Set-theoretic Issues
In general, given a category C and some class w of morphisms in the category, there is some question as to whether it is possible to form a localization w-1 C by inverting all the morphisms in w. The typical procedure for constructing the localization might result in a pair of objects with a proper class of morphisms between them. Avoiding such set-theoretic issues is one of the initial reasons for the development of the theory of model categories.
Read more about this topic: Localization Of A Category
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