Do quasi-categories have a `completion'?

Question

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!

Answer 1

Maybe this is not exactly what you are looking for, but it might be a place to start.

There is an analog of the Yoneda embedding for quasi-categories which takes the form $$j : \mathcal C \rightarrow Map(\mathcal C^{op}, \mathcal S)$$ where here $\mathcal S$ is the quasicategory of spaces (that is, the coherent nerve of the category of Kan complexes.) This map is an embedding which can be regarded as the free completion of $\mathcal C$ with respect to colimits, just as in ordinary category theory, and moreover preserves all limits which exist in $\mathcal C$. This is of course quite a bit more than just finitely bicomplete, but Lurie's book on Higher Topos Theory outlines a method for adjoining any class of colimits, and presumably this would work for finite colimits as well if that's what you want.

Answer 2

About completion of (simple) categories see (my try) in:

Equalizer completion

In Adamek, J. and Rosicky, J., Locally presentable and accessible categories, Cambridge University Press. (1994)

you find interesting kind of completions (category).