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Luke
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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+GS2GZ1+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

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+GS2 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

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

Source Link
Luke
Luke
  • 453
  • 2
  • 7

What is the correct definition of localisation of a category?

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+GS2 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