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| Shades |
Friday, May 31, 2013
Thursday, May 30, 2013
Wednesday, May 29, 2013
Thursday, May 16, 2013
Wednesday, May 15, 2013
The face of my heart
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| The face of my heart |
The face of my heart…
Look into the eye,
see,
see,
see,
sea,
sea of never-ending love for you; be for me; us
Wednesday, May 8, 2013
Functor Categories as Topoi
Consider the category of functors between two categories $E:=D^C$, with D being Sets. It may be shown that E is complete and co-complete due to the fact Sets is; thinking of each object in E as a map from C to Sets, in the sense of evaluation, it can be shown that $(\varinjlim F_i) (C)= \varinjlim F_i(C)$ where $F_i$ are objects in E. In particular $-\times E: E \rightarrow E$ commultes with both types of limits.
Essentially E inheriting Sets' properties, similarly it can be shown that it is abelian, with enough injective objects (again see See Tamme).
On objects A of E, PE(A) defined to be the set of subfunctors $Hom_E(-,A)xE$ is a functor with values in E, and the subfunctors FxE are in bijection with the morphisms in Nat(F,PE). Hence E is a topos.
The Functor category E as described above is called a presheaf, and in the case of a locally small category, via a Yoneda Embedding it may be though of as enlarging an arbitrary category C in such a way, that the enriched C will be complete, co-complete, abelian and posses enough injectives.
Tuesday, May 7, 2013
A little something on Topoi
A Topos E is a mathematical object called a locally small Category, which has limits and a power object. That is to say that for any object A of E the functor Sub($- \times A$) of objects which factor through $- \times A$ is non-empty member of the category $Sets$. Moreover there is an object of E called the power object of A for which Hom(-,PA)$\cong$ Sub(-,-$\times$ A).
Note: recall since E is locally small Hom(-PA) is a set so the equivalence of objets above is a bijection in Sets.
Some examples of Topoi would be the category sets; here for any object X PX is $\mathscr{P}$(X) its power set. With the natural equivalence of $\phi$: Hom(-,PA)$\cong$ Sub(-,-$\times$ A) to be $\phi$:Hom(1,PB)$\rightarrow$Sub(B).
Note: recall since E is locally small Hom(-PA) is a set so the equivalence of objets above is a bijection in Sets.
Some examples of Topoi would be the category sets; here for any object X PX is $\mathscr{P}$(X) its power set. With the natural equivalence of $\phi$: Hom(-,PA)$\cong$ Sub(-,-$\times$ A) to be $\phi$:Hom(1,PB)$\rightarrow$Sub(B).
Monday, May 6, 2013
Sunday, May 5, 2013
Contemporary Sound, Radio-Show Broadcast !!!!
Contemporary Sound Internet-Radio Broadcast (Its on when I'm on)
If I;m listening on Grooveshark.. then if you wish; you can listen to my current mix :))
(this is available only when I'm on, so tune in to find out :))) )
If I;m listening on Grooveshark.. then if you wish; you can listen to my current mix :))
(this is available only when I'm on, so tune in to find out :))) )
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