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?

2. And if yes, why one tries whenever possible to impose conditions of $\mathcal{C}, \mathcal{W}$ to assure that the localization $\mathcal{C}[\mathcal{W}^{-1}]$ stays locally small?

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?

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?