Size issues in localization $\mathcal{C}[\mathcal{W}^{-1}]$ category

Question

When one starts with a locally small category $\mathcal{C}$ and wants to localize it at an appropriate choosen collection of morphisms $\mathcal{W}$, then in general one faces some size issues in the localized category $\mathcal{C}[\mathcal{W}^{-1}]$. The point is that even if $\mathcal{C}[\mathcal{W}^{-1}]$ exists, it might show bad behaviour with respect to it's size, eg it might be not more locally small.

It seems that one is eager to avoid this size phenomena wheneven possible, eg one can say that development of rather deep concepts like model categories can be regarded as attept to address this issue.

My questions are rather basic:

1. When one says (as in Charles Weibel's Homological Algebra) that one tries to ignore the "size issues" around the localization $\mathcal{C}[\mathcal{W}^{-1}]$, does one really only refer to the aim that "in nice situations" one wants $\mathcal{C}[\mathcal{W}^{-1}]$ to keep locally small, that's all? Or is there "more" involved?

2. And, why one tries whenever possible to impose conditions on $\mathcal{C}, \mathcal{W}$ to assure that the localization $\mathcal{C}[\mathcal{W}^{-1}]$ stays locally small? If not, in what kind of troubles one is running?

Is it just because in the case $\mathcal{C}[\mathcal{W}^{-1}]$ locally small it is just easier to handle/ work with from pure "practical" point of view, ie that it has "simpler" structure (eg see Gabriel-Zisman Theorem in Weibel's book) or are there structural reasons involved, in the sense that in case $\mathcal{C}[\mathcal{W}^{-1}]$ not locally small it fail abstractly to satisfy some important category theoretical properties?

For example I saw often that one often wants that the homotopy category (which is a special case of a localization procedure) stays locally small, but why is it so important at the end of the day?

Answer 1

People do always mention the size issue with localizations, but in this particular case I think it’s not so much about size; the failure of local smallness is more primarily a way of expressing how incredibly awful $C[W^{-1}]$ might be in general. If you knew nothing about some localization other than that it turned out to be locally small, I can’t really think of anything that would do for you. The payoff of the theories of calculi of fractions, model categories, etc is not only to get a locally small localization but really to get a localization that can actually be worked with.

Answer 2

This is really an extended comment, to clarify a (potentially tangential) point.

Working in a foundation like $ZFC$, if you have a locally small category $\mathcal{C}$ and a collection of morphisms $\mathcal{W}\subseteq{\bf Hom}_\mathcal{C}$ it is certainly possible that $\mathcal{C}[\mathcal{W}^{-1}]$ will be a large category, but even in this situation there is hope without drastically altering the concepts under consideration; rather, we can shift the context we consider them in.

Note: everything I'm about to do can be done with inaccessible cardinals or Grothendieck universes.

Working in higher order set theory ($HOST$), the defining formula of the sets and functions defining in $\mathcal{C}$ can be used to define a category $\mathcal{C}$ in $HOST$ which is locally $0$-small and $1$-small. The localization $\mathcal{C}[\mathcal{W}^{-1}]$ may then be $0$-large, but it will always be $1$-small and we can manipulate $1$-small categories just as easily as $0$-small ones (traditional "small categories"), so long as we remember to pay attention to which level of size we're working with.

Similarly, any category defined in any foundation with finitely many levels of collection which ends up being 'too large' can be reinterpreted in $HOST$ to obtain a version of the same category, appropriately contextualized to make it malleable for our purposes.

Yes, there are still structural facts this approach will never get around (Freyd's classic 'any large cocomplete large category is thin', for example), but all of these facts take on a new light in this setting. Sticking with Freyd, his observation now takes the form "any $n$-cocomplete $n$-large category is thin". We can have (for example) $1$-complete $1$-locally small non-thin categories -- these won't be definable in $ZFC$, but in this setting they are just as well-behaved as the familiar $0$-complete $0$-locally small categories we all know and love.

Yes, we can still ask about categories that are 'large' in this setting in the sense that no level of collection contains them completely, like the category of all collections at all levels, but this situation can once again be ameliorated by shifting foundations (as suggested by the final axiom in my note above). Adding this axiom we can construct a category of $n$-collections for all $n$ (which isn't possible without it), but we still have other large categories -- these can once again be made appropriately small by shifting foundations again, in a way that is clearest for me to think about using the multiversal foundation laid out in another note I wrote last year.