Timeline for Linearity of a dg category $C$ over $HH^0(C)$
Current License: CC BY-SA 4.0
5 events
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| Apr 23, 2020 at 18:43 | comment | added | Daniel Pomerleano | After your prodding, I dug around and found Proposition 7.1.3.13 of Lurie's Higher Algebra which proves the existence of connective covers for E_n algebras. My case is just a special case of this because an E_n structure on a discrete object is just a commutative ring. The proof is along the lines you described (demonstrating the compatibility of the t-structure with the monoidal structure). Thanks! | |
| Apr 23, 2020 at 16:12 | comment | added | Dmitry Vaintrob | Ah I see your question. You can use that the category of coconnective complexes is equivalent (as a symmetric monoidal infinity category) to the full subcategory of the category of unbounded complexes with cohomology in nonnegative degrees. This means an E_n object in one is equivalent to an E_n object in the other. | |
| Apr 23, 2020 at 1:54 | comment | added | Daniel Pomerleano | @DmitryVaintrob: This looks promising, thanks! Sorry for being slow, but how are you defining your map H^0(A) \to A? Just by choosing lifts of the cohomology classes? | |
| Apr 22, 2020 at 22:54 | comment | added | Dmitry Vaintrob | This is certainly true, no obstruction theory needed. As you point out, specifying an $R$-linear structure is equivalent to constructing an $E_2$ homomorphism $R\to CH^*(C)$. Now for any coconnective $E_n$ algebra $A$, the map $H^0(A)\to A$ is a map of $E_n$ algebras (indeed, uniquely so). There are many ways to prove this fact, which essentially amounts to showing that the natural transformation $H^0(X)\to X$ is a symmetric monoidal transformation between symmetric monoidal endofunctors of the category of coconnective complexes. | |
| Apr 22, 2020 at 17:38 | history | asked | Daniel Pomerleano | CC BY-SA 4.0 |