Does anyone know a characterization of those flat morphisms of qcqs schemes $f:X→Y$ such that $f_*:D(X)→D(Y)$ is fully faithful? Is flat monomorphism enough?

Examples are inclusions of quasi-compact opens and maps $\operatorname{Spec}\mathcal{O}_{X,x}→X$ for $x∊X$.

@DenisNardin If it's a flat monomorphism then $f^*f_* = id$ by flat base change, right?

@RizaHawkeye Ah that seems to be doing the trick. Thank you!

let C be a quasicategory given as a limit of a diagram of quasicategories (I care about the sequential case). what's the easiest way of seeing that the mapping spaces in C are limits of the mapping spaces in the components of the diagram?

@BrunoStonek Let $C$ be a quasi-category, and, $x,y \in C_0$ be two objects. Then $C(x,y)$ can be modelled as $\{(x,y)\} \times_{C \times C} C^{\Delta^1}$. If you consider a limit of quasi-categories, then the hom space in this limit is a limit of pullbacks.

thanks. that's what I was thinking of but was afraid I might be ignoring some subtlety...

limits of quasicategories make me nervous

Limits of q-categories are fine. It's colimits that ought to make you nervous :)

what's the deeper reason, quasicategorically? in complete segal spaces, I get it that they're a localization of simplicial spaces, so that explains why limits are easy

(the reason for my feeling uneasy is because I feel I get too close to foundational, model-dependent stuff, which always makes me uncomfortable!)

I think Adrian's explanation is as foundational as it gets (and it's pretty much the same explanation as in CSS)

@BrunoStonek Hmmm... well here is a subtlety: You want your diagram to be sufficiently fibrant so that its limit is a homotopy limit. You need to think about why you also get a good diagram of hom-spaces.

But I guess you could say that the functor from "doubly pointed q-cats" to spaces sending $(C,x,y)$ to $Map_C(x,y)$ has a left adjoint given by sending a space $X$ to the q-category with two objects, only identity endomorphisms and $X$ as a mapping space

@AdrianClough Nah. the formula you gave is a homotopy pullback, so everything is good

@DenisNardin Great!

@BrunoStonek A deeper reason could be this: Categories are in some sense algebraic objects, and limits of algebraic objects are usually easier than colimits. In classical situations you will have some forgetful functor to sets from groups (rings, modules, etc.) which commutes with limits but not with most colimits. If you want to compute colimits you may observe that these are simple on free objects, and that arbitrary colimits can be obtained from colimits of free objects cont.

(consider e.g. the free product of groups). The same is true for quasi-categories. As these are fibrant objects they are closed under limits. Taking colimits amounts to taking colimits in the category of all simplicial sets, and here we view simplicial sets as specifying generators and relations for an $\infty$-category. Thus when you compute a (homotopy) colimit you end up with some thing that is not fibrant, and you need to fibrantly replace, in order to get the quasi-category you want.

This all reminds me of a thing I have been wondering about: It is true that filtered colimits in groups (say) commute with the forgetful functor to sets. It turns out that filtered colimits in simplicial sets with the Joyal model structure are automatically homotopy colimits. This, a priori, has mostly to do with the fact that this model category is combinatorial. However, it could still be that these two facts are related. Has anyone thought about this?

Btw. since this started with a question on hom sets, it might be worth pointing out that hom sets of filtered colimits of $\infty$-categories are filtered colimits of hom sets as discussed in people.math.harvard.edu/~gaitsgde/GL/colimits.pdf

@AdrianClough This follows also from the formula you gave above, since Δ¹ and Δ° are compact objects (in fact compact generators) in Cat

Or, I guess, by the variant Map(x,y)={(x,y)}×_{iC×iC} Map(Δ¹,C), but it's pretty much the same idea

@DenisNardin Neet!

I wish we had an axiomatic framework for ∞-categories where all these facts could be just derived from first principles. Oh well, someone will develop that eventually

General question: Suppose C is an ordinary category with a model structure and D is its infinity category formed by localizing the weak equivalences. If F : C -> Set is a representable functor, are there known sufficient conditions to ensure that its left derived functor LF : D -> sSet, where of course sSet is the infinity category of simplicial sets is also representable?

In this later case, I suppose it is only meaningful to ask this for objects of D, that is derived Yoneda would not kick in if one asked only that there is an object X of D so LF(A) = Hom_D( A, X) for all objects A in C, but if I'm mistaken and this sort of weak form of Yoneda is interesting or valuable, I'd love to know.