Timeline for Why isn't the localization $C[W^{-1}]$ (locally) small when $C$ is small and $W$ admits a calculus of (right) fractions?
Current License: CC BY-SA 4.0
7 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jan 17, 2019 at 20:14 | vote | accept | Tyler Bryson | ||
| Jan 17, 2019 at 20:07 | comment | added | Harry Gindi | That's because when performing localization, the motivating examples are large and come from homotopy theory. | |
| Jan 17, 2019 at 20:05 | answer | added | Harry Gindi | timeline score: 13 | |
| Jan 17, 2019 at 20:05 | comment | added | Tyler Bryson | There are lots of warnings scattered across the nLab pages about local smallness in localizations...including explicit warnings about smallness on both the localization page and the calculus of fractions page. Just doing my due diligence in case I am missing something serious. | |
| Jan 17, 2019 at 20:00 | comment | added | Harry Gindi | Where did you read that it wasn't small? I think everything is okay when C is small. The problem is that if C is large but locally small, the localization isn't locally small. Usually when you perform localization, it's done with respect to large locally small categories. The way that you escape smallness is that the zig-zags can generally range over all objects of C, so you get a locally large thing because C is large. | |
| Jan 17, 2019 at 19:59 | comment | added | Tyler Bryson | Some motivation for the question: on the relevant nLab page mentions a result guaranteeing local smallness provided: - $W$ admits a calculus of right fractions - $C$ admits small filtered colimits - $W/c$ is cofinally small In context of the question, this seems unnecessary. | |
| Jan 17, 2019 at 19:46 | history | asked | Tyler Bryson | CC BY-SA 4.0 |