Size-issues in the definition of categorical limits

Question

I know (at least) two definitions of categorical limits. The setting is always as follows. Let $F:J\rightarrow C$ be a functor between not necessarily small nor locally small categories.

For the first definition, denote by $\Delta: C\rightarrow C^J$ the diagonal functor that sends an object $c\in C$ to the composite $\Delta(c): J\rightarrow \ast \rightarrow C$, where $\ast \rightarrow C$ maps the unique object in the terminal category $\ast$ to $c$. A morphism $f:c\rightarrow d$ is sent to the natural transformation $\Delta(c)\Rightarrow \Delta(d)$, which has $f$ as its components if $J$ is inhabited and is empty otherwise.

1. A limit of $F$ is a (terminal) universal morphism from $\Delta$ to $F$, i.e. a terminal object in the comma category $(\Delta \downarrow F)$. (This could also be rephrased in less fancy language as 'a limit is a terminal object in the category of cones over $F$'.)

It seems to me that in this definition one does not have to impose any size restrictions on $C$ nor $J$, correct?

Another definition is:

2. Assume that $J$ is (essentially) small and that $C$ is locally small. Consider the functor $\operatorname{Cone}(-,F): C^{op}\rightarrow Set$ which sends an object $c$ to its set of cones. A limit of $F$ is a representation of this functor (i.e. a representing object $l_F$ together with a choice of natural isomorphism $\operatorname{Cone}(-,F)\cong \operatorname{Hom}(l_F,-)$).

Here we require that $J$ is (essentially) small and that $C$ is locally small so that $\operatorname{Cone}(-,F)$ is $Set$-valued, right? So that we can speak about representable functors, right?

Are there any other reasons why some authors consider limits only over an (essentially) small indexing category and a locally small target category? What are the respective advantages/disadvantages of the two definitions of a limit I gave? Why would one even have the second definition if it is less general than the first?

Answer 1

I'm not exactly qualified to talk about size issues but, here goes.

Assuming both $(1)$ and $(2)$ make sense, they will describe isomorphic objects. To me the real, barebones definition is:

$(3)$ A limit of $F$ (regardless of how large $J,C$ are) is an object $L$ and a $J$-indexed family of arrows $(\phi_j:L\to F(j))$ satisfying the cone axiom such that any other cone $(\psi_j:K\to F(j))$ uniquely factors through the cone $\phi$.

Then $(1)$, $(2)$ and:

$(4)$ A limit of $F$ is a terminal object in a suitable category of cones

$(5)$ A limit of $F$ is an evaluation $R(F)$ where $R:C^J\to C$ is a right adjoint to $\Delta:C\to C^J$

$(6)$ A limit of $F$ is a representing object for the functor $\mathsf{Set}^J(\ast,C(-,F)):C\to\mathsf{Set}$

Are all variations on the same theme. We want (a) a way to restate $(3)$ as efficiently and cleanly as possible, because having clean organising language is an objective of category theory and (b) having different perspectives on the same thing, by viewing it as part of various different structures, is good because it broadens our mental horizons and allows us to apply theorems about said structures to our object of interest.

For example, I wouldn't ordinarily care about $(2)$ or its extremely close cousin $(6)$ but recently I encountered the concept of weighted limit. The perspective of $(6)$ allows us to immediately ask the question - what if we replace the trivial functor $\ast$ with a more interesting one?

A $W$-weighted limit of $F$ is a representing object for the functor $\mathsf{Set}^J(W,C(-,F))$.

That definition, and the resultant theory, wouldn't have been particularly possible if we had kept our eyes fixed on the barebones definition $(3)$. So I hope that's a sort of answer to the second part of your question.

As for size issues, $(3)$ is the essential idea; $(2)$ is a convenient way of encoding that idea, but it has the unfortunate drawback that in particular foundations it might not make sense for all categories. You're right it requires local smallness to have representability. If you're in a "large" situation you can still use $(3)$, which is morally the same but doesn't require you to construct a $\mathsf{Set}$-valued functor. Similarly the right adjoint might not always exist in general for $(5)$ and $(6)$ suffers from the same size issue... but all that does is make us state "limit" in the more boring manner of $(3)$, we don't really lose anything. The perspectives offered by the other definitions can still live in your head, even if verbatim they don't make sense. Note that even $(1)$ could be susceptible to size issues (though I don't know for sure) - does the comma category $(\Delta|F)$ exist? Is it small enough?

Possible reasons for restricting to limits in locally small categories is to (a) enjoy being able to literally state things like $(2),(6)$ and (b) because this "is usually true in practice". Also we usually only expect small limits to exist - that's implicit in the definition of complete category, for instance - because only this is true in our favourite prototypical category, $\mathsf{Set}$. But limits definitely can be, and should be, thought of in a size independent way. Just as long as you can talk about class functions and functors, you should be able to discuss $(3)$.

Answer 2

Are there any other reasons why some authors consider limits only over an (essentially) small indexing category and a locally small target category?

1. Nearly every category that arises in practice is locally small, so that restriction is basically harmless. Personally for me "category" means "locally small category."

2. Local smallness is required for the Yoneda embedding to exist, and probably a bunch of other stuff.

3. There is a theorem due to Freyd: if a category $C$ has all large limits or all large colimits then it is a preorder. So large limits and colimits are rare and we restrict our attention to small limits and colimits, e.g. in the definition of a complete or cocomplete category we only ask for small limits and colimits. This has important implications, e.g. it is responsible for the size conditions in the adjoint functor theorem.

Generally there are several different equivalent ways to think about limits and colimits and different ones are valuable in different situations. There's no reason to throw any one of them out. I personally prefer to think in terms of representing functors most of the time, but also sometimes in terms of adjunctions, and almost never in terms of cones or cocones, but others will have different tastes.

For lots more on the size issues you can consult Mike Shulman's Set theory for category theory. Note that in the statement and proof of the adjoint functor theorem there, local smallness is crucial because one needs to form a product indexed by a homset.