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comment Quasi-isomorphism from “almost acyclic” complex to its homology
In fact, what does "$X$ is quasi-isomorphic to $Y$" mean?
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answered History of category theory
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comment On pushouts and mapping cylinders in exact categories
For the prospective reader, I might add that the easy observation that the forgetful functor from $C\mathcal N$ to the category of graded objects over $\mathcal N$ reflects exactness is implicitly being used.
Jun
17
comment On pushouts and mapping cylinders in exact categories
Great! In fact, not too many days ago I managed to solve the problem with help from my supervisor, and I was planning to post it here very soon. I got the same thing as you though I had to use elements to be convinced that the $P$ you define is effectively the pushout, etc. Thanks a lot for your effort on an already old and quite ignored question.
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accepted On pushouts and mapping cylinders in exact categories
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accepted Is $\Omega \tilde X \simeq \Omega_0 X$?
May
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comment Is $\Omega \tilde X \simeq \Omega_0 X$?
Thanks! I'll eagerly read your expanded answer.
May
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comment Is $\Omega \tilde X \simeq \Omega_0 X$?
Nice! I really like the argument, very clean. Didn't Milnor prove a theorem that loop spaces of CW-complexes have the homotopy type of a CW complex? (I didn't really ever read the actual formal statement), if that's the case then if $X$ is a CW-complex the proof works, and I'm interested in CW complexes so that's that :)
May
30
comment Is $\Omega \tilde X \simeq \Omega_0 X$?
Well ok, this is essentially the statement of the unique path lifting property, disregarding continuity issues, I think. What I still don't really get is why $\Omega_0$ appears.