Timeline for Do pretopoi have cohomology and homotopy groups?
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5 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jan 25, 2022 at 20:32 | comment | added | D.-C. Cisinski | If we restrict with absolute Kan extensions (as resolutions give us), this gives immediate compatibilities between derived functors defined via $(\infty,1)$-categories and those defined via $1$-categories. | |
| Jan 25, 2022 at 20:32 | comment | added | D.-C. Cisinski | About Kan extensions and $(\infty,1)$-categories: if $E$ is small, than the $\infty$-category $D(E)$ obtained by inverting quasi-isomorphisms is small. For any $(\infty,1)$-category $C$ with small colimits, we can left Kan extend along any functor between small $(\infty,1)$-category $A\to B$ any functor $A\to C$. Take $A=Ch(E)$ and $D(E)$. If $C$ is a $1$-category, all this will factor through the homotopy categories of $A$ and $B$ for free. | |
| Jan 25, 2022 at 20:24 | comment | added | D.-C. Cisinski | RHom always exist: consider the dg category of cochain complexes, take the Drinfeld dg quotient by acyclic complexes and RHom will be the Hom of the latter. Taking (the spaces of the) connective cover of such Rhom will always coincide with the mapping space of the $(\infty,1)$-category obtained by inverting quasi-isomorphisms in the 1-category of cochain complexes. This is a story you can tell with the derived category of any abelian category (no need of projectives/injectives). Smilarly, $Kan(E)$ has an $(\infty,1)$-theoretic localization which is perfectly understandable. | |
| Jan 25, 2022 at 16:53 | vote | accept | user475784 | ||
| Jan 25, 2022 at 14:32 | history | answered | Zhen Lin | CC BY-SA 4.0 |