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Ali Caglayan
Ali Caglayan
3:45 PM
@JonasFrey I don't think that is true, did you mean smash product (historically known as reduced join) instead of join? Forgetting about bundles, let $S^V$ denote the sphere obtained from the 1-point compactification of $V$, there is a natural homeomorphism $S^V \wedge S^W \simeq S^{V \oplus W}$. Sphere bundles are just 1-point compactifications of vector bundles which is why I don't think what you have written is true.

In terms of the question I don't think there is any operation on spheres which will give you what you want. As in I cannot see any way of getting $S^{V \otimes W}$ from $S^…
 
 
3 hours later…
Dedalus
Dedalus
7:04 PM
Here is a possibly very stupid question: If I understand correctly, Cisinski, and Hoang Kim Nguyen have, by different methods obtained a version of straightening / unstraightening. Namely, it can be shown that functors into the infinity category of qcats are identified with cocartesian fibrations (if I recall correctly) . The method is quite natural. My question is whether Lurie’s version of straightening is still needed.
Of course, needed is a loaded term. I am more curious as to whether Lurie’s approach is, at the moment, more practical to use.
 
Jonas Frey
Jonas Frey
7:21 PM
@AliCaglayan OK I should have been more precise. I didn't mean the sphere obtained by 1-point compactifying a vector space, but the one obtained by removing the origin. removing 0 from Rn gives S(n-1), and S(n-1)*S(n-1) = S(n+m-1).
 

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