Timeline for Does a fully faithful functor always preserve limits and colimits?
Current License: CC BY-SA 4.0
10 events
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| Sep 1, 2020 at 18:40 | history | edited | Zach Goldthorpe | CC BY-SA 4.0 |
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| Sep 1, 2020 at 18:34 | comment | added | Zhaoting Wei | It seems that I misunderstand the term "preserve". I will modify my question. Thank you. | |
| Sep 1, 2020 at 18:33 | comment | added | Zach Goldthorpe | What you are doing is the opposite, which is regarding the property of preserving a limit. You're correct that a limit cone $\ell$ in $C$ is not necessarily a limit cone $F\ell$ in $D$ even if $F$ is fully faithful, but I even provide an example of this in my answer. | |
| Sep 1, 2020 at 18:31 | comment | added | Zach Goldthorpe | In the definition of a reflected limit, we have that $F$ reflects limits if $\ell$ is a limit in $C$ whenever $F\ell$ is a limit in $D$. In particular, this means that when testing if $F$ reflects limits, you are already assuming that any cone $d$ in $D$ admits a unique factoring morphism $d\to F\ell$. We then only care about those cones of the form $Fc$ for some cone $c$ in $C$. | |
| Sep 1, 2020 at 18:23 | comment | added | Zhaoting Wei | Let $l$ be a limit cone in $C$. Although for any cone $c$ in $C$, there is a unique morphism $Fc\to F l$, this does not mean that for any cone $d$ in $D$ there is a unique morphism $d\to F l$ since $d$ is not necessarily of the form $Fc$. | |
| Sep 1, 2020 at 18:21 | comment | added | Zhaoting Wei | Reflecting limits means that "cones in $C$ that are limits in $C$ become limits in $D$". | |
| Sep 1, 2020 at 18:09 | comment | added | Zach Goldthorpe | Reflecting limits means that "cones in $C$ that are limits in $D$ become limits in $C$" | |
| Sep 1, 2020 at 18:07 | comment | added | Zach Goldthorpe | When you're talking about reflecting limits, you only care about cones that come from $C$. What you know is that the image of $\ell$ is a limit cone in $D$, so fully faithfulness allows you to move the problem to $D$ (where you know by assumption that there is a factoring morphism) and then bring it back to $C$. | |
| Sep 1, 2020 at 18:04 | comment | added | Zhaoting Wei | The problem is that not all cones in $D$ are of the form $Fc$. | |
| Sep 1, 2020 at 17:58 | history | answered | Zach Goldthorpe | CC BY-SA 4.0 |