Timeline for Is the tensor product of chain complexes a Day convolution?
Current License: CC BY-SA 4.0
10 events
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| May 5, 2020 at 6:45 | vote | accept | Emily | ||
| May 5, 2020 at 6:45 | comment | added | Emily | This is great! Thanks, Alexander! (And everyone else!) | |
| May 4, 2020 at 17:47 | comment | added | Tim Campion | A chain complex in any $Ab$-enriched category is an $Ab$-enriched functor $\mathcal A$ $\mathsf C \to \mathcal A$. If $\mathcal A$ is additive, then this is equivalently an additive functor $\bar{\mathsf C} \to \mathcal A$, where $\bar{\mathsf C}$ is the additive envelope of $\mathsf C$ (i.e. its completion under direct sums). I haven't given this much thought, but I think that $\bar{\mathsf C}$ is closed under tensor product in $Ch(Ab)$. So by passing to a slightly larger base category, we get $\otimes$ as the Day convolution of a monoidal rather than a promonoidal structure. | |
| May 4, 2020 at 12:55 | comment | added | Phil Tosteson | In the terminology of my answer, this profunctor is the bimodule. I didn't realize this was the original generality of Day convolution! This generality makes more sense because it is Morita invariant. | |
| May 4, 2020 at 12:09 | history | edited | Alexander Campbell | CC BY-SA 4.0 |
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| May 4, 2020 at 11:33 | history | edited | Alexander Campbell | CC BY-SA 4.0 |
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| May 4, 2020 at 11:21 | history | edited | Alexander Campbell | CC BY-SA 4.0 |
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| May 4, 2020 at 11:10 | history | edited | Alexander Campbell | CC BY-SA 4.0 |
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| May 4, 2020 at 10:55 | history | edited | Alexander Campbell | CC BY-SA 4.0 |
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| May 4, 2020 at 10:49 | history | answered | Alexander Campbell | CC BY-SA 4.0 |