Take the 2-minute tour ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

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!

share|improve this question
add comment

2 Answers

up vote 3 down vote accepted

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.

share|improve this answer
    
Excellent! Precisely the type of thing I was looking for. Thanks much! –  Dylan Wilson Nov 15 '10 at 4:35
add comment

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).

share|improve this answer
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.