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A category is called balanced if every bimorphism is an isomorphism.

Consider a concrete category such that every bijective morphism is a isomorphism. Does the category is balanced? Does converse is true?

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2 Answers 2

up vote 3 down vote accepted

In the category of rings, every bijective morphism is an isomorphism, but the inclusion of $\mathbb{Z}$ in $\mathbb{Q}$ is a bimorphism that is not an isomorphism, so the category of rings is not balanced.

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The converse is true, of course: a bijective morphism in a concrete category must be a bimorphism and hence if a concrete is balanced, all its bijective morphisms are isomorphisms. –  Rob Arthan Aug 5 '13 at 16:58
    
Is there a general name for concrete categories where isomorphisms are exactly bijective morphisms? –  Corvus Aug 5 '13 at 17:40
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Generally speaking, whenever you see a definition that says: A thing X is called Y if.....

what they really mean is: A thing X is called Y iff...

In other words the definition is a logical equivalence