9
$\begingroup$

Let $C, D$ be categories with finite limits, $F:C\to D$ be a essentially surjective functor that commutes with finite limits, and let $S$ be the set of morphisms of $C$ that become isomorphisms in $D$ under $F$. It can be proved that $S$ is a right multiplicative system (as defined here). Then we have a functor $S^{-1}F: S^{-1}C \to D$. Are there criterion for $S^{-1}F$ to be an equivalence?

(To avoid set-theoretic issues, let's assume $C, D$ are essentially small.)

$\endgroup$
5
$\begingroup$

The criterion for $S^{-1}F$ to be an equivalence is that $F$ satisfies the definition of the localisation, as in Definition 7.1.1 in Categories and Sheaves.

In the above settings, since $F$ is essentially surjective, so is $S^{-1}F$. Then, assuming the axiom of choice, the question degenerates to when is $S^{-1}F$ is fully faithful. The question's assumptions do not seem to simplify the general criterion, given in the definition. One needs hom-sets in $D$ to be isomorphic to hom-sets in $S^{-1}C$, and there are two ways (or a combination of both) in which that may fail:

  1. If hom-sets in $D$ are quotients of those of $S^{-1} C$:

    Let $C$ be an essentially small category with finite limits (e.g. small finite sets), let $D$ be the terminal category (with one object and one morphism), and let $F:C\to D$ be the terminal functor. $D$ has finite limits, and $F$ is essentially surjective and commutes with finite limits. $S$ consists of all morphisms in $C$, and hence $S^{-1}C$ is the groupoidification of $C$.

  2. If hom-sets in $D$ are 'bigger' than those of $S^{-1}C$:

    Let $F:C\to D$ be the functor $$\mathbb{R}\otimes_{\mathbb{Z}}-:\mathbb{Z}\text{-}\mathbf{Mod}\to\mathbb{R}\text{-}\mathbf{Mod}.$$ Both $\mathbb{Z}\text{-}\mathbf{Mod}$ and $\mathbb{R}\text{-}\mathbf{Mod}$ have finite limits, $F$ commutes with finite limits that $\mathbb{R}$ is a flat $\mathbb{Z}$ module, and $F$ is essentially surjective as each $\mathbb{R}$-module (i.e. $\mathbb{R}$-vector space) with a base $B$ is isomorphic to the image of $\mathbb{Z}B=\bigoplus_B \mathbb{Z}$ along $F$. The functor, $$S^{-1}F:S^{-1}C\to D,$$ that is, $$\mathbb{R}\otimes_{\mathbb{Q}}-:\mathbb{Q}\text{-}\mathbf{Mod}\to \mathbb{R}\text{-}\mathbf{Mod}$$ is not fully faithful, as $\mathbb{Q}\text{-}\mathbf{Mod}(\mathbb{Q},\mathbb{Q})\cong \mathbb{Q}\ncong\mathbb{R}\cong \mathbb{R}\text{-}\mathbf{Mod}(\mathbb{R},\mathbb{R})$.

The above two examples are the trivial cases where $S^{-1}F$ fails to be an equivalence of categories. More generally, one may have a combination of such deficiencies.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.