Limits in an $(\infty,1)$-category

Question

In ordinary category theory, the notion of limit in a category $C$ is usually formulated with a category (of indices) $J$ and a functor $F:J\to C$ (a diagram in $C$), and a limit of this diagram is something satisfying some universal property.

In the context of quasi-categories (I looked in the nLab and in Higher Topos Theory), the category $J$ is replaced by a simplicial set $K$ and the functor $F$ is replaced by a map of simplicial sets $K\to C$ (a quasi-category is a particular simplicial set). But this definition is specific of quasi-categories, if you take another model for $(\infty,1)$-categories, it does not make sense to talk about a map between a simplicial set and an $(\infty,1)$-category.

Is there a definition of limits for $(\infty,1)$-categories independent of the model?

In particular, can we replace $K$ by an $(\infty,1)$-category and $F$ with an $(\infty,1)$-functor? If we can, why does everybody take a simplicial set instead (for quasi-categories)?

Answer 1

Definitions

I think there is a definition that should fit into most models of $(\infty,1)$-categories. If you want an "elevator speech" answer, it's:

Definition. A limit of a diagram $\mathcal D \to \mathcal C$ is a terminal object in the $(\infty,1)$-category of objects living over $\mathcal D$.

This is the (almost naive) generalization of one definition for limits in usual category theory. But let me elaborate on this definition.

(Full disclosure, I almost always work with quasicategories, so I anticipate that a better answer can be given by someone who's worked with all models. I hope this answer will be helpful regardless. Also, when you say "limit," I assume you mean homotopy limit. I sometimes distinguish between the two since not everybody is happy when I use classical terminology with an implicit "$\infty$" or "homotopy" before every word.)

Morally speaking, a good model for "$(\infty,1)$-categories" should have definitions for the following ideas:

1. Mapping spaces. That is, given an $(\infty,1)$-category $\mathcal C$, between any two objects $X,Y$ of $\mathcal C$, a topological space of morphisms ${\mathcal C}(X,Y)$. This is a fairly obvious pre-requisite because $(\infty,1)$-categories are supposed to be like categories enriched in spaces.
2. Terminal objects. Morally, these are objects $\ast$ such that for any other object $Y$ in your $(\infty,1)$-category $\mathcal C$, the mapping space ${\mathcal C}(Y,\ast)$ is contractible. There may be more subtle issues involved in defining terminal objects properly, depending on your model, but at least in the case of quasi-categories, it turns out this moral definition is perfectly fine as an actual one. (See Corollary 1.2.12.5 of HTT.)
3. Under/Over-Categories, aka Cone Categories. Given two $(\infty,1)$-categories $\mathcal C$ and $\mathcal D$, an $(\infty,1)$-category of ( $(\infty,1)$-) functors between them. And in our discussion, we specifically want the following: Given a diagram ${\mathcal D} \to \mathcal C$, a good notion of an ( $(\infty,1)$-)category whose objects are functors from $\ast \star {\mathcal D}$ to $\mathcal C$, where $\ast \star {\mathcal D}$ is the category obtained by affixing an initial object to $\mathcal D$. This is the same thing as the category of objects of $\mathcal C$ equipped with a map to the diagram $\mathcal D \to \mathcal C$.

You can see why this third point, about cone categories, is so simple in the quasi-category model. It is as simple as defining the join of simplicial sets, and knowing what the mapping space is between simplicial sets.

Anyhow, if you believe that your model (whatever it is) has definitions for the above three things, you can define a limit to be a terminal object in a cone category. You can dually define colimits as initial objects in an undercategory.

Actually proving that (co)limits are preserved.

I assume you wanted an answer that was more specific about actually computing (homotopy) limits using different models (complete Segal spaces, quasi-categories, Kan simplicial categories, et cetera) but I'm afraid I don't know much about comparing homotopy limit computations in different models. Lurie does, however, prove in HTT (Theorem 4.2.4.1) that the usual homotopy (co)limits you'd compute in a category enriched over Kan complexes will agree with the homotopy (co)limits you'd compute in the quasi-category model. So that's a good start! And if you believe in the equivalences between different models of $(\infty,1)$-categories (see for instance Julie Bergner's "A Survey of $(\infty,1)$-Categories") then the equivalences should preserve initial objects of cone categories, so this would be an argument that all models preserve (homotopy) (co)limits.

Why "everybody" takes simplicial sets.

Actually, a lot of people prefer to use other models like the Segal space model. But you can see that with the combinatorics of quasi-categories, a lot of things can be defined and proved fairly cleanly, as I pointed out in some of my commentary above. So that's one advantage of Joyal's quasi-category model. But there are many situations in which the space of objects is so naturally a space that you might prefer a model which isn't based on weak Kan complexes. For instance, in Galatius-Madsen-Tillman-Weiss, they think of the category of cobordisms as a category with a space of objects and a space of morphisms. This model might make it easier, for instance, to compute the classifying space of an $(\infty,1)$-category. And if you were interested in computing a (co)limit of a functor mapping such an $(\infty,1)$-category into another, you wouldn't want to say that your diagram comes from a simplicial set.

