Let $\mathcal{C}$ Be A Category and $S$ a class of morphisms (let us call these weak equivalences) of $\mathcal{C}$. One often defines the localization of $\mathcal{C}$ with respect to $S$ is the category, $S^{-1}\mathcal{C}$ that is initial among all functors that turn the elements of $S$ into isomorphisms. This definition depends on the choice of weak equivalences. What may happen is that their is some other class, $S'$ such that $S^{-1}\mathcal{C}$ and $S'^{-1}\mathcal{C}$ are equivalent categories. Thus, $S$ could be thought of as a presentation of the localization. The question is: is their a presentation-free definition of localization.

More exactly Fill in the blank: A localization of a category $\mathcal{C}$ is a functor, $F:\mathcal{C}\rightarrow\mathcal{D}$ such that ** ________**.

Any partial results are perfectly fine, but not quite what I am looking for.

`$S$`

with respect to which you localize. Namely`$S = \{ f \mid F(f) \textrm{ is an isomorphism} \}$`

. – Karol Szumiło Apr 6 '12 at 11:14`$S$`

is any class of morphisms and`$F : \mathcal{C} \to \mathcal{D}$`

a localization with respect to`$S$`

, then`$F$`

is also a localization with respect to`$S_F = \{ f \mid F(f) \textrm{ is an isomorphism} \}$`

. A class`$S$`

such that`$S = S_F$`

is often called saturated. – Karol Szumiło Apr 6 '12 at 19:04