Timeline for Example(s) of monoidal symmetric closed category with NNO without infinite coproducts?
Current License: CC BY-SA 3.0
15 events
| when toggle format | what | by | license | comment | |
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| Jun 27, 2014 at 4:57 | answer | added | Mike Shulman | timeline score: 3 | |
| Jun 26, 2014 at 21:02 | vote | accept | Cyrille Corpet | ||
| Jun 26, 2014 at 20:23 | comment | added | Cyrille Corpet | You're right, sorry. The restriction to contractions is only needed to have $\ell_1(\mathbb R)$ as a countable coproduct of $I=\mathbb R$. | |
| Jun 26, 2014 at 15:00 | history | edited | Cyrille Corpet | CC BY-SA 3.0 |
edited title
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| Jun 26, 2014 at 13:14 | comment | added | Yemon Choi | @CyrilleCorpet I think the usual (projective) tensor product of Banach spaces gives you smc structure, since you still have Hom(X \otimes Y, Z) naturally iso to Hom ( X, Hom(Y,Z) ) | |
| Jun 26, 2014 at 13:07 | comment | added | Cyrille Corpet | @YemonChoi: is it really closed ? The usual closed category of Banach is the one with contracting linear maps (which has countable coproducts). | |
| Jun 26, 2014 at 12:33 | comment | added | Yemon Choi | Could you add "with NNO" to the title of your question? (A counterexample to the question in your title, but which I think has no NNO anyway, is the category of Banach spaces and continuous linear maps between them.) | |
| Jun 26, 2014 at 9:24 | answer | added | Zhen Lin | timeline score: 7 | |
| Jun 26, 2014 at 9:09 | comment | added | Cyrille Corpet | This definition allows to define sequences in $X$ by recursion: given a first "element" $1\to X$ and a "recursion law" $X\to X$, it defines a sequence $N\to N$. Nicer properties are given by the symmetric closed structure, for instance, addition is given as a map $N\otimes N\to N$ which is defined by adjunction by a map $N\to \operatorname{Hom}(N,N)$, itself defined by the usual recursive definition in $\operatorname{\mathbf{Set}}$. | |
| Jun 26, 2014 at 9:03 | history | edited | Cyrille Corpet | CC BY-SA 3.0 |
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| Jun 26, 2014 at 8:59 | comment | added | Fernando Muro | I don't get your definition of NNO. It looks like incomplete. | |
| Jun 26, 2014 at 8:24 | history | edited | Cyrille Corpet | CC BY-SA 3.0 |
added 111 characters in body
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| Jun 26, 2014 at 8:23 | comment | added | Cyrille Corpet | Actually, I would like a category where a Natural Number Object could exist. I have edited my question accordingly. | |
| Jun 26, 2014 at 8:14 | comment | added | Zhen Lin | Trivial example: the category of finite sets with the cartesian product... | |
| Jun 26, 2014 at 8:04 | history | asked | Cyrille Corpet | CC BY-SA 3.0 |