# On limits and Colimits

I want to ask a stupid question. I wonder whether following morphism exists in general

Let I be an infinite set. i belongs to I

Hom(A,colimBi)--->limHom(A,Bi) and

ColimHom(A,Bi)---->Hom(A,colimBi)

What I know is if we replace lim by infinite product and colim by infinite coproduct. It exists,but I am not sure in this general case above

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I would suggest that you reread what you have written and correct the misprints, for starters. The first would-be morphism does not make much sense due to some mixup between lim and colim, apparently. In the second one, one "Hom" is missing and there is a misprint "B1". –  Leonid Positselski Dec 12 '09 at 12:44
I am sorry for the misprints,I have corrected –  Shizhuo Zhang Dec 12 '09 at 12:48
No, he is asking about morphisms, not isomorphisms. So the first formula is indeed wrong, but the morphism in the second formula does actually exist. Neither it matters whether the colimit in the second formula is finite or infinite. The morphism exists in both cases, and it is not an isomorphism, in general, in both cases. –  Leonid Positselski Dec 12 '09 at 12:55
The existence of that morphism is trivial and I'm pretty sure that's not what he's looking for. –  Harry Gindi Dec 12 '09 at 12:59
@Shizhou, if you intend these objecs to be abelian groups, you should say so in the question. (cf your comment below.) –  Scott Morrison Dec 12 '09 at 16:40

For any diagram B_i and an object A in a category, there are natural maps of sets:

1. colim Hom(A,B_i) --> Hom(A, colim B_i)
2. colim Hom(B_i,A) --> Hom(lim B_i, A)

These maps need not be isomorphisms, in general (neither even when the diagram is filtered, nor when it is finite). Nor are they isomorphisms for infinite products and coproducts, in general (for finite products and coproducts in an additive category they are isomorphisms, though).

Besides, for any diagram B_i and an object A there are natural isomorphisms of sets:

1. Hom(A, lim B_i) = lim Hom(A,B_i)
2. Hom(colim B_i, A) = lim Hom(B_i,A)

These isomorphisms hold for any diagram (it does not have to be filtered, nor does it have to be finite). Actually, they hold by the definition of lim and colim.

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