Timeline for Relative cocompletion of a category
Current License: CC BY-SA 4.0
9 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| May 16, 2021 at 21:44 | comment | added | varkor | @Adrien: ah, of course! | |
| May 16, 2021 at 21:40 | vote | accept | Adrien | ||
| May 16, 2021 at 21:10 | comment | added | Adrien | Sure, but $A$ is assumed to be finitely cocomplete, so if $\iota$ is the identity then so is $B$ :) Thanks for clarifying, that makes sense. | |
| May 16, 2021 at 20:55 | comment | added | varkor | @Adrien: I think I misunderstood what you meant initially. But in this case, say that $B$ has only some finite colimits, and take $\iota$ to be the identity. Then $B_A = B$, but this doesn't have all finite colimits, so is not correct. By "the class of colimits in the image of $\iota$", I mean the diagrams with colimits in $B$ that arise by postcomposing diagrams with colimits in $A$ by $\iota$. | |
| May 16, 2021 at 16:09 | comment | added | Adrien | Yes, but then I suggest to take those that become representable upon restriction along $\iota$, so in the case $A=B$ and $\iota=id$ it will give back $B$ as it should, i think ? For your second point then could you clarify what you mean exactly by "the class of colimits in the image of $\iota$" ? Do you actually mean finite diagrams valued in the image of $\iota$ ? Sorry if I'm being daft. | |
| May 16, 2021 at 12:23 | comment | added | varkor | @Adrien: if I'm not misunderstanding your description, I think it is not correct. For instance, take $\iota$ to be the identity on $B$. Then the finite colimits of representables of $B$ form the cocompletion under finite colimits, which ignores all existing finite colimits in $B$. In answer to your second question, you ought not to need any requirement on $\iota$, since $\iota$ itself is not crucial: the only important data is which colimits are in the image of $\iota$. | |
| May 16, 2021 at 12:15 | history | edited | varkor | CC BY-SA 4.0 |
edited body
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| May 16, 2021 at 12:08 | comment | added | Adrien | Thanks this is very useful ! Just to be clear: are you saying the description I sketched in my question is correct ? Also, I'm a bit unclear on whether your construction requires, say, $\iota$ to reflect finite colimits. E.g. I'm not, I think, making the assumption that the image in $B$ of a diagram in $A$ has a colimit in $B$, nor am I assming this colimit is the image of the one computed in $A$. | |
| May 15, 2021 at 12:19 | history | answered | varkor | CC BY-SA 4.0 |