Simplicial Sets as the Diagram

Also, it seems you're interested in why Lurie takes as the diagram a map $\mathcal{D} \to \mathcal C$ in which $\mathcal D$ is a simplicial set. I don't think I would take "simplicial set" as the important idea here; I think it's more that $\mathcal D$ is an $(\infty,1)$-category, so for Lurie, it's a weak Kan complex, and in particular a simplicial set. I'm not sure (as Moosbrugger alludes to above) why the generality of an arbitrary simplicial set is so important, but in general I think we would take a diagram $\mathcal D \to \mathcal C$ to be a map of $(\infty,1)$-categories. That is, $\mathcal D$ being a simplicial set isn't so important for this definition. It just needs to be an $(\infty,1)$-category in whatever model you're using, and in the Lurie example, you should probably think of it as a quasi-category, rather than just an arbitrary simplicial set. And you can always replace an arbitrary simplicial set with a weak Kan complex (these are the fibrant-cofibrant objects in the Joyal model category.)

Answer 2

I just want to make the point that (co)limits in $(\infty, 1)$-categories are not that scary. In classical category theory, it's well known that once you know how to take limits in sets, then you know how to take limits in any category -- or at least, how to recognize them. For instance, if you want to take the product of $X$ and $Y$ in an arbitrary category, then homming into $X \times Y$ is the same as homming into both $X$ and $Y$. I.e., $\hom(Z, X) \simeq \hom(Z, X) \times \hom(Z, Y)$ as functors. What happens in $\mathbf{Sets}$ determines everything else.

For $(\infty, 1)$-categories, what happens in $\mathbf{Spaces}$ determines everything else, except you have to replace "limit of spaces" by "homotopy limit of spaces." Granted, these are a bit more exotic than ordinary limits, but you can at least compute them (using things like the Bousfield-Kan formula). So, for instance, if you have a fiber product diagram $X \to Z, Y \to Z$ in some $(\infty, 1)$-category, then the fiber product $X \times_Z Y$ is the one that represents the space-valued functor $\hom(., X) \times_{\hom(., Z)} \hom(., Y)$ where the product there is the homotopy fiber product of spaces. This determines your object up to equivalence: it has to be this way, since homotopy limits are defined only up to equivalence (though sometimes one chooses canonical choices, as with Bousfield-Kan).

Notice I'm using the property that mapping spaces exist, which is supposed to be a defining property of $(\infty, 1)$-categories. Note also that this actually works if you compare simplicially enriched categories (well, at least fibrant ones, I think) with quasi-categories.

In practice, my impression is that one usually takes limits or colimits over the nerve of ordinary categories anyway, even in this higher categorical setting.

Answer 3

One notion which is sometimes convenient, and can be considered to "underlie" all notions of $(\infty,1)$-category, is a derivator. A prederivator is just a 2-functor $Cat^{op} \to Cat$ (modulo size questions). And any $(\infty,1)$-category, in any model, will have an underlying "representable" prederivator defined by $X\mapsto Ho(C^X)$, where $C^X$ is the "functor $(\infty,1)$-category" in whatever model you've chosen and $Ho$ denotes the homotopy category of an $(\infty,1)$-category.

A derivator is a prederivator satisfying axioms saying that (among other things) all limits and colimits exist. But it is also possible to characterize when a particular limit or colimit exists in a prederivator; see for instance here. One expects, though I don't know to what extent it has been proven, that a limit in any model for $(\infty,1)$-categories will give rise to a limit in the underlying prederivator. I don't know about the converse, but for many applications of limits and colimits, the structure of a derivator is sufficient.

Answer 4

Here's how you describe $K$-indexed diagrams in an arbitrary ``theory of $\infty$-categories."

Suppose you have a "theory of $\infty$-categories" (i.e., $(\infty,1)$) in whatever sense. All I need for this discussion is that it contains usual categories, that there's an $\infty$-category of $\infty$-categories, and a notion of (homotopy) colimit (though obviously one would prefer more axioms for a more serious axiomatization).

Then attached to any simplicial set $K$ there will be an $\infty$-category $\mathscr{C}_K$ defined as follows. $K$ is a functor $\Delta^{op}\to\operatorname{Sets}$, and we can consider sets as particular kinds of $\infty$-categories (disrete groupoids, if you like). The colimit of the induced functor $\Delta^{op}\to\operatorname{Cat} _{\infty}$ is our $\mathscr{C}_K$.

Then in Lurie's picture, the $\infty$-category you attach (in your fixed theory) to the quasi-category of $K$-diagrams in a quasi-category $\mathscr{D}$ should be equivalent to diagram indexed by $\mathscr{C}_K$ in whatever corresponds to $\mathscr{D}$ in your fixed theory of $\infty$-categories.

So the point is that it can only possibly provide a little bit of convenience to allow such things, and probably the convenience is not so terribly great. E.g., it allows you to speak of diagrams indexed by something like $\Delta^1\underset{\Delta^0}{\times}\Delta^1$, which obviously isn't a quasi-category, but is combinatorially much simpler than whatever alternative.