How should I think about presentable $\infty$-categories? 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.


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*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? 


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*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?


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*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?
 A: Just to be a little bit contrary, let me point out one concrete reason that locally presentable categories are not aesthetic: it is unclear whether they have any analogue in constructive mathematics.  Most of basic category theory is entirely constructive (the classic quip is "have you ever seen anyone prove that a diagram commutes by assuming that it doesn't and deriving a contradiction?").  But once you move into locally-presentable world, all that goes out the window: the well-ordering and transfinite iteration techniques, and the theory of ordinal and cardinal numbers that underlie them, rely relentlessly not only on the law of excluded middle but also the axiom of choice.  It's not impossible that there is an analogous theory constructively, but as far as I know, no one has ever managed to write one down.
A: A "cleaner" form of the definition: Under Vopenka's principle the following are equivalent for a category $\mathcal C$:


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*$\mathcal C$ is locally presentable;

*$\mathcal C$ is cocomplete, and equivalent to a full subcategory of the category of presheaves on a small category;

*$\mathcal C$ is complete, and equivalent to a full subcategory of the category of presheaves on a small category.


Aside: You might reasonably object that assuming Vopenka's prinicple just gets us mired in more set theory, but think of it like this: Grothendieck introduced his universe axiom (rediscovering the fundamental notion of an inaccessible cardinal) precisely to minimize the amount of set theory one must think about. Likewise, assuming Vopenka's principle here allows us to talk about locally presentable categories without mentioning regular cardinals. End aside.
I'm not sure an analogous theorem has been proven $\infty$-categorically, but the intuition stands: a category $\mathcal C$ is locally presentable if it is cocomplete (equivalently, complete) and has a small subcategory $\mathcal G$ which "controls" $\mathcal C$ to the extent that the restricted Yoneda embedding $\mathcal C \to Psh(\mathcal G)$ is fully faithful -- this is called being a densely generating subcategory (though I think unfortunately the term strong generator has been used in the $\infty$-categorical literature, which classically means something a bit weaker). Another way to think about this is that every object of $\mathcal C$ is a colimit of objects of $\mathcal G$ in a canonical way.
Which categories are locally presentable? In ordinary categories, "most" complete / cocomplete categories you see are locally presentable, with the exception of the category $Top$ of topological spaces and categories built from it. In $\infty$-categories, we rarely consider such examples, and the most notable exception to the rule "presentable = (co)complete" is the $\infty$-category $Pr^L$ of presentable $\infty$-categories itself, which fails to be presentable for the more basic reason that it is not locally small.
The role of (higher) filteredness: To open the box a hair more, you may be familiar with arguments of the form "homotopy groups commute with filtered colimits, and therefore..." or "A map from a finite complex into filtered colimit must factor through some stage of the colimit, and therefore...". All the business with regular cardinals and $\kappa$-filtered colimits in the precise definition of a locally presentable category is just to give a general framework for such arguments. Much of the time, when you see "Let $\kappa$ be a regular cardinal" in the context of locally presentable categories, you can assume $\kappa = \aleph_0$ and the statements will become more familiar. Categories of modules, etc. are locally $\aleph_0$-presentable i.e. "locally finitely presentable". Higher values of $\kappa$ are needed to consider e.g. Banach algebras, or sheaves on large spaces.
The role of regularity: As Pedro Sanchez Terraf points out in the comments, a cardinal $\kappa$ is regular iff the class of sets of cardinality $<\kappa$ is closed under $<\kappa$-sized unions, making them a sort of "weak universe". I tend to think of a cardinal $\kappa$ as just an avatar of the "weak universe" of sets of cardinality $<\kappa$.
 A diagram is $\kappa$-filtered if and only if it is $cf(\kappa)$-filtered, where $cf(\kappa)$ is the cofinality of $\kappa$ (defined to be the smallest cardinal $\lambda$ such that there is an unbounded map $\lambda \to \kappa$), which is always a regular cardinal. So for example, an $\aleph_\omega$-filtered diagram is the same thing as an $\omega$-filtered diagram, i.e. a filtered diagram. Basically if you wanted to drop "regular", you'd just have to talk about cofinality a lot more, and you'd be mired in more set theory. I think of it like this: $\aleph_\omega$ is the smallest singular (= non-regular) cardinal. I don't want to think about the set-theoretical intricacies that come up when thinking about $\aleph_\omega$. So I restrict to regular cardinals when I can to sidestep all that set theory.
Gabriel-Ulmer Duality
There is an equivalence of categories between the category $Cocts_\kappa$ of small categories with all $\kappa$-small colimits (and functors preserving these colimits) and the category $Pres_\kappa$ of $\kappa$-presentable categories (and left adjoint functors whose right adjoints preserve $\kappa$-filtered colimits). The equivalence sends a small category $\mathcal C$ to the category $Ind_\kappa(\mathcal C) = Fun^\kappa(\mathcal C^{op}, Set)$ of functors $\mathcal C^{op} \to Set$ which preserve $\kappa$-small limits. In the other direction, we send a locally $\kappa$-presentable category to its full subcateogry of $\kappa$-presentable objects.
So a locally presentable category can be viewed as just a small category with a certain amount of colimits, plus "all the natural infinitary constructions" you could build from it.
$Pr^L$ as a sui generis object The category of all presentable categories (and left adjoint functors) is also a very nice category.


