Timeline for Can every weighted colimit in a $\mathbf{Pos}$-enriched category be rephrased as a conical colimit?
Current License: CC BY-SA 4.0
13 events
| when toggle format | what | by | license | comment | |
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| Jul 29 at 14:15 | vote | accept | Nick Hu | ||
| Jan 23 at 12:25 | comment | added | Giacomo | Simon makes a good point. In my head the question is asking whether the class of all weighted colimits is the "saturation" of the class of conical colimits (in the sense of Kelly-Schmitt). That is, whether every $\mathcal V$-category with conical colimits also has all weighted colimits, and every $\mathcal V$-functor preserving conical colimits, also preserves all weighted ones. The answer is negative for $\mathcal V=\mathbf{Pos}$ since discrete posets form a $\mathbf{Pos}$-category that has all conical colimits but lacks the weighted ones. | |
| Jan 22 at 20:11 | comment | added | Simon Henry | Note that this is important as for example you can always express a Weigthed colimit as a conical colimit of tensor. | |
| Jan 22 at 20:09 | comment | added | Simon Henry | I might be nitpicking, but I think the question is too vague: Every weighted colimit $X= colim_W D(i)$ in $C$ is the colimit of the constant functor $* \to C$, $F(*) = X$. This is clearly not the answer you expect, but this answers the question which is asked. Given that the actual question probably has a negative answer, I feel the details of how one gets ride of this sort of trivial answer is actually important. For e.g. Giacomo's answer is making the additional assumption that the replacement needs to take value in the same full subcategory as the original weighted diagram. | |
| Jan 22 at 16:25 | answer | added | Giacomo | timeline score: 6 | |
| Jan 14 at 6:36 | comment | added | Alexander Campbell | The example in your Observation 1 is not relevant to the problem of expressing weighted limits as conical limits. It is an example of a diagram in an enriched category which admits a limit in the underlying category but not in the enriched category. | |
| Jan 13 at 23:02 | history | edited | Nick Hu | CC BY-SA 4.0 |
I meant equaliser not coequaliser
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| Jan 13 at 22:52 | comment | added | Jonas Frey | Pos-cotensors in Pos itself are just exponentials in the cartesian closed sense. I wouldn't even know how to express a cotensor with the nontrivial 2-element poset as a conical limit. But maybe that's possible since Kelly goes a more complicated route for his counterexample in Cat? Does anyone know how? | |
| Jan 13 at 18:29 | history | edited | Nick Hu | CC BY-SA 4.0 |
edited title
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| Jan 13 at 18:27 | history | edited | Nick Hu | CC BY-SA 4.0 |
fix formatting
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| Jan 13 at 18:24 | history | edited | Nick Hu | CC BY-SA 4.0 |
added 54 characters in body; edited title
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| S Jan 13 at 18:19 | review | First questions | |||
| Jan 13 at 18:52 | |||||
| S Jan 13 at 18:19 | history | asked | Nick Hu | CC BY-SA 4.0 |