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hom-functor [2014/06/23 22:57] nikolaj |
hom-functor [2014/06/26 14:30] nikolaj |
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| Since locally compact groups have the Haar measure, one can form the space of integrable functions $G\to\mathbb C$ and carry it along with these functors. This procedure is the Fourier transform and it really works for all groups in this category! E.g. for $U(1)$ itself, i.e. the periodic interval, we find that the dual group is $\mathbb Z$ and the associated transform is the Fourier series. | Since locally compact groups have the Haar measure, one can form the space of integrable functions $G\to\mathbb C$ and carry it along with these functors. This procedure is the Fourier transform and it really works for all groups in this category! E.g. for $U(1)$ itself, i.e. the periodic interval, we find that the dual group is $\mathbb Z$ and the associated transform is the Fourier series. | ||
| - | Extending this to a category of non-commutative groups is a research subject. Tannakian categories ... Grothendieck stuff ... we see that it's desirable for the Hom-sets to have their own algebraic structure (in the above, the Hom-set was again a group) and this is where many ideas come from. | + | == More general cases == |
| + | - Extending this to a category of non-commutative groups is a research subject. Tannakian categories ... Grothendieck stuff ... we see that it's desirable for the Hom-sets to have their own algebraic structure (in the above, the Hom-set was again a group) and this is where many ideas come from. | ||
| + | - A similar construction works if we consider commutative Banach algebra and their maps maps to $\mathbb C$. This is the Gelfand transform business. It reduces to the Fourier transform if we consider the space $L^1(\mathbb R)$ and convolution as multiplication. | ||
| === Yoneda === | === Yoneda === | ||
