strict localization

Given a class WW of arrows in a category, the localization Q W:CC[W 1]Q_W: C\to C[W^{-1}] (if it exists) inverts all the arrows in WW, i.e. Q W(w)Q_W(w) is an isomorphism in C[W 1]C[W^{-1}] and is universal with respect to all functors with such a property. The universality makes it a 2-categorical colimit called a coinverter in the meta-2-category of large 2-categories. The universality says that every WW-inverting functor factors through Q WQ_W up to an iso 2-cell; i.e. for every F:CDF:C\to D which is inverting all arrows in WW, there exist a functor G:C[W 1]DG:C[W^{-1}]\to D such that FGQ WF \cong G\circ Q_W; in addition, for all categories DD, the functor Fun(Q W,D)Fun(Q_W,D) of precomposing with Q WQ_W is fully faithful functor from Fun(C[W 1],D)Fun(C,D)Fun(C[W^{-1}],D)\to Fun(C,D) (it follows that GG is unique up to unique iso). It follows that the image of Fun(Q W,D)Fun(Q_W,D) is precisely the subcategory of Fun(C,D)Fun(C,D) consisting of WW-inverting functors.

We say that Q WQ_W is a strict localization (functor) if for all FF, the corresponding GG can be chosen in a unique way such that F=GQ WF = G\circ Q_W. It follows that every two strict localizations are not only equivalent but in fact isomorphic. For example, when WW admits the calculus of (say, left) fractions then both the large construction via the path category and the construction via one sided fractions give the isomorphic categories. The additional requirement that for each DD, Fun(Q W,D)Fun(Q_W, D) is fully faithful automatically follows essentially from the uniqueness of GG satisfying strict equality F=GQ WF = G\circ Q_W (or, alternatively, for small categories, using lemma 1.2 in Gabriel-Zisman which claims this for the construction of the localized category via the path category, and the fact that every two strict localizations are isomorphic).

Created on July 24, 2011 at 12:46:45. See the history of this page for a list of all contributions to it.