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Mar
18
answered Why is $\pi_1(\Bbb{R}^n,x_0)$ the trivial group in $\Bbb{R}^n$?
Mar
17
comment If a morphism of pushouts of complexes (with one arrow monic) is composed of quasi-isos, then the induced arrow is one also
@Sunny: you're right, I wasn't clear enough. I mean one arrow monic in both pushout diagrams, yes.
Mar
16
comment How to prove a Set category with at least one element is a separator?
Also, be careful, you say "a separator object for $f_1,f_2$", but that's not right. A separator is a separator for the category, and it satisfies the condition you wrote for every $f_1,f_2$.
Mar
16
comment How to prove a Set category with at least one element is a separator?
@Problemania: $f(a)\neq g(a) \Rightarrow f\circ h(x)\neq g\circ h (x)$ since $a=h(x)$. Hence the two functions $f\circ h$ and $g\circ h$ are different at the element $x\in X$. If two functions differ in one point, then they are different, by definition of equality of functions! As for your second comment, yes, exactly, that's actually the definition of generator (or a slight rewording of it).
Mar
15
comment How to prove a Set category with at least one element is a separator?
I'm not familiar with Lawvere's book, but I take it that he, as Johnstone, takes "separator" to mean what other categorical sources call "generator" . If it is not the case, then tell me and I'll delete the answer.
Mar
15
answered How to prove a Set category with at least one element is a separator?
Mar
15
reviewed Reject Why is the partial derivative $f_x' = 0 $ is not continous?
Mar
15
revised If a morphism of pushouts of complexes (with one arrow monic) is composed of quasi-isos, then the induced arrow is one also
added 526 characters in body; edited title
Mar
14
comment If a morphism of pushouts of complexes (with one arrow monic) is composed of quasi-isos, then the induced arrow is one also
Thank you. I will read your answer with detail later. I was just aware of "homotopy colimits" in the category of topological spaces, not in any more general sense.
Mar
14
comment If a morphism of pushouts of complexes (with one arrow monic) is composed of quasi-isos, then the induced arrow is one also
@ZhenLin: Thanks. I'll roll up my sleeves and diagram-chase in that particular case then, I guess. I'm intrigued about your comment about homotopy colimits though, if you would like to expand on it (maybe as an answer?) it would be great :)
Mar
14
comment If a morphism of pushouts of complexes (with one arrow monic) is composed of quasi-isos, then the induced arrow is one also
@Zhen: What if one of the arrows is a monomorphism (in the case of the pushout)?
Mar
14
asked If a morphism of pushouts of complexes (with one arrow monic) is composed of quasi-isos, then the induced arrow is one also
Mar
14
comment Mayer-Vietoris Type Sequence For Pushouts
One little comment. A nice reference for homotopy pushouts/pullbacks is Arkowitz's Introduction to homotopy theory. In particular, the second sentence of your second paragraph is proposition 6.2.6.
Mar
12
reviewed Approve Probability of at least one male and one female sharing the same birthday
Mar
10
comment Simply connected reduced suspension on path connected X
@JuanS: I haven't read the posts in detail, but in the first one the counterexample he produces involved the Hawaiian earring which does not satisfy the hypotheses of Freudenthal's suspension theorem.
Mar
6
awarded  Nice Answer
Feb
24
reviewed Reject What makes a limit 'go away'?
Feb
24
reviewed Reject truth tables and validity of arguments
Feb
23
comment The fundamental group of a topological group is abelian
Hey, cute abstract nonsense proof! Thanks for posting it! (+1)
Feb
21
accepted Is an integer a sum of two rational squares iff it is a sum of two integer squares?