Timeline for Notion of $\kappa$-sifted categories?
Current License: CC BY-SA 4.0
5 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jul 1 at 17:09 | comment | added | Georg Lehner | For future readers: It was established on discord that there are two working definitions of $\kappa$-sifted: - Either the weak version, as defined by @Z. M above in terms of the diagonal functors being cofinal. - The strong version, as used by Adámek et al. defining $C$ to be $\kappa$-sifted if $C$-indexed colimits in Sets (1-categorically) or Spaces ($\infty$-categorically) commute with $\kappa$-small products. The weak version does not need to imply the strong version - This is where the confusion comes from. | |
| Jul 1 at 9:41 | comment | added | Z. M | The real subtlety is that the doctrine of $\kappa$-small sets is probably not sound, that is to say, filtered and weakly filtered categories with respect to this doctrine might not coincide (I am referring the terminology in Adámek–Borceaux–Lack–Rosický and its $\infty$-categorical analogue explained in Rezk's draft). My definition corresponds to being weakly filtered. | |
| Jun 30 at 20:38 | comment | added | Z. M | It seems that this definition is different from mine. Fix an uncountable regular cardinal $\kappa$. Let $R$ be a ring with a monogenic non-flat $R$-module $M=R/r$. Then the category $\mathcal C$ of free $R$-modules $F$ of rank $<\kappa$ equipped with a map $F\to M$ of $R$-modules has $\kappa$-small coproducts, thus it is $\kappa$-sifted in my definition. However, it does not seem to be filtered, let alone being $\kappa$-filtered: the colimit of the functor $\mathcal C\to\operatorname{Mod}_R,N\mapsto N$ seems to be $M$, but by Lazard's theorem, if $\mathcal C$ is filtered, then $M$ is flat. | |
| Jun 30 at 19:12 | vote | accept | Z. M | ||
| Jun 30 at 20:30 | |||||
| Jun 30 at 18:33 | history | answered | varkor | CC BY-SA 4.0 |