Timeline for Reflective exponential ideals in presheaf categories
Current License: CC BY-SA 4.0
7 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Oct 2 at 22:55 | comment | added | varkor | I agree that's an improvement. | |
| Oct 2 at 22:21 | comment | added | Ivan Di Liberti | Finally, I am perfectly aware that I added the hypothesis of completeness when answering your question. At the moment I can't find a sharper argument than this one. | |
| Oct 2 at 22:20 | comment | added | Ivan Di Liberti | The fact that $y(c)^y(a)$ is indeed $y(c^a)$ follows by direct inspection. Indeed $P(A)(F \times y(a), y(c)) = P(A)(\text{colim} y(d_i) \times y(a), y(c)) = \text{lim} P(A)( y(d_i) \times y(a), y(c)) = \text{lim} A( d_i \times a, c)...$. | |
| Oct 2 at 22:02 | comment | added | Ivan Di Liberti | Sure. Let $A$ be a bicomplete cartesian closed category and consider the Yoneda embedding into small presheaves $y: A \to P(A)$. Now, the embedding preserves finite products and it is easy to see that because $A$ is cartesian closed, $A$ must be an exponential ideal in $P(A)$, indeed $y(c)^d = y(c)^{\text{colim} yc_i} = \text{lim} y(c)^{y(c_i)} = y(\text{lim} c^{c_i})$. So now we can apply Sketches A4.3.1. | |
| Oct 2 at 19:01 | comment | added | varkor | @IvanDiLiberti: I don't see you obtain finite product preservation by the reflector in this case. Perhaps you could elaborate in a second answer? | |
| Oct 2 at 18:51 | comment | added | Ivan Di Liberti | Better is true, no? Every cocomplete cartesian closed category is reflective in its category of small presheaves and the reflector preserves finite products (the argument is the same of Day's). Notice that finite products indeed do exists in the category of small presheaves by Remark 3.9 in Rosicky-Adamek "How nice are are free completions?" | |
| Oct 2 at 18:31 | history | answered | varkor | CC BY-SA 4.0 |