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Is there a nice example of an additive category $C$ and a family of morphisms $S\subset Mor(C)$ such that $C[S^{-1}]$ is no longer additive?

I know that in general localization of categories behaves badly (see examples here), but I haven't seen an example of $C[S^{-1}]$ not being even additive.

Also, it is known that if $S$ satisfies Ore conditions, then the localization $C[S^{-1}]$ will be additive, and the localization functor $Q\colon C\to C[S^{-1}]$ will also be additive. So one should search for required examples among ones where $S$ is not Ore.

Thank you very much for your help!

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I think the point is that in general you can't prove anything at all about the localization. It might not even exist in the same universe. Finding a counterexample will be very difficult I think. –  Adeel Feb 7 at 8:23

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up vote 4 down vote accepted

Let's let $C$ be the category of complex vector spaces, and $S$ be the set consisting entirely of the projection operator $p:\Bbb C \to 0$.

I claim that the localization $C[S^{-1}]$ is a new category where we've simply imposed an equivalence relation on Hom-sets: $$ Hom_{C[S^{-1}]}(V,W) = Hom_C(V,W) / \{\text{rank 1 maps} \sim 0\} $$ First, since a composite of any rank $\leq 1$ map with any other map has rank $\leq 1$, this does define a category $D$.

First, the map in $S$ maps to an isomorphism in $D$. In this category $D$ the identity and zero map $\Bbb C \to \Bbb C$ are the same, so the projection $p$ is an isomorphism with inverse $0: 0 \to \Bbb C$.

Moreover, any rank $1$ map in $C$ has a factorization $V \stackrel{f}{\to} \Bbb C \stackrel{g}{\to} W$, and in $C[S^{-1}]$ we have: $$ gf = gp^{-1} p f = g p^{-1} 0 = g p^{-1} p 0 = g 0 = 0. $$ This actually tells us that any functor $C \to C'$ sending $p$ to an isomorphism actually factors uniquely through $D$. This universal property makes $D$ into a localization of $C$ with respect to $S$.

However, the addition on $C$ doesn't descend to $D$ at all (any map is a sum of rank-1 maps).

(In this case, we've chosen a localization that doesn't really interact with the additive structure of $C$ at all, which is the source of the problem. I don't really know if it's in the spirit of the question you asked.)

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Very cool answer! –  Brian Fitzpatrick Feb 14 at 19:22

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