Timeline for Is ΩΣ in {simplicial commutative monoids} group completion?
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| Apr 13, 2017 at 12:57 | history | edited | CommunityBot |
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| Feb 1, 2010 at 3:24 | vote | accept | Reid Barton | ||
| Jan 31, 2010 at 17:38 | answer | added | Charles Rezk | timeline score: 10 | |
| Jan 31, 2010 at 5:52 | comment | added | Reid Barton | @Charles: Oh yes, thanks! That's what I was missing. | |
| Jan 31, 2010 at 5:38 | comment | added | Charles Rezk | Well, simplicial commutative monoids is cotensored over pointed simplicial sets. If I use this to define $\Sigma_C M$ to mean $(\Delta^1/\partial \Delta^1)\otimes M$, then I think this is exactly a model for $BM$, using the fact that finite comproducts of commutative monoids are set theoretic products. And my "cotensor suspension" is really the same as your "pushout suspension", since $C$ is a proper simplicial model category. Or am I missing the point here? | |
| Jan 31, 2010 at 5:15 | comment | added | Tyler Lawson | I think that's basically an equivalent question; Quillen ("On the group completion of a simplicial monoid") showed that the homotopy group completion $\Omega BM$ is weakly equivalent to levelwise group completion for cofibrant objects. So I think identifying $\Sigma_C M$ with $BM$ is the key issue. My best guess is to show it's equivalent for objects that are free on a simplicial set (since the free functor commutes with suspension) and then bootstrap it up to cofibrant objects using pushout diagrams. | |
| Jan 31, 2010 at 4:58 | comment | added | Reid Barton | That's part of my question, and the other part is to identify $\Sigma_C M$ with $BM$ (is that obvious? I don't see it right away.) I tried to clarify exactly what my notation was intended to mean. | |
| Jan 31, 2010 at 4:45 | history | edited | Reid Barton | CC BY-SA 2.5 |
added 312 characters in body
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| Jan 31, 2010 at 4:22 | comment | added | Charles Rezk | So "$\Sigma M$" is really a model for the classifying space $BM$? So you are really asking: what does $BM$ of a discrete commutative monoid $M$ look like? In particular, does it have non-trivial homotopy groups in dimensions greater than $1$? (That's how I read your question, anyway.) | |
| Jan 31, 2010 at 0:07 | comment | added | Reid Barton | The underlying space of $\Sigma M$ won't be the suspension of the underlying space of $M$, because the forgetful functor from C to Spaces doesn't commute with colimits. For instance, if I replaced "commutative monoids" with "abelian groups", then C would be the category of nonnegatively-graded chain complexes, $\Sigma$ would be a shift so that $(\Sigma X)_n = X_{n-1}$, $\Omega$ would be a shift in the other direction, and $\Omega \Sigma$ would be the identity functor. | |
| Jan 31, 2010 at 0:01 | comment | added | Ryan Budney | This doesn't work at the level of $\pi_0$. If $M$ is such that $\pi_0 M$ is say the natural numbers you want its group completion to satisfy $\pi_0$ is the integers, right? But $\pi_0 \Omega \Sigma$ of any topological monoid equivalent to the naturals should be huge, a free group on infinitely many generators. Or am I misunderstanding your question? | |
| Jan 30, 2010 at 22:43 | history | asked | Reid Barton | CC BY-SA 2.5 |