Timeline for Are lax functor categories into a cartesian closed 2-category cartesian closed?
Current License: CC BY-SA 3.0
10 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Apr 28, 2015 at 2:05 | comment | added | Tim Campion | Ah, yes, I too misread the question | |
| Apr 27, 2015 at 7:44 | comment | added | Chris Schommer-Pries | @TimCampion That is a good idea, but I am not quite sure how it solves my problem. It is somehow important in my case that the objects of $Lax(I, C)$ are the strict functors (not lax), but that the morphisms are lax transformations. (This funny hybrid is the thing which is adjoint to the Gray tensor product, if that helps). | |
| Apr 27, 2015 at 4:20 | comment | added | Tim Campion | For your purposes, it would suffice to construct a small 2-category $I'$ such that $Strict(I',C) \cong Lax(I,C)$. Probably an Australian could quote you a general result that does this, but if you just need $I$ to be the arrow category, you can probably do it by hand -- after all, when $I$ is the point, then $I'$ would be the walking monad $\mathbf{B}\Delta_+$. | |
| Apr 23, 2015 at 17:07 | answer | added | Theo Johnson-Freyd | timeline score: 1 | |
| Apr 23, 2015 at 13:40 | comment | added | Chris Schommer-Pries | On the otherhand your example does give a monoidal 2-category, just not cartesian monoidal. In that case the lax functor category does indeed happen to be monoidal, just not Cartesian monoidal, as expected. | |
| Apr 23, 2015 at 13:39 | comment | added | Chris Schommer-Pries | For simplicity we might suppose that C has strict 2-products, and that this gives the monoidal structure. Your example doesn't have that unless C is trivial. | |
| Apr 23, 2015 at 13:32 | comment | added | მამუკა ჯიბლაძე | Well in certain sense it is in that example. Could you add what kind of compatibility do you mean? | |
| Apr 23, 2015 at 13:25 | comment | added | Chris Schommer-Pries | So the monoidal structure on C must be compatible with the 2-category structure on C. go ahead and assume that. | |
| Apr 23, 2015 at 13:21 | comment | added | მამუკა ჯიბლაძე | Consider the case when the underlying category of $C$ is trivial, so 2-endomorphisms of the identity form a commutative monoid. Then the underlying category of $\mathrm{Lax}(0\to1,C)$ will have single object with that endomorphism monoid, and it hardly ever has binary products. | |
| Apr 23, 2015 at 12:44 | history | asked | Chris Schommer-Pries | CC BY-SA 3.0 |