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Aug
1
awarded  Talkative
Jul
2
awarded  Curious
Jun
12
comment deRham cohomology of $S^n$ without Mayer-Vietoris
Is your final step for the $k=n$ case necessary? Isn’t it sufficient to observe that the space of top forms is rank 1, and the nowhere vanishing volume form must be nontrivial in cohomology since its integral is nonzero, and therefore it must be a generator of the top degree cohomology?
Jun
12
comment deRham cohomology of $S^n$ without Mayer-Vietoris
In the $k=1$ case, $\alpha$ and $\beta$ differ by a constant. Why does this suffice to extend them to all of $S^n$ for $n>1$? This fails for $n=1$, of course. I guess it’s that $\alpha - \beta$ is only locally constant, and only in the $n>1$ case where $S^{n-1}$ is connected do we get an actually constant difference, which allows gluing to a global function.
May
8
awarded  Organizer
May
8
revised what does BG classify? i.e. what is a principal fibration?
add tag for classifying spaces
May
8
suggested approved edit on what does BG classify? i.e. what is a principal fibration?
May
8
comment Are some numbers more irrational than others?
If your criterion is ease of proof of irrationality, normal numbers are particularly easy to see irrationality of.
Apr
30
awarded  Yearling
Mar
22
awarded  Informed
Mar
10
comment In what generality does Eilenberg-Watts hold?
Thanks for the tip about dense generators, Zhen. That cleared everything up for me. So Eilenberg-Watts lemma is just another statement of the Yoneda lemma, basically? That's neat.
Mar
8
comment In what generality does Eilenberg-Watts hold?
@ZhenLin: Yes, the representables are generators of the presheaf category. But I still don't see how to proceed by analogy with the ring module case without some addition lemmas. In an arbitrary category with generator $P$, can we write every object as a colimit of a constant functor at $P$? I guess that would be sufficient to give me what I want, but it seems stronger than the definition of generator I know (either $\hom(P,-)$ is injective on arrows or every object receives an epimorphism from $\coprod P$)
Mar
8
asked In what generality does Eilenberg-Watts hold?
Jan
29
revised comparison between two monadic definitions for an operad
reword title slightly
Jan
29
asked comparison between two monadic definitions for an operad
Jan
20
comment Maximal ideals in the ring of continuous real-valued functions on R
So why doesn't this apply to the maximal ideal containing the compactly supported functions? The residue field is not $\mathbb{R}$? What is it? The field of functions which don't vanish at infinity?
Jan
12
awarded  Popular Question
Jan
6
awarded  Excavator
Jan
6
revised Weighted limits and completeness
Mike's link to his Chicago homepage is dead. I update URL to same PDF on his San Diego webpage.
Jan
6
suggested approved edit on Weighted limits and completeness