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John Rognes
John Rognes
2:14 AM
@MarkPenney Peter May discussed the grouplike case in Theorem 2.3 of his paper "E_\infty-spaces, group completion, and permutative categories" math.uchicago.edu/~may/PAPERS/13.pdf , published shortly after "The Geometry of Iterated Loop Spaces".
 
 
9 hours later…
user616128
user616128
10:50 AM
1) The category of pointed topological spaces with weak equivalences being homotopy equivalences, fibrations being serre fibrations and cofibrations satisfying the LLP w.r.t fibrations is a model category
2) It doesn't seem that $X\times I$ is actually a cylinder object (meaning there isn't a cofibration $X\coprod X\to X\times I$ where the codiagonal factors) unless $X$ is a CW-complex?
3) Assuming 1) is actually a model category, then the homotopy category is triangulated. Then the translation functor is the (reduced) suspension functor i.e. $X[1]=\Sigma X$ and thus the adjoint $\Omega$ te…
 
Denis Nardin
Denis Nardin
I think your question (2) might become sensible about what is a topological space for you. But the cylinder object should be X∧I_+=X×I/*×I
Moreover, your question (3) is based on a false premise: it is not true that the homotopy category of every model category is triangulated
It is true only for stable model categories, which this is not an example of (precisely because X→ΩΣX is not an equivalence)
 
user616128
user616128
Thanks, that gives me some direction!
 
Denis Nardin
Denis Nardin
If it helps your intuition, you can check that continuous maps out of X∧I_+ are exactly the pointed homotopies of maps
Note also that the coproduct in pointed spaces is given by the wedge $X∨Y = X⊔Y/(\ast⊔\ast)$
 
 
2 hours later…
Harry Gindi
Harry Gindi
12:39 PM
If you have a proper cartesian model category, can you always equip the category of pointed objects with the obvious smash?
like, will it again be a monoidal model cat
I think this is true
 
Denis Nardin
Denis Nardin
You probably need it to be cartesian closed of some kind
You'll need that your cartesian product commutes with taking the cofiber
Otherwise the smash product is not even associative
For a case where this does not happen, take chain complexes over a field
 
Harry Gindi
Harry Gindi
That's what I meant by Cartesian model cat
I don't know any way to define a monoidal model cat without it being at least biclosed
 
Denis Nardin
Denis Nardin
Oh you mean that it's cartesian closed. Sorry, then it should work. It certainly works ∞-categorically, and I hope there are no snags with the presentation of the model structure
 
Harry Gindi
Harry Gindi
You might need to assume blah blah combinatorial
 
 
3 hours later…
Tom Bachmann
Tom Bachmann
3:31 PM
Is there a reference for the adams-novikov spectral sequence of MSU that is more readable than novikov's original paper?
 
 
2 hours later…
Harry Gindi
Harry Gindi
5:22 PM
Hey, does anyone have a copy of Barwick's thesis?
 
Denis Nardin
Denis Nardin
I do, but it's a scanned copy
He lost the original pdf file in a computer crash
 
Harry Gindi
Harry Gindi
Would you mind sending it to me?
 
Denis Nardin
Denis Nardin
Oh also I just remember I owe you an email
 
Harry Gindi
Harry Gindi
thanks =)
 
Denis Nardin
Denis Nardin
Sorry, I'm really terrible at these things
 
Harry Gindi
Harry Gindi
5:27 PM
No worries
I want to see if his techniques go through fine if you let n be infinite
trying to construct some explicit functor to complicial sets, no clue if it'll work though
thanks!
 
Denis Nardin
Denis Nardin
No problem and sorry for the delay with the other email
 
Harry Gindi
Harry Gindi
not a problem, take your time =)
It seems like the term 'delooping machine' has fallen out of fashion
is this basically an operad?
shockingly, there's no page on the nLab for it
 
Denis Nardin
Denis Nardin
5:54 PM
It's a structure you can put on a space to get a delooping of it
Really, it's what's written on the label :)
I assume it's fallen out of fashion because we now understand the situation pretty well
Although you might hear the words "motivic delooping machines" from time to time, since for motivic spectra the situation is a bit murky
Even that is now a lot clearer than a few years ago
 
Harry Gindi
Harry Gindi
ah
 
Harry Gindi
Harry Gindi
6:16 PM
Is there any literature on a model category of ω-fold complete Segal spaces with somewhat explicit generating anodynes?
I'm sure you can do the usual trick of taking a colimit in locpres, which is the limit in the locpres categories with the right adjoints, then flip it back in again to a colimit of Δ^n->Δ^n+1 and then taking the union of all of the maps you localize against, but it's not the greatest thing to try to compute anything with
 

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