Timeline for Categorical duals in Banach spaces
Current License: CC BY-SA 2.5
8 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jan 5, 2010 at 18:35 | vote | accept | Reid Barton | ||
| Jan 5, 2010 at 18:35 | vote | accept | Reid Barton | ||
| Jan 5, 2010 at 18:35 | |||||
| Jan 5, 2010 at 18:30 | comment | added | Reid Barton | (Just to spell out exactly how this is related to my original question: Asking for a dual for an object $X$ of a closed symmetric monoidal category is the same as asking for an object $X^*$ and an identification $\mathbf{Hom}(X, {-}) = X^* \otimes {-}$. In particular (using closedness again) $\mathbf{Hom}(X, {-})$ must commute with colimits. In my case $X = (\mathbb{R}^2)_1 = \mathbb{R} \amalg \mathbb{R}$, and so $\mathbf{Hom}(X, V) = V \times V$. | |
| Jan 5, 2010 at 18:15 | comment | added | Reid Barton | Here is a concrete counterexample for someone to sanity check. It seems that "the set of extreme points of the unit ball" is an isomorphism invariant of a Banach space, and that it preserves coproducts and products, at least the ones used in forming $(R^{\amalg n})^{\times 2}$ and $(R^{\times 2})^{\amalg n}$. It then sends these spaces to sets of cardinality $(2n)^2$ and $4n$, respectively, which are distinct for $n \ge 2$. This shows that $V \mapsto V \times V$ doesn't commute with coproducts. | |
| Jan 5, 2010 at 18:02 | comment | added | Yemon Choi | I think so; see the updated entry. | |
| Jan 5, 2010 at 18:02 | history | edited | Yemon Choi | CC BY-SA 2.5 |
response to comment/query
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| Jan 5, 2010 at 17:46 | comment | added | Reid Barton | And the injective and projective tensor products are non-isometric in this case for general V, right? | |
| Jan 5, 2010 at 17:25 | history | answered | Yemon Choi | CC BY-SA 2.5 |