nLab iterated localization

Idea

Iterated localizations are, to a first approximation, functors right orthogonal to conservative functors, i.e. the class of functors obtained from factoring any functor as some functor followed by a conservative functor.

For the correct statement one needs to regard $\mathbf{Cat}$ as a 2-category, or at least as a model category.

Definition

Given a small category $C$ and a set of morphisms $S \subset Mor(C)$, we can always construct the localization category $C[S^{-1}]$ which is universal in the sense that every functor $F \colon C \to D$ which inverts all the morphisms in $S$ factors through it (see localization):

On the other hand, it is often interesting to look at which morphisms of $C$ become isomorphisms under the action of a functor $F \colon C \to D$ (e.g. $F$ might be a cohomology theory and we want to assess which maps in $C$ are quasi-isomorphisms).

Call $\ker F$ the class of maps in $C$ which are inverted by $F$ (ndr: this isn’t standard notation). Then we can localize $C$ at $\ker F$ and thus factor $F$ as a localization $L_1$ followed by a functor $F_1:C[\ker F^{-1}] \to D$:

It turns out that this $F_1$, in general, still inverts further morphisms. Therefore it makes sense to iterate this process to obtain the following sequence of functors:

where $C_{n+1} \coloneqq C[\ker F_n^{-1}]$.

Now consider the colimit

$$C_\omega \,\coloneqq\, \colim_n C_n$$

of this sequence of localizations. Clearly there is a canonical functor $L \colon C \to C_\omega$ and also a mediating map

(1)$$K \,\colon\, C_\omega \to D$$

such that $K\circ L = F$. In fact, one can prove that $K$ is conservative.

Definition

A functor $F \colon C \to D$ is an iterated localization if $K$ (1) is an equivalence.

This makes $\big(\text{iterated localizations}, \text{conservative functors}\big)$ a(n homotopy) factorization system on $\mathbf{Cat}$ (considered as a 1-category with weak equivalences given by equivalences of categories).

See also

• localization of a category, category of fractions

• orthogonal factorization systems

• conservative functor

References

The above is pretty much directly taken from:

• Joyal‘s CatLab, Ex. 6.12 in: Factorisation systems

• André Joyal, §11.14 and pp. 70 in: Notes on quasi-categories (2008) [pdf, pdf]