Easy on the higher categorical language... I'm good with categories, and I'm good with simplicial sets. If you need to use something bigger, then please give a reference, or a gentle description. Thanks!