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I am trying to understand the concept of duality in category theory, but I am having a problem, well illustrated by the following situation.

Let $H$ be any nontrivial subgroup of the alternating group $G=A_5$. Thus there exists a group monomorphism $H \to G$. By duality, there exists a group epimorphism $G \to H$. This implies $H \cong G/N$ for some normal subgroup $N$ in $G$. This is ridiculous, as $A_5$ is simple. What is wrong here?

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What is $G$? Why do you think there is an epimorphism by duality? –  Thomas Andrews Aug 3 '13 at 1:58
    
Ah, in the dual category there is an epimporphism. That's just because an epimorphism in the dual category is just a monomorphism in the original category. There is no magic here. The dual of the group category is not the group category. –  Thomas Andrews Aug 3 '13 at 2:00
    
Yes, that was my problem. Thanks! –  Just Some Old Man Aug 3 '13 at 2:01
    
The key is to realize the dual category is a sort of "abstract nonsense" that shows up often enough in real problems that we stop thinking of it as nonsense. :) –  Thomas Andrews Aug 3 '13 at 2:04

1 Answer 1

up vote 6 down vote accepted

That is not how duality works. The correct statement is that there is an epimorphism $G \to H$ in the opposite category $\text{Grp}^{op}$.

I don't know a nice explicit description of this category, but we can be much more explicit if we restrict our attention to abelian groups. There the functor $A \mapsto \text{Hom}(A, S^1)$ gives an equivalence between $\text{Ab}^{op}$ and the category of compact Hausdorff abelian groups (Pontrjagin duality), and so in particular a monomorphism $A \to B$ between abelian groups is sent to an epimorphism $\hat{B} \to \hat{A}$ between their Pontrjagin duals. For example, the monomorphism

$$\mathbb{Z} \ni n \mapsto 2n \in \mathbb{Z}$$

is sent to the epimorphism

$$S^1 \ni z \mapsto z^2 \in S^1.$$

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I made the mistake of thinking the category of groups and its opposite are equivalent. Duality does not induce an equivalence of categories. Flipping arrows isn't as strong as I thought! Thank you –  Just Some Old Man Aug 3 '13 at 2:00
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@Joseph: the main point of duality arguments is that you often prove a statement that's true in all categories, but then instantiating it in a given category $C$ gives a different statement about the category $C^{op}$ which is then also true in all categories (e.g. you prove something about monomorphisms and this automatically gives a dual statement about epimorphisms). The secondary point is that often different categories you care about are equivalent to each other's opposites in interesting ways. But again, in this case I don't know an interesting description of $\text{Grp}^{op}$. –  Qiaochu Yuan Aug 3 '13 at 2:02
    
That is very powerful and you stated it well! I am at the point of my course where we are leaving specific algebraic objects and entering general category theory, so this is new to me, but I can understand its importance. –  Just Some Old Man Aug 3 '13 at 2:07

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