Timeline for Are compact objects in presheaf categories finite colimits of representables?
Current License: CC BY-SA 4.0
7 events
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| Dec 26, 2022 at 2:38 | comment | added | Tim Campion | @FShrike There's no functor out of any product category in general. In this case, you would need to. lift an idempotent on $T = \varinjlim_i T_i$ to an idempotent on each $T_i$, natural in $i$. There's no reason for such a lift to exist in general. | |
| Dec 25, 2022 at 23:44 | comment | added | FShrike | @TimCampion What am I missing here? By the Fubini theorem for categorical limits and colimits, any finite colimit of finite colimits is equivalent to a colimit out of the product indexing category - which, as a product of finite categories, is finite - so we see a finite colimit of representables pop out at the end. I’m not sure why we need more complicated arguments | |
| May 27, 2020 at 17:10 | comment | added | Tim Campion | I think I see how this works for finite coproducts -- but could you spell out the coequalizers for me? | |
| May 27, 2020 at 6:01 | comment | added | Aurélien Djament | The class of presheaves which are finite colimit of representables is stable under finite colimits: the stability under finite coproducts is obvious and the stability under coequalisers is easily seen by hand using the universal property of colimits and the fact that Hom(x,-) commutes with colimits for x representable. | |
| May 27, 2020 at 5:55 | comment | added | Aurélien Djament | The class of presheaves which are finite colimit of representables is stable under finite colimits: | |
| May 26, 2020 at 20:38 | comment | added | Tim Campion | Hang on -- this shows that $X$ is a finite colimit of (finite colimits of representables) -- a "2-fold" finite colimit of representables. But how does one turn this into an actual finite colimit of representables? I believe that any retract of $\kappa$-small colimits of representables is a $\kappa$-small colimit of representables when $\kappa$ is an uncountable regular cardinal, but I'm not sure about the case $\kappa = \aleph_0$... | |
| May 25, 2020 at 6:28 | history | answered | Aurélien Djament | CC BY-SA 4.0 |