Let me start out with a confession. I have never cared much for set-theoretic size issues, for they seem not to cause much trouble in my day-to-day mathematical life. Despite that, I have always been uncomfortable with their existence, although this discomfort is mostly rooted in a personal aesthetic ideal: I see technical digressions on cardinals, universes, and transfiniteness as a stain on otherwise clean mathematical theories.
In the literature on $\infty$-categories, a great deal of attention appears to be given to so-called presentable $\infty$-categories. As a reminder, we say an $\infty$-category $\mathcal{C}$ is presentable if it has all small colimits, and there exists a regular cardinal $\kappa$ so that $\mathcal{C}$ can be realised as the category of $\kappa$-small $\operatorname{Ind}$-objects of some small $\infty$-category. While in most mathematical areas I am interested in, size contraints are of only minor interest, in higher category theory, they appear to arise a lot, usually in the form of requiring certain $\infty$-categories to be presentable. As an example, in Lurie's Higher Algebra, the word 'presentable $\infty$-category' appears hundreds of times. Being such a common occurrence, I can no longer sweep these constraints under the rug: I have to face reality.
Having swallowed this set-theoretic pill, the solution appears simple: I shall make myself remember the definition of presentability, and apply the truths as decreed by Lurie and others only to those $\infty$-categories which either Google tells me are presentable, or which I have verified to be presentable myself. Several problems arise, however, and they are the motivation for the questions (or rather, three perspectives on a single question) that I wish to ask.
- I fail to truly understand the definition. I have only a limited conception of what a cardinal really is, let alone 'regular cardinal'. I have neither feeling for the definition, nor for determining whether a given $\infty$-category satisfies it.
Question 1. What is the intuitive idea behind presentability? What does it mean to presentable? How do I recognise if a given $\infty$-category is presentable?
- I fail to truly understand the motivation behind the definition. I am not well-versed enough in the concepts involved to understand what goes wrong if we do not assume presentability, nor do I understand why presentability is defined the way it is. What if we drop 'regular' in 'regular cardinal'? Why not realise $\mathcal{C}$ as $\operatorname{Pro}$-objects?
Question 2. Why is the definition of presentable the way it is, and not something slightly different?
- I fail to see the beauty in the definition. Although this one is purely subjective, I hope someone feels what I feel. There are these beautifully clean theorems in higher category theory — theorems which have nothing to do with size — that convince me straight away that $\infty$-categories truly are natural and intrinsically simple things, but on top of that one has size constraints all over the place. They simply feel in dissonance with an otherwise flawless theory.
Question 3 (optional). In your opinion, why is presentability a natural, and aesthetic definition?
Best Answer
Presentable $\infty$-categories can be understood without every having to think about cardinals. An $\infty$-category is presentable iff it is equivalent to one of the form $\mathcal{P}(C,R)$, where
That's it. The conditions that $C$ is small and $R$ is a set allow you to show that the inclusion $\mathcal{P}(C,R)\to \mathrm{PSh}(C)$ admits a left adjoint, which implies that $\mathcal{P}(C,R)$ is complete and cocomplete, which is something you definitely want.
"Presentable" should be thought of in terms of "presentation", analogous to presentations of a group. In some sense $\mathcal{P}(C,R)$ is "freely generated under colimits by $C$, subject to relations $R$". More precisely, there is an equivalence between (i) colimit preserving functors $\mathcal{P}(C,R)\to D$ to cocomplete $\infty$-category $D$, and (i) a certain full subcategory of all functors $F\colon C\to D$ that "send relations to isomorphisms" (precisely: those $F$ such that $\widehat{F}(f)$ is iso for all $f\in R$, where $\widehat{F}\colon \mathrm{PSh}(C)\to D$ is the left Kan extension of $F$ along $C\to \mathrm{Psh}(C)$).
So its easy to construct colimit preserving functors from presentable categories (and all such functors turn out to be left adjoints).