What is the correct definition of localisation of a category?

Question

Disclaimer: I wasn't sure if this was an appropriate question for MathOverflow, and so I've also asked this on StackExchange.

There appears to be a discrepancy in the literature regarding the definition of a localisation of a category. Let $\mathcal{C}$ be a category and let $S$ be a class of morphisms. The classical literature, for example Gabriel & Zisman, define a localisation of $\mathcal{C}$ by $S$ as follows:

GZ1) A category $\mathcal{C}[S^{-1}]$ along with a functor $Q: \mathcal{C} \longrightarrow \mathcal{C}[S^{-1}]$ which makes the elements of $S$ isomorphisms.

GZ2) If a functor $F: \mathcal{C} \longrightarrow \mathcal{D}$ makes the elements of $S$ isomorphisms, then there is a functor $G: \mathcal{C}[S^{-1}] \longrightarrow \mathcal{D}$ such that $F$ factors through $Q$ in the sense that $F = G \circ Q$. Moreover, $G$ is unique up to natural isomorphism.

Gabriel & Zisman then state the following lemma:

For each category $\mathcal{D}$, the functor $$- \circ Q: \text{Fun}(\mathcal{C}[S^{-1}], \mathcal{D}) \longrightarrow \text{Fun}(\mathcal{C}, \mathcal{D})$$ is an isomorphism of categories from $\text{Fun}(\mathcal{C}[S^{-1}], \mathcal{D})$ to the full subcategory of $\text{Fun}(\mathcal{C}, \mathcal{D})$ consisting of functors which make elements of $S$ invertible.

Gabriel & Zisman then claims that this lemma is just a restatement of the conditions GS1 and GS2 above in more precise terms.

On the other hand, Kashiwara & Shapira define a localisation of the category $\mathcal{C}$ by $S$ as follows:

KS1) A category $\mathcal{C}[S^{-1}]$ along with a functor $Q: \mathcal{C} \longrightarrow \mathcal{C}[S^{-1}]$ which makes the elements of $S$ isomorphisms;

KS2) If a functor $F: \mathcal{C} \longrightarrow \mathcal{D}$ makes the elements of $S$ isomorphisms, then there is a functor $G: \mathcal{C}[S^{-1}] \longrightarrow \mathcal{D}$ such that $F$ factors through $Q$ in the sense that $F = G \circ Q$.

KS3) If $G_{1}$ and $G_{2}$ are two objects of $\text{Fun}(\mathcal{C}[S^{-1}], \mathcal{D})$, then the natural map $$ - \circ Q: \text{Hom}_{\text{Fun}(\mathcal{C}[S^{-1}], \mathcal{D})}(G_{1}, G_{2}) \longrightarrow \text{Hom}_{\text{Fun}(\mathcal{C}, \mathcal{D})}(G_{1} \circ Q, G_{2} \circ Q) $$ is a bijection.

However, Kashiwara & Shapira then make the claim that condition KS3 implies that the $G$ in KS2 is unique up to unique isomorphism.

These seem to be contradictory. Gabriel & Zisman claims that their definition makes $G$ unique up to isomorphism. Kashiwara & Shaipira claims that their definition makes $G$ unique up to unique isomorphism.

Ordinarily this wouldn't be a problem - obviously one is free to define your terms however you please. But on the face of it, it would appear that GZ1+GZ2 is equivalent to KS1+KS2+KS3, yet each text makes a different claim about the uniqueness of G. When I attempted to prove the uniqueness of G, I was able to show that it was unique up to isomorphism, but not that the isomorphism was unique as Kashiwara & Shapira claim. In fact, if $F$ has an automorphism besides the identity, then it would seem that there would necessarily be multiple distinct isomorphisms between $G_{1}$ and $G_{2}$.

This is something that I have seen in a number of other texts besides these two. And even worse, some texts seem to use one definition or the other in their proofs.

Am I just missing something and these are equivalent? Any clarification here is appreciated.

Thanks

Answer 1

Actually, both of these definitions look weird to me.

I would say there are two natural ways to define the localization $C[S^{-1}]$ by a universal property, as follows. For any category $D$, let ${\rm Fun}_S(C,D)$ denote the full subcategory of the functor category ${\rm Fun}(C,D)$ spanned by those functors that send the morphisms in $S$ to isomorphisms. Let $Q:C\to C[S^{-1}]$ be a functor that sends the morphisms in $S$ to isomorphisms; then there is an induced functor $(-\circ Q): {\rm Fun}(C[S^{-1}],D) \to {\rm Fun}_S(C,D)$. I would say that $Q$ is a strict localization if this functor $(-\circ Q)$ is an isomorphism of categories, and a weak localization if $(-\circ Q)$ is an equivalence of categories.

If we unravel that, then being a strict localization means that

1. For any functor $F:C\to D$ sending $S$-morphisms to isomorphisms, there exists a literally unique functor $G:C[S^{-1}]\to D$ such that $F = G\circ Q$, and
2. condition KS3.

while being a weak localization means that

1. For any functor $F:C\to D$ sending $S$-morphisms to isomorphisms, there exists some functor $G:C[S^{-1}]\to D$ such that $F \cong G\circ Q$ (isomorphic, not equal!), and
2. condition KS3.

The definition you quoted from KS lies somewhere in between these two: it asks only that $G$ exists rather than being unique, but it asks this $G$ to factor $F$ strictly. This corresponds to asking that the functor $(-\circ Q)$ be a surjective-on-objects equivalence, which is a rather odd condition. (The nlab gives the "weak localization" definition.

Note that being a strict localization is the same as the lemma you quoted from GZ, but that it is actually strictly stronger than their definition as you quoted it: saying that $(-\circ Q)$ is an isomorphism requires $G$ to be literally unique given $F$, not merely unique up to isomorphism. Moreover, the up-to-isomorphism GZ definition, as Simon says, is not "categorically correct", and should not be written as the definition of localization. (I believe that GZ were writing before questions such as the difference between isomorphism and equivalence of categories was widely appreciated, though, which may somewhat excuse this sloppiness.)

However, I believe that in this special case, it happens to be true that if a strict localization exists (which, in $\rm Cat$, it always does), then any GZ-localization is equivalent to this strict localization. For applying both universal properties yields functors back and forth comparing the two localizations, which commute strictly with the localization functors from $C$. Then the second clauses of the two universal properties imply that both composites of these functors are isomorphic to identities, hence form an equivalence of categories. So if you "only care about determining categories up to equivalence", then the two definitions are in fact equivalent — but if you only care about determining categories up to equivalence, then you should really be using the definition of weak localization, since it is the only one that's invariant under replacing everything by something equivalent.

Regarding your problem with showing uniqueness of the isomorphism, note that "unique up to unique isomorphism" doesn't mean that there is a unique isomorphism $G_1\cong G_2$ period, it means that there is a unique such isomorphism that commutes with all the other data. The latter condition removes any dependency on automorphisms of the input. For instance, a product object $A\times B$ is unique up to unique isomorphism, even though any automorphism of $A$ or $B$ induces an automorphism of $A\times B$: such nontrivial automorphisms don't commute with the projection maps. In the case of localization, the weak localization condition should be enough to show that there is a unique isomorphism that commutes with the other data.

Answer 2

See Chapter 6 of the book Derived Categories for various notions of localization, including the Ore conditions. (Also available at the arXiv at https://arxiv.org/abs/1610.09640.)

Here's a sample of the type of stuff in loc. cit. It captures the 2-categorical nature of localization