Homotopy Theory

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5:51 AM

@NarukiMasuda Note that when Lurie defines a distinguished triangle, he specifies also the map $Z→X[1]$, the requirement I wrote above is just another way of saying that this is the canonical map arising from the Puppe sequence and not some random thing

2 hours later…

Saal Hardali

7:38 AM

Is there a noetherian local ring $R$ of dimension $n$ s.t. the punctured spectrum $Spec R - \{ 0 \}$ can be covered by less than $n$ affine open subsets?

If we add the condition that the open affines are also "basic" then this can't happen because this is an equivalent definition of dimension, i'm wondering whether this condition can be dropped. I highly suspect that it can't be dropped but I have no counter example.

1 hour later…

Andrea Marino

8:44 AM

Hi guiys, another question for you :) Lurie define in higher algebra both free algebras and operadic Kan extensions, apparently as two different instances. The main thing one has to figure out to see the equivalence is the following: given a functor $A \to B$ of infinity operads and an element $b \in B$, one can do two things:
1. Take the relative $X \to \Delta^1 \times \Nerve(\Fin)$ and then compute $ A \times_{X} X_{/b}$;
2. Take $A \times_B B_{/b}$;
In the case of operadic extensions one uses the first version, and in the case of free algberas one takes the second. They seems to be equiv…

Andrea Marino

9:14 AM

ok, solved with an explicit model of the relative nerve :S quite messy but works...

12 hours later…