last day (2471 days later) » 

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23:44
Thanks for helping me
Long
Long
yea np
do you follow a lecture or textbook ?
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Topos
I've been looking at several different algebra books (e.g. Hungerford)
I understand how you found the bijection, but to show that it is natural I need to find a natural transformation to make the square commute, right?
Long
Long
no, the bijection IS the natural transformation
first things first
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but don't I separately need to show that it is natural?
Long
Long
wait a sec, let's clarify some things before we march on :D
can you identify the /functors/, between which I constructed the natural transformations?
functors
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23:48
wait, wasn't the bijection between the sets of morphisms B(F(A),B) and A(A,G(B))?
Long
Long
let's use lowercase letters for objects in A and B
because F(A) doesnt make sense
F : A -> B
so you can only apply F to an object a \in A
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ok, then the sets of morphisms B(F(a), b) and A(a, G(b))
for any fixed a, b
Long
Long
yes, but then again not every bijection will do, because we need this /naturality/ thing
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Topos
yeah, that's what I don't understand
we have two sets, and we have a bijection, but how do we show that it is natural?
Long
Long
the concept we need is /natural transformation/
can you define it?
so the recipe goes as follows
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23:53
it is a map between functors, and it has two maps, F(a) \arrow G(a') and G(a) \arrow G(a') such that the diagram commutes
Long
Long
not quite
the functors cannot be arbitrary, they need to have the same source and target
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Topos
yes
Long
Long
so you have say two functors F : A -> B, H : A -> B
and a natural transformation between them assigns to each object a \in A a morphism F(a) -> H(a), which is a morphism in B
not just two
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and then we also need the diagrams of the morphisms to commute, correct?
Long
Long
yes, so you get a diagram for each morphism in A
you can try to work out this example:
(later)
the assignment V -> V^** maps a vector space to its double-dual
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23:58
i'll try that later
but the original problem dealt with sets, which is what confused me
the isomorphism was between sets, I mean
Long
Long
no, we have functors
actually
so thats the next thing
in order to have a natural transformation we need to identify the functors
these are:

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