Timeline for When is the opposite of the category of algebras of a Lawvere theory extensive?
Current License: CC BY-SA 4.0
9 events
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| Dec 28, 2021 at 22:22 | comment | added | Simon Henry | It's not clear to me. It hands you some operations - to be honest similar to the ones in Broodryk's second paper - they look more general than the example we have, but I don't quite see how to cook up mew interesting example. | |
| Dec 28, 2021 at 22:13 | comment | added | John Baez | This answer is easier for me to understand than Broodryk's paper. Can someone use this answer to cook up an example of a Lawvere theory with a coextensive category of models that's quite different from the theory of commutative rings, or commutative rigs, or other variations on that theme? Or does the co-étaleness of $\Delta : T \to T \times T$ and $T \to 1$ practically hand us a commutative multiplication and unit on a silver platter? | |
| Dec 28, 2021 at 22:07 | history | edited | John Baez | CC BY-SA 4.0 |
fixed spelling errors / typos
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| Dec 28, 2021 at 17:09 | comment | added | Ivan Di Liberti | A couple of remarks on my comment. (1) The theory $2$ is something like the free category with products over the category $\{\cdot \cdot\}$. (2) I think Lawvere theories are tensored over Cat so that $T \otimes 2$ is actually the same of $T \circ \{\cdot \cdot\}$. (3) If anything, I think my comment hints that your $T \times T$ could be $T \coprod T$, of course this would follow if Law is itself extensive. | |
| Dec 28, 2021 at 16:25 | comment | added | Simon Henry | Interesting point. I was thinking of Lawere theory as category with finite product and I think I really mean $T \times T$ as categories with finite product... But that should be equivalent to the tensor product with $2$.. and this might give a better way to think about it as this might avoid the problem I discuss at the end... | |
| Dec 28, 2021 at 16:20 | comment | added | Ivan Di Liberti | I think that your $T \times T$ is really $T \otimes 2$, where $\otimes$ is the tensor product of theories and $2$ is the Lawvere theory classifying of pairs of sets. | |
| Dec 28, 2021 at 16:19 | history | edited | Simon Henry | CC BY-SA 4.0 |
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| Dec 28, 2021 at 16:12 | history | edited | Simon Henry | CC BY-SA 4.0 |
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| Dec 28, 2021 at 16:01 | history | answered | Simon Henry | CC BY-SA 4.0 |