Timeline for Monad, algebras and reflexive coequalizer
Current License: CC BY-SA 4.0
9 events
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| Feb 25, 2019 at 17:59 | history | edited | Mike Shulman | CC BY-SA 4.0 |
grammar
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| Feb 25, 2019 at 13:25 | answer | added | Todd Trimble | timeline score: 4 | |
| Feb 25, 2019 at 12:53 | comment | added | Todd Trimble | Fosco, $\Delta$ (the augmented simplex category) is initial among strict monoidal categories equipped with a monoid. So given a monad $T = UF$, i.e., a monoid in the endofunctor category $[M, M]$, we get an induced cosimplicial object $\Delta \to [M, M]$, and dually we get a simplicial object $\Delta^{op} \to [N, N]$ which upon evaluation at $x \in Ob(N)$ gives a simplicial object $\Delta^{op} \to N$. | |
| Feb 25, 2019 at 12:34 | history | edited | Paris | CC BY-SA 4.0 |
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| Feb 25, 2019 at 12:33 | history | edited | David Roberts♦ | CC BY-SA 4.0 |
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| Feb 25, 2019 at 12:29 | history | edited | Paris | CC BY-SA 4.0 |
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| Feb 25, 2019 at 11:07 | comment | added | Harry Gindi | @FoscoLorengian You get both from a comonad. The canonical resolution arising from a comonad gives a simplicial object, while the cobar construction gives a cosimplicial object. An example of a simplicial object coming from a comonad is the left adjoint to the coherent nerve restricted to nerves of categories. This produces a simplicial object from the free-forgetful comonad on Cat. See: ncatlab.org/nlab/show/canonical+resolution | |
| Feb 25, 2019 at 10:45 | comment | added | fosco | $G$ is a comonad, I think you get a cosimplicial object $G : \Delta\to N$. | |
| Feb 25, 2019 at 9:53 | history | asked | Paris | CC BY-SA 4.0 |