Timeline for Monoidal model category structure on a functor category.
Current License: CC BY-SA 3.0
6 events
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| Oct 2, 2013 at 23:33 | history | edited | David White | CC BY-SA 3.0 |
added 157 characters in body
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| Feb 1, 2013 at 21:01 | comment | added | Ricardo Andrade | @David: It does. Thank you very much. | |
| Feb 1, 2013 at 18:12 | comment | added | David White | Okay, I just dug out the preprint. He uses the projective model structure on $Arr(M)$. You don't need cofibrant objects to be flat. You just need $M$ to be cofibrantly generated and a closed symmetric monoidal model category. The diagram category you get is also cofibrantly generated and a symmetric monoidal model category. If $M$ has the monoid axiom, then so does $Arr(M)$. Hope that helps | |
| Feb 1, 2013 at 5:05 | comment | added | David White | I feel uncomfortable going into detail on his unpublished work, and I also don't have the preprint in front of me. I remember it's quite general, like $M$ probably only needs to be a cofibrantly generated monoidal model category. It definitely doesn't need to be combinatorial. I know he later assumes the monoid axiom, but I don't think he needs it for the proof that the pushout product axiom is inherited. He may use the hypothesis that cofibrant objects are flat, since that's one of his favorite hypotheses to place in a situation like this. | |
| Feb 1, 2013 at 4:55 | comment | added | Ricardo Andrade | The box product on $\textrm{Fun}(A,M)$ you mention is simply the Day convolution product coming from the symmetric monoidal structure on $A=(0\to 1)$ given by $a\otimes b=\min(a,b)=a\cdot b=a\wedge b$. Thus, under some conditions, the result you state that $\textrm{Fun}(A,M)$ inherits the pushout-product and monoid axioms from $M$ follows from propositions 2.2.15 and 2.2.16 in Sam Isaacson's thesis (linked in my answer). Do you know in what generality Mark Hovey proves this result? | |
| Jan 31, 2013 at 14:05 | history | answered | David White | CC BY-SA 3.0 |