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*It has a tensor product where $\mathcal K \otimes \mathcal L = Fun^R(\mathcal K^{op}, \mathcal L)$ is the category of "$\mathcal L$-valued sheaves on $\mathcal K$", i.e. functors $\mathcal K^{op} \to \mathcal L$ which preserve limits and sufficiently-filtered colimits (equivalently, right adjoint functors $\mathcal K^{op} \to \mathcal L$).

*It has an internal hom which is the obvious thing.

*Limits are computed as in $Cat$.

*Colimits are computed by passing to the right adjoints of everything involved and then taking a limit in $Cat$.
The nice properties of $Pr^L$ provide motivation for considering all locally presentable categories together, rather than just fixing a $\kappa$ once and for all to work with.
A: 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


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*$C$ is a small $\infty$-category,

*$R=\{f_i\colon X_i\to Y_i\}$ is a set of maps in $\mathrm{PSh}(C)=\mathrm{Fun}(C^\mathrm{op}, \mathrm{Gpd}_\infty)$, and 

*$\mathcal{P}(C,R)$ is the full subcategory of $\mathrm{PSh}(C)$ spanned by $F$ such that $\mathrm{Map}(f,F)$ is an isomorphism of $\infty$-groupoids for all $f\in R$. 


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).
A: Here is a naive answer. Set theoretical issues play a role, sure, but that's not how I think of presentable $\infty$-categories (or presentable categories!). Really, the key feature of presentable ($\infty$-)categories is that their objects are "presentable". You have a set of generators, and every object of you category can be written as a "small" colimit – with a nice shape – of these generators. Because colimits are what they are, you can easily compute morphisms out of such a colimit. And because the generators are "compact", you can compute morphisms out of a generator into the nicely-shaped colimits easily too. So in the end, all you need to care about are your generators: any object is a nice colimit of the generators, and any morphism can be expressed in terms of morphisms between the generators.
Maybe examples are best. The category of sets is presentable. Any set is a nice colimit of its finite subsets, and any map from a finite set into a nice colimit factors through one of the factors of the colimit. So to study sets, you can just study finite sets, and only think about finite subsets of bigger sets (that's what we do all the time!). Another example would be modules: any module is a nice colimit of finite-rank free modules. You can do a lot of stuff with finite-rank free modules, and only then think about bigger modules and quotients.
Of course, size issues play a big role; if you don't impose them, then nothing would prevent you from choosing all objects as generators, which is somewhat useless...
The nLab article is nicely written if you want more details.
Presentable $\infty$-categories also have a very strong point in their favor from my limited perspective: they are exactly the one that come from combinatorial simplicial model categories (see e.g. Proposition A.3.7.6 in Lurie's HTT). For people interested in homotopy theory, that's pretty sweet. You can in some sense think that the $\infty$-category in question is "presented" by the model category: using all the machinery, you can explicitly compute hom spaces that perhaps you couldn't with just the $\infty$-category. And if you are convinced that model categories are interesting, and if you are also convinced that locally presentable categories are interesting (since combinatorial model categories are locally presentable by definition) then surely you would agree that this makes presentable $\infty$-category interesting, the previous statement being an if and only if.
PS: Some care about the terminology is needed. A presentable ($\infty-$)category is not a presentable object in the category of categories. Rather, the objects of the category are presentable, and many authors say "locally presentable category" instead (like "locally small category": the category isn't small, but if you focus on two objects at a time, it is).
