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Timeline for What is the symmetric monoidal structure on the $(\infty,1)$-category of spectra?

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Dec 29, 2017 at 14:44 comment added Anton Fetisov As others have answered, the product itself is constructed very easily and explicitly, and coincides with the classic products on the category of spectra. The bulk of Lurie's proof goes to construct the $E_\infty$-structure, i.e. provide all required associators, commutators, twistors and higher coherences between different products. Even stating it precisely isn't simple, thus a very abstract approach is required. The classical theories encode $E_\infty$-action in explicit geometric objects like actions of orthogonal or symmetric groups, so remaining structure is commutative "on the nose".
Dec 29, 2017 at 11:05 answer added Rune Haugseng timeline score: 6
Dec 29, 2017 at 6:55 history edited Exit path CC BY-SA 3.0
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Dec 29, 2017 at 6:48 answer added Denis Nardin timeline score: 11
Dec 29, 2017 at 4:34 vote accept Exit path
Dec 29, 2017 at 4:19 answer added Dylan Wilson timeline score: 35
Dec 29, 2017 at 3:51 history edited YCor
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Dec 29, 2017 at 3:25 comment added Exit path @QiaochuYuan A sufficient answer would be some kind of concrete description of the object $X \wedge Y$, or at least what it looks like in the homotopy category (is it just the ordinary smash product for symmetric spectra?)
Dec 29, 2017 at 3:22 comment added Qiaochu Yuan On objects it's determined by the condition that it preserves ($\infty$-)colimits in each variable and has unit the sphere spectrum. This is tightly analogous to the way the tensor product of abelian groups is determined by the condition that it preserves colimits in each variable and has unit $\mathbb{Z}$. Is this the sort of thing you're asking for? I don't really know what would qualify as an answer to this question.
Dec 29, 2017 at 3:09 history asked Exit path CC BY-SA 3.0