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 hom-functor [2014/11/01 16:48]nikolaj hom-functor [2015/02/26 11:55] (current)nikolaj Both sides previous revision Previous revision 2015/02/26 11:55 nikolaj 2014/11/01 16:48 nikolaj 2014/06/26 14:30 nikolaj 2014/06/26 14:30 nikolaj 2014/06/23 22:57 nikolaj 2014/06/23 22:52 nikolaj 2014/06/23 22:45 nikolaj 2014/06/23 22:43 nikolaj 2014/06/23 22:35 nikolaj 2014/06/23 22:33 nikolaj 2014/06/23 18:13 nikolaj 2014/06/23 18:06 nikolaj 2014/06/23 18:04 nikolaj 2014/06/23 18:02 nikolaj 2014/06/23 17:30 nikolaj 2014/06/23 17:29 nikolaj 2014/06/23 17:29 nikolaj 2014/06/23 17:22 nikolaj 2014/06/23 17:17 nikolaj 2014/06/23 17:17 nikolaj 2014/06/23 17:16 nikolaj 2014/06/23 10:56 nikolaj old revision restored (2014/06/23 10:46)2014/06/23 10:54 nikolaj 2014/06/23 10:53 nikolaj 2014/06/23 10:52 nikolaj 2014/06/23 10:51 nikolaj 2014/06/23 10:50 nikolaj 2015/02/26 11:55 nikolaj 2014/11/01 16:48 nikolaj 2014/06/26 14:30 nikolaj 2014/06/26 14:30 nikolaj 2014/06/23 22:57 nikolaj 2014/06/23 22:52 nikolaj 2014/06/23 22:45 nikolaj 2014/06/23 22:43 nikolaj 2014/06/23 22:35 nikolaj 2014/06/23 22:33 nikolaj 2014/06/23 18:13 nikolaj 2014/06/23 18:06 nikolaj 2014/06/23 18:04 nikolaj 2014/06/23 18:02 nikolaj 2014/06/23 17:30 nikolaj 2014/06/23 17:29 nikolaj 2014/06/23 17:29 nikolaj 2014/06/23 17:22 nikolaj 2014/06/23 17:17 nikolaj 2014/06/23 17:17 nikolaj 2014/06/23 17:16 nikolaj 2014/06/23 10:56 nikolaj old revision restored (2014/06/23 10:46)2014/06/23 10:54 nikolaj 2014/06/23 10:53 nikolaj 2014/06/23 10:52 nikolaj 2014/06/23 10:51 nikolaj 2014/06/23 10:50 nikolaj 2014/06/23 10:50 nikolaj 2014/06/23 10:46 nikolaj 2014/06/23 10:30 nikolaj 2014/06/23 10:30 nikolaj 2014/06/23 10:28 nikolaj 2014/06/23 10:27 nikolaj 2014/06/23 10:26 nikolaj 2014/06/23 10:25 nikolaj 2014/06/22 18:44 nikolaj 2014/06/22 18:42 nikolaj 2014/06/22 18:35 nikolaj 2014/06/22 18:35 nikolaj 2014/06/22 18:34 nikolaj 2014/06/22 18:33 nikolaj 2014/06/22 18:32 nikolaj 2014/06/22 18:22 nikolaj 2014/06/22 18:22 nikolaj 2014/06/22 18:22 nikolaj 2014/06/22 18:21 nikolaj 2014/06/22 18:09 nikolaj 2014/06/22 18:09 nikolaj created Line 17: Line 17: === Contravariant hom-functor === === Contravariant hom-functor === - {{ hom.png?​X250}} + >Maybe I'll do a seperate entry later, although that's a little tiresome >Maybe I'll do a seperate entry later, although that's a little tiresome Line 36: Line 36: == More general cases == == 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. + 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 === + - Motivation: The following is very similar to what we did with the maps to $U(1)$ above. Consider a vector space $\mathcal V$ over some field $\mathbb K$ (e.g. $\mathbb R^3$) and some functionals $\mu$ and $\nu$, which are elements of the function space ${\mathbb K}^{\mathcal V}$. We can defined their addition $\mu+'​\nu$ by + - + - $(\mu+'​\nu)(v):​=\mu(v)+\nu(v)$ + - + - simply because their values lie in the field $\mathbb K$, which already has addition. And as a side note, if we restrict ourselves to the linear functionals,​ then this functional space becomes the so called dual vector space. + - + - Now back to the general case. Instead of working with the category ${\bf C}$, one can work with the category of set valued covariant functors ${\bf Set}^{{\bf C}^\mathrm{op}}$ (called [[Presheaf category]]). One substitutes an object $A$ with the contravariant functor $\mathrm{Hom}_{\bf C}(-,A)$ (=[[Yoneda embedding]]) and the arrows actually are the same/​isomorphic to the old ones (=Yoneda lemma). + - The advantage of this is that that new category ​${\bf Set}^{{\bf ​C}^\mathrm{op}}$ has more objects than ${\bf C}$, namely some non-representable functors $F$. (Representable means $F$ is isomorphic ​to some hom-functor anyway.) For example, ​the category ${\bf C}$ might not have products $\times$, but because ${\bf Set}$ has products, ​the category ​${\bf Set}^{{\bf C}^\mathrm{op}}$ always has them. + 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. === Reference === === Reference ===