Timeline for Enriched cartesian closed categories
Current License: CC BY-SA 4.0
15 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| S Feb 10, 2019 at 23:41 | history | bounty ended | Mike Shulman | ||
| S Feb 10, 2019 at 23:41 | history | notice removed | Mike Shulman | ||
| Feb 10, 2019 at 23:41 | vote | accept | Mike Shulman | ||
| Feb 10, 2019 at 6:29 | answer | added | Reid Barton | timeline score: 5 | |
| Feb 6, 2019 at 9:08 | answer | added | Karol Szumiło | timeline score: 8 | |
| S Feb 3, 2019 at 12:04 | history | bounty started | Mike Shulman | ||
| S Feb 3, 2019 at 12:04 | history | notice added | Mike Shulman | Draw attention | |
| Jan 31, 2019 at 9:38 | answer | added | Mike Shulman | timeline score: 5 | |
| Jan 31, 2019 at 9:31 | comment | added | Mike Shulman | Hmm, actually maybe that isn't quite enough: the large category we embed it in will still have $(A\otimes 1)\times Y \ncong A\otimes Y$, but it may not be cartesian closed. However, I've now constructed such a $D$, which seems to me to be a big step towards a negative answer, so I'll post it below. | |
| Jan 31, 2019 at 6:09 | comment | added | Mike Shulman | @TheoJohnson-Freyd Maybe local presentability of $V$ might matter; but as I said, I'm happy to assume anything even up to its being a presheaf topos. Given $V$, though, I don't think local presentability of $C$ can matter: if it fails for some $C$, then it fails for some specific $A\in V$ and $Y\in C$, which means there is some small subcategory of $C$ in which it fails, and we can restrict to that subcategory and then embed it in a locally presentable $V$-category preserving the relevant limits and colimits to get a "good" $C$ where it still fails. | |
| Jan 31, 2019 at 6:06 | comment | added | Mike Shulman | @TheoJohnson-Freyd Good point about $X=1$. | |
| Jan 31, 2019 at 1:02 | comment | added | Theo Johnson-Freyd | Which is in turn equivalent to asking whether $(A\otimes 1) \times Y \overset?= A\otimes Y$, for $A \in V$ and $Y \in C$. | |
| Jan 31, 2019 at 1:00 | comment | added | Theo Johnson-Freyd | It is not hard to reduce to the case $X=1$. Your question is equivalent to asking whether $C(Y,Z) \overset?= C(1,Z^Y)$. | |
| Jan 31, 2019 at 0:59 | comment | added | Theo Johnson-Freyd | My spidey sense says that the only way this could fail would be for stupid set theoretic reasons, if your categories are too far from being locally presentable. (I don't make a wager about that case --- perhaps there are no set theoretic obstructions.) But I agree I cannot answer the question. | |
| Jan 30, 2019 at 20:56 | history | asked | Mike Shulman | CC BY-SA 4.0 |