Where is the problem when a category is not locally small?

Question

The point of departure is the following : There is a very simple way to construct stable homotopy categories for small categories by forming a category of fractions (for example, used by Higson in his construction of E-theory). On the other hand, to construct such categories in a more general setting, one uses triangulated categories or the Spanier-Whitehead construction.

Now in principle, if one just uses categories of fractions, one does not get that the homomorphisms between fixed objects form a class. Why exactly is this a problem?

Answer 1

The key issue is the failure of Bousfield localisation. Suppose we start with a locally small homotopy category $\mathcal{C}$ (so $\mathcal{C}(X,Y)$ is a set for all $X$ and $Y$) and a class of morphisms $S$. We can then form the category of fractions $\mathcal{C}[S^{-1}]$, but it need not be locally small. Given an object $Y\in\mathcal{C}$, one can ask whether there exists an object $Y[S^{-1}]\in\mathcal{C}$ with a natural isomorphism $$\mathcal{C}(X,Y[S^{-1}])\simeq\mathcal{C}[S^{-1}] (X,Y)$$ for all $X\in\mathcal{C}$. It is clear that if $Y[S^{-1}]$ exists for all $Y$, then $\mathcal{C}[S^{-1}]$ must be locally small. There are various good theorems saying that this condition is sufficient as well as necessary, under certain auxiliary assumptions. Functors of the form $Y\mapsto Y[S^{-1}]$ are widely used in homotopy theory, so this issue is important. Quillen's theory of model categories gives a useful framework in which one can often show that $\mathcal{C}[S^{-1}]$ is locally small.