Timeline for Strongly compact categories (reference request)
Current License: CC BY-SA 4.0
18 events
| when toggle format | what | by | license | comment | |
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| Jun 1, 2021 at 21:23 | comment | added | varkor | I don't know of any references for the variant without small-cocompleteness! | |
| Jun 1, 2021 at 21:23 | history | edited | varkor | CC BY-SA 4.0 |
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| Jun 1, 2021 at 21:21 | comment | added | varkor | Thanks. I had interpreted Theorem 8 as showing total categories to be equivalent to compact categories, but this is not true, because Ulmer's "Yoneda embedding" is really a restricted variant. | |
| Jun 1, 2021 at 20:59 | comment | added | Martin Brandenburg | Compact categories don't have to be cocomplete, see the mentioned paper "Compact and hypercomplete categories", Example 3.15. (I guess that this example is also strongly compact.) I don't see why Ulmer's paper should imply this. | |
| Jun 1, 2021 at 16:36 | comment | added | varkor | @MartinBrandenburg: compact categories are cocomplete and complete (by virtue of Ulmer's theorem), so it seems reasonable to also assume cocompleteness for strongly compact categories (it also seems a natural condition to impose to ask for preservation of small colimits from $\mathcal C$). Ulmer requires small-cocompleteness, for instance. I don't know whether the same results will hold if you relax that assumption. | |
| Jun 1, 2021 at 16:27 | history | edited | varkor | CC BY-SA 4.0 |
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| Jun 1, 2021 at 16:09 | comment | added | Martin Brandenburg | I don't understand the second sentence in your answer, since strongly compact categories don't need to be cocomplete (right?) - they are just complete. | |
| Jun 1, 2021 at 14:50 | comment | added | varkor | @MartinBrandenburg: I did indeed, thanks :) | |
| Jun 1, 2021 at 14:49 | history | edited | varkor | CC BY-SA 4.0 |
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| Jun 1, 2021 at 14:46 | comment | added | Martin Brandenburg | Great! In the first paragraph, do you mean "is a left adjoint"? | |
| Jun 1, 2021 at 14:01 | history | edited | varkor | CC BY-SA 4.0 |
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| Jun 1, 2021 at 13:56 | comment | added | varkor | I've updated my answer to correct the mistake Ivan pointed out. | |
| Jun 1, 2021 at 13:55 | history | edited | varkor | CC BY-SA 4.0 |
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| Jun 1, 2021 at 13:46 | comment | added | varkor | @IvanDiLiberti: I overlooked that subtlety. Thanks. This isn't quite the right concept. | |
| Jun 1, 2021 at 13:44 | history | edited | varkor | CC BY-SA 4.0 |
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| Jun 1, 2021 at 13:42 | comment | added | Ivan Di Liberti | A category is $\mathcal{C}$ is strongly compact if and only if every COCONTINUOUS functor satisfies the solution set condition. This makes things different. The notion of strong compactness should be seen as a kind of totality. | |
| Jun 1, 2021 at 12:58 | comment | added | Martin Brandenburg | Thanks. Can you perhaps add the definition of "petit with respect to small presheaves" in the special case relevant to my question? The paper deals with more general objects. | |
| Jun 1, 2021 at 12:03 | history | answered | varkor | CC BY-SA 4.0 |