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2:29 AM
does anyone here have experience counting small structures using computers?
I can't find help with this problem
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Q: How many such families exist? (Placing pigs into pens)

Modded BearConsider the set $[n]$ and a positive integer $k$. Now consider the set $F$ of families of subsets of $[n]$ exist such that: There are exactly $k$ elements in $F$ Subsets can appear more than once Each subset has an odd number of elements Two intersecting subsets $A$ and $B$ satisfy $A\subsete...

I have an idea for a weird algorithm that may work, but I don't know how to implement it.
 
This is the wrong room to ask.
 
:(
do you know if there is a right room?
 
@ModdedBear there is a general math.stackexchange room and a general mathoverflow room that would at least make more sense than here. but in general if nobody is responding to your question ON the forum, spamming it in the chatrooms isn't going to help your cause
 
2:48 AM
ok, thanks.
 
3:05 AM
hi @ModdedBear, you should look up "the twelve-fold way" -- i think your question falls into this rubric
there is a fairly explicit formula for the quantity you're trying to compute
 
3:21 AM
@AaronMazel-Gee I don't think the twelve-fold way gives me what I want
 
 
14 hours later…
5:44 PM
Anyone here know whether there's any work reconciling cohomology via comonads (à la Barr–Beck) and cohomology via homotopy theory (à la Quillen–Verdier)?
 
I thought Quillen said something...
Homotopical algebra Chapter 2 section 5 seems like it might be relevant
Specifically, Theorem 5
 
huh, so it works as well as one might expect
thanks!
 
 
2 hours later…
8:15 PM
If I want to avoid computation in secondary cohomology operation, is there any other way to show the correspondence between Kervaire invariant and h_i^2 in Adams 2-line?
 
Anyone recall whether or not geometric realization of simplicial sets is a monoidal functor (w/r/t the cartesian monoidal structure)?
In other words, is the geometric realization of an "H-space" an actual H-space?
 
yes, geometric realisation preserves finite products – provided you go to a suitable category like CGHaus
 
Ah okay. Is this just an adjoint-functor thing?
 
no
There is some abstract nonsense reason why geometric realisation preserves finite limits at the level of sets, but you have to do something extra to get the topology right
 
ah i see. thanks! I just found this bit of the nlab
hm, so it would seem then that for an E_2-ring R, |BGL_1(R)| (the geometric realization of the \infty-groupoid) should be an H-space, right?
 
8:28 PM
well, if you say that you only need geometric realisation to preserve finite products to conclude that...
 
well that's not quite true.
i mean, that doesn't immediately imply that, i don't think
one in particular needs to check that one has the various diagrams "up to homotopy"
 
if I recall correctly, the property of being a H-space is something that is entirely in terms of the homotopy category
geometric realisation descends to an autoequivalence of the homotopy category, so there really shouldn't be a problem
 
ah yeah. i guess i've forgotten about the homotopy category, haha.
 
9:10 PM
jon, why the heck are you worrying about models for spaces?? ;o)
 
=P
@AaronMazel-Gee it turns out that sometimes you have to get your hands dirty.
in particular i'm trying to show that a certain thing is $A_\infty$, which is kind of hard, but I have a model for it in spaces which is strictly associative.
So I go down to spaces, get the relevant diagrams and then apply the simplicial nerve to get an $A_\infty$ object.
(in the $\infty$-category of spaces)
There are probably easier ways to do this, but I'm a genius at overcomplicating things.
 

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