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Stream: theory: category theory

Topic: Example where localisation doesn't commute with taking presh

Fernando Yamauti (Oct 02 2025 at 16:54):

Hi. Does anyone know any explicit example of small cat $\mathcal{C}$ together with a subcat $\mathcal{W}$ such that $\widehat{\mathcal{C}}[\mathcal{W}^{-1}] \cong \widehat{\mathcal{C}[\mathcal{W}^{-1}]}$ fails?

I mean, of course, that's always true when taking localisations internal to locally pres cats. But here I'm not requiring the localisation functor nor the codomain of the localisation to satisfy anything special.

I also tried computing some finite examples and they all seem to satisfy that. So perhaps failure would only happen with infinite $\mathcal{C}$...

fosco (Oct 02 2025 at 17:12):

Fernando Yamauti said:

Hi. Does anyone know any explicit example of small cat $\mathcal{C}$ such that $\widehat{\mathcal{C}}[W^{-1}] \cong \widehat{\mathcal{C}[W^{-1}]}$ fails?

...basically any conceivable example? If "hat" is presheaves, the right hand side of the purported equation is cocomplete. The lhs is very rarely even finitely cocomplete.
Maybe there's something I don't understand?

fosco (Oct 02 2025 at 17:14):

(I guess there's also a small, harmless abuse of notation, $W$ on the LHS is the image of $W$ on the other side, via Yoneda?)

Fernando Yamauti (Oct 02 2025 at 17:32):

fosco said:

(I guess there's also a small, harmless abuse of notation, $W$ on the LHS is the image of $W$ on the other side, via Yoneda?)

Yes. My bad for the poor writing.

Fernando Yamauti (Oct 02 2025 at 17:36):

fosco said:

Fernando Yamauti said:

Hi. Does anyone know any explicit example of small cat $\mathcal{C}$ such that $\widehat{\mathcal{C}}[W^{-1}] \cong \widehat{\mathcal{C}[W^{-1}]}$ fails?

...basically any conceivable example? If "hat" is presheaves, the right hand side of the purported equation is cocomplete. The lhs is very rarely even finitely cocomplete.

The lhs doesn't have to be a priori cocomplete. But it turned out to be in several finite examples I tried computing. So what's is this any conceivable example you mention?

Mike Shulman (Oct 02 2025 at 18:42):

In looking for a concrete counterexample I would try the "smallest possible" one first: $C = W = 2$, so that $C[W^{-1}] = 1$ and $\widehat{C[W^{-1}]} = \mathrm{Set}$, while $\widehat{C}$ is the Sierpinski topos $\mathrm{Set}^2$. Then to show that $\widehat{C}[W^{-1}] \not\simeq \mathrm{Set}$ (or at least that the canonical functor there isn't an equivalence), find a functor $\widehat{C} \to D$ for some category $D$ that inverts $W$ but doesn't factor through Set.

Fernando Yamauti (Oct 02 2025 at 19:11):

Mike Shulman said:

In looking for a concrete counterexample I would try the "smallest possible" one first: $C = W = 2$, so that $C[W^{-1}] = 1$ and $\widehat{C[W^{-1}]} = \mathrm{Set}$, while $\widehat{C}$ is the Sierpinski topos $\mathrm{Set}^2$. Then to show that $\widehat{C}[W^{-1}] \not\simeq \mathrm{Set}$ (or at least that the canonical functor there isn't an equivalence), find a functor $\widehat{C} \to D$ for some category $D$ that inverts $W$ but doesn't factor through Set.

Hmm...right. I thought I had a proof that $X^{h_0} \to X^{h_1}$ was also a weak equivalence for any presheaf $X$, but I think I made a mistake. So perhaps that example might work...

fosco (Oct 02 2025 at 20:18):

Fernando Yamauti said:

what's is this any conceivable example you mention?

...besides those where the localization is nice enough to keep the category cocomplete (for example, when it's a reflective localization, Borceux I, §5.3). But in general, even a cocomplete reflective localization will not have the same colimits of the category before localization (take, for example, sheaves on a topology or a site). I can't think of a more minimal example than the one Mike provided. I would look for other counterexamples in, say, localizing at all morphisms, or taking a very small category of topological spaces: its homotopy category in the sense of "hom-sets modulo homotopy" will not have, say, all coequalizers.

fosco (Oct 02 2025 at 20:21):

I am too tired to think about it now, but I hope someone else will provide other examples. I realized I never asked myself what happens when you localize $\widehat C$ at the image of Yoneda... you're asking: for what $C$'s is it true that $\widehat C[(yC)^{-1}]$ is equivalent to the presheaves over the "localize all" category $C[C_1^{-1}]$?

fosco (Oct 02 2025 at 20:22):

I suspect: almost none

Kevin Carlson (Oct 06 2025 at 20:24):

Won't $\hat{C}[(yC)^{-1}]$ be $\hat{C}[\hat{C}^{-1}],$ by density of Yoneda? Then it's a groupoid and indeed will never be presheaves over $C[C^{-1}]$ unless $C$ is empty.

Fernando Yamauti (Oct 06 2025 at 20:35):

Kevin Carlson said:

Won't $\hat{C}[(yC)^{-1}]$ be $\hat{C}[\hat{C}^{-1}],$ by density of Yoneda? Then it's a groupoid and indeed will never be presheaves over $C[C^{-1}]$ unless $C$ is empty.

The localisation functor doesn't necessarily preserve any (co)limits and, also, the internal hom $X^{(-)}$ for a fixed presheaf $X$ might fail to send weak equiv to weak equiv. So I don't see how you can get from Yoneda that it's the same as inverting everything...

Kevin Carlson (Oct 06 2025 at 20:40):

I was thinking that $\hat{C}(X,Y)=\lim_{x\in \int X}\mathrm{colim}_{y\in\int Y}\hat{C}(\hat x,\hat y),$ and all the elements of the inner set are being made invertible. But I didn't think any further, and now I see that there's no obvious way to take inverses on the components of some such map to assemble a map in the other direction; maybe I was intuitively thinking more of a double colimit than a limit of colimits.