Jacob Lurie
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Member for 14 years, 5 months
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Last seen more than 3 years ago
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Example of a (presentable $k$-linear $\infty$-)category which is dualizable but not compactly generated?
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Cotangent complex of perfect algebra over a perfect field
The vector space $\mathbf{R}[i] \otimes_{ \mathbf{R} } \mathbf{R}[j]$ does not have a unique $\mathbf{R}$-algebra structure, even if you require the inclusion maps $\mathbf{R}[i] \hookrightarrow \mathbf{R}[i] \otimes_{ \mathbf{R} } \mathbf{R}[j] \hookleftarrow \mathbf{R}[j]$ to be maps of $\mathbf{R}$-algebras. There's a unique algebra structure in which $\mathbf{R}[i]$ and $\mathbf{R}[j]$ commute with each other, but also algebra structures where they do not (like $\mathbf{H}$). The ring spectrum $B^{+}$ is like the latter.
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Cotangent complex of perfect algebra over a perfect field
The ring of quaternions $\mathbf{H}$ contains commutative subfields $\mathbf{R}[i]$ and $\mathbf{R}[j]$, and the inclusion maps $\mathbf{R}[i] \hookrightarrow \mathbf{H} \hookleftarrow \mathbf{R}[j]$ induce an isomorphism $\mathbf{R}[i] \otimes_{\mathbf{R}} \mathbf{R}[j] \simeq \mathbf{H}$. But $\mathbf{H}$ is not commutative.
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The universal property of the unseparated derived category
@Yonatan It's not part of the definition, but it does happen automatically (every topos can be realized as the category of sheaves with respect to the canonical topology on itself).
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The universal property of the unseparated derived category
@Yonatan I would say that the $\infty$-topos analogue of the unseparated derived category is the construction which carries a $1$-topos $\mathcal{X}$ to the $\infty$-category of space-valued sheaves on $\mathcal{X}$, where you equip $\mathcal{X}$ with the canonical Grothendieck topology. So the universal property is: coproducts go to coproducts, and Cech nerves of effective epimorphisms go to colimit diagrams. (If you ask this for general hypercoverings, you get the $\infty$-topos of hypercomplete sheaves. This is more like the analogue of the usual derived category.)
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The universal property of the unseparated derived category
@Denis: I don't see how say anything about that (except in the case where $\mathcal{C}$ is stable with a nice t-structure, and the functors are required to be right t-exact).
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The universal property of the unseparated derived category
Added some missing subscripts and coproduct-preservation hypotheses that I think are needed when the target category is not prestable.
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What is homology anyway?
@Saal: In response to the question "Why isn't there a six-functor formalism for homology?", I think what I'm saying is "There is, and it is identical to the usual six-functor formalism": the latter is no more about "cohomology" than it is about "homology". If your last question is interpreted as "Can one define some version of homology, in terms of sheaf theory, for spaces that are not locally compact?", then I don't know. But that's venturing pretty far away from the usual context in which one talks about six functors, constructibility, Verdier duality, etcetera.
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What is homology anyway?
I'm not sure what you're asking. You can contemplate the Verdier dual of any (complex of) sheaves $F$ on any (locally compact) space $X$. You can then take the compactly supported cohomology of $DF$, which will be (pre)dual to the cohomology of $F$ under some mild assumptions. This takes algebras with respect to the usual tensor product to coalgebras with respect to the $!$-tensor product (this again requires some finiteness, but that's already present in the case where $X$ is a point: the dual of a coalgebra is always an algebra, but the dual of an algebra is not always a coalgebra).