Are quasicategories (resp. kan complexes) (co)reflective in simplicial sets?

Question

This is definitely well known to experts, but I'm struggling to find a reference.

It seems intuitive that we should be able to "complete" any simplicial set into a quasicategory or a kan complex by "adding in" all the missing (inner) horns. This should be analogous to getting a groupoid by inverting all the arrows in a category, and so should be a reflector.

The question, then, is whether this can be made precise. Is the category of quasicategories (resp. kan complexes) a reflective subcategory of the category of simplicial sets?

As a ~bonus question~, groupoids are also coreflective in categories, since we can take the core of a category instead of the groupoid completion. Can we similarly restrict to some "core" of a simplicial set in order to see quasicategories or kan complexes as a coreflective subcategory of $s\mathsf{Set}$? I suspect the answer to this is "no", which is why I'm leaving it as a bonus question.

Thanks in advance!

Answer 1

No. This cannot be true, since neither Kan complexes nor quasicategories are closed under limits or colimits in simplicial sets. For limits, this can be simply seen by the fact that not every sub-simplicial set of a Kan complex or of a quasicategory has the same property. For colimits, the quotient identifying the vertices of the nerve of the groupoid representing an isomorphism is not a Kan complex or a quasicategory. The problem with your proposed reflector is that it gives a construction unique only up to weak equivalence, not up to isomorphism.

What is true is that Kan complexes are reflective and coreflective in the $(\infty,1)$-category of quasicategories. This is directly analogous to the situation for categories and for groupoids. Less intuitively, the completion of a simplicial set into a quasicategory or a Kan complex can be seen as the fibrant replacement functor in the relevant model categories.

Answer 2

There is an important error in your intuition that quasicategories or Kan complexes should be a reflective subcategory. You can, indeed, take a simplicial set and formally adjoin fillers for every (inner) horn. The problem, though, is that horn fillers are not unique, so this formal construction will not have the universal property you want. For instance, say you start with the horn $\Lambda^2_1$ and then formally adjoint a $2$-simplex filling the horn. If you map $\Lambda^2_1$ into a quasicategory with two different fillers for this horn, which one are you going to map your formally adjoined $2$-simplex to?

This is a common pattern: if you have subcategory of objects characterized by the existence of a structure that is not unique, it will typically not be reflective. Other examples of this include divisible abelian groups as a subcategory of abelian groups (in contrast with uniquely divisible abelian groups which do have a reflector given by tensoring with $\mathbb{Q}$) and path-connected topological spaces as a subcategory of topological spaces.

Of course, this on its own does not prove that the adjoint does not exist, but it strongly suggests that you should not expect it to exist. Typically, the easiest way to turn an idea like this into an actual proof is to get your hands dirty and find an explicit example of a limit that is not preserved, as in Kevin Arlin's answer.