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.)


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.


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