Timeline for Generalizing indexed coproduct from $\mathrm{Set}$ to other monoidal categories
Current License: CC BY-SA 3.0
13 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Apr 29, 2014 at 23:47 | vote | accept | Brent Yorgey | ||
| Apr 29, 2014 at 21:55 | answer | added | Peter LeFanu Lumsdaine | timeline score: 5 | |
| Apr 29, 2014 at 21:50 | comment | added | Peter LeFanu Lumsdaine | @BrentYorgey: you say in the first comment that you don't want to use the "pick some ordering of J, then use this to take the J-iterated monoidal product" approach, because you're working in a constructive setting. However, what definition of "finite" are you using for J? In such settings there are several options for this? If you assume that J is cardinal-finite, then you can use the indexed-monoidal-product approach without difficulty. In fact, I may expand this into an answer. | |
| Apr 29, 2014 at 20:48 | comment | added | Michal R. Przybylek | Because your grupoid $\mathbb{B}$ consists of all objects from $\mathbf{FinSet}$, and every functor preserves isomorphisms, this (strong) monoidal structure restricts to a monoidal structure on $\mathbb{B}$. Now, what kind of generalization are you looking for? Obviously, you may replace $\mathbf{FinSet}$ by any category with any (strong) monoidal structure, and everything will remain true. | |
| Apr 29, 2014 at 20:48 | comment | added | Michal R. Przybylek | Brent, I have no idea of what you are trying to achieve, but I can tell you what you are doing. First, forget about $\mathbf{Set}$, because you are talking about finite sets. So, you have observed that the category of finite sets $\mathbf{FinSet}$ has finite products. Finite products in $\mathbf{FinSet}$ are generated by the (strong) monidal structure $\langle 1, \times \rangle$ on $\mathbf{FinSet}$. (cont) | |
| Apr 29, 2014 at 20:30 | comment | added | Brent Yorgey | Yes, in my use case $\mathcal J$ is always finite. | |
| Apr 29, 2014 at 20:25 | comment | added | Chris Heunen | In the example with $\mathbb{B}$ you're taking $\mathcal{J}$ finite. Is this always the assumption? | |
| Apr 29, 2014 at 19:49 | comment | added | Jacques Carette | Care to expand on how copowers fit in? As far as I can tell, the main theorems that show that they exist fail in Brent's setup as described above. [I can see that they are related, just not that they actually answer the question as posed.] | |
| Apr 29, 2014 at 17:09 | comment | added | John Wiltshire-Gordon | Check out ncatlab.org/nlab/show/copower | |
| Apr 29, 2014 at 17:08 | review | First posts | |||
| Apr 29, 2014 at 17:12 | |||||
| Apr 29, 2014 at 17:01 | comment | added | Brent Yorgey | I should also note that if $\oplus$ is symmetric, one can construct a functor $\mathcal C^{\mathcal J} \to \mathcal C$ by invoking the axiom of choice---for example, by arbitrarily choosing an ordering on the objects of $\mathcal J$, taking the product in the given order, and then showing that the ordering does not matter up to isomorphism. However, I want to avoid the axiom of choice, because I am working in a computational/constructive setting. | |
| Apr 29, 2014 at 16:58 | history | edited | Brent Yorgey | CC BY-SA 3.0 |
$\mathcal J$ must have a finite set of objects
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| Apr 29, 2014 at 16:51 | history | asked | Brent Yorgey | CC BY-SA 3.0 |