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hom-functor [2014/11/01 16:48]
nikolaj
hom-functor [2015/02/26 11:55] (current)
nikolaj
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 === 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
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 == 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 ===
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