The Stacks project

Lemma 12.8.2. Let $\mathcal{C}$ be an additive category. Let $S$ be a left or right multiplicative system. Then $S^{-1}\mathcal{C}$ is an additive category and the localization functor $Q : \mathcal{C} \to S^{-1}\mathcal{C}$ is additive.

Proof. By Lemma 12.8.1 we see that $S^{-1}\mathcal{C}$ is preadditive and that $Q$ is additive. Recall that the functor $Q$ commutes with finite colimits (resp. finite limits), see Categories, Lemmas 4.27.9 and 4.27.17. We conclude that $S^{-1}\mathcal{C}$ has a zero object and direct sums, see Lemmas 12.3.2 and 12.3.4. $\square$

Comments (4)

Comment #8867 by on

I think this proof is a little tricky whenever is not locally small (for non-small ) because of the size issues I discuss here. Nevertheless, one can avoid these size issues and show existence of finite products via this argument.

Comment #9219 by on

All categories are small unless mentioned otherwise... discussed elsewhere...

Comment #9222 by on

@#9219 I am aware about the SP convention that everything is a small category (unless otherwise stated). My point is that inside the SP itself there may be an application of this result to some non-small category, like, say, the derived category of some non-small abelian category (like -modules or some category of modules over a scheme). It is not far-fetched that something like this might already appear (although I don't know any instance throughout the SP now).

Comment #9224 by on

That's not a comment on this lemma however, so I am going to leave this lemma as is.

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  • 2 comment(s) on Section 12.8: Localization

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