Timeline for Can filtered colimits be computed in the homotopy category?
Current License: CC BY-SA 4.0
10 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Feb 10, 2020 at 15:59 | comment | added | Tim Campion | Haha! I can't believe I forgot such a striking fact! | |
| Feb 10, 2020 at 14:56 | comment | added | Kevin Carlson | Nope! As, er, tcamps recalls in the comments here: math.stackexchange.com/a/2128004/31228... the functor is roughly “the set of split subobjects with a disjoint basepoint.” In particular any object lacking nontrivial split subobjects will be mapped to a two-point set, so the functor is not injective on isomorphism classes even when applied to Set itself. If there are a large number of such objects, then the functor will have a large fiber on isomorphism classes while still being conservative. | |
| Feb 10, 2020 at 7:54 | comment | added | Tim Campion | @KevinCarlson Whoa -- that's surprising! When you say "every locally small $C$ admits a functor $C \to Set$", do we have to assume that $C$ has no more isomorphism classes of objects than the size of the universe? | |
| Feb 10, 2020 at 7:32 | comment | added | Kevin Carlson | Fun fact, in case you intended the implication in your last point that $h\mathcal S$ admits no conservative functor whatsoever to Set: actually every locally small category admits such a functor! I didn’t believe that when I heard it, but it’s yet another coup de Freyd. Of course there is still a big gap between $h\mathcal S$ and $D(R)$ in admitting such a functor which is a coproduct of representables. | |
| Feb 10, 2020 at 4:52 | history | edited | Tim Campion | CC BY-SA 4.0 |
added 521 characters in body
|
| Feb 10, 2020 at 4:21 | history | edited | Tim Campion | CC BY-SA 4.0 |
added 235 characters in body
|
| Feb 10, 2020 at 4:12 | history | edited | Tim Campion | CC BY-SA 4.0 |
added 65 characters in body
|
| Feb 10, 2020 at 4:02 | history | edited | Tim Campion | CC BY-SA 4.0 |
added 225 characters in body
|
| Feb 10, 2020 at 3:48 | history | edited | Tim Campion | CC BY-SA 4.0 |
added 225 characters in body
|
| Feb 10, 2020 at 3:42 | history | answered | Tim Campion | CC BY-SA 4.0 |