Remark 10.1.1.
Recall from Remark 9.5.2 that we are in the situation of having to construct one category from another by “formally inverting” some morphisms. We are familiar with processes of these type from algebra, such as the group completion of a monoid (e.g., passage from positive integers to arbitrary integers) or the localization of a ring at a multiplicative subset (e.g., passage from integers to rational numbers). The category-theoretic situation is similar but rather fraught with arrows, and somewhat complicated by the fact that composition of morphisms is not commutative. Similar (but a bit less fraught) considerations apply to localization in a noncommutative ring.
To isolate a key difficulty, imagine trying to define a morphism in the localization category as a formal composition \(g^{-1} \circ f\) where \(f\) is a morphism and \(g^{-1}\) is the “formal inverse” of another morphism. Then the composition of two such morphisms would have the form \(g_1^{-1} \circ f_1 \circ g_2^{-1} \circ f_2\) and we would then need to rewrite the inner composition \(f_1 \circ g_2^{-1}\) as a composition \(g_3^{-1} \circ f_3\) in the opposite order. Then the total composition would become
\begin{equation*}
g_1^{-1} \circ g_3^{-1} \circ f_3 \circ f_2 = (g_3 \circ g_1)^{-1} \circ (f_2 \circ f_3)
\end{equation*}
which has the right form.
We give only a brief summary of the formalism needed to make this idea work. See [117], tag 04VB for further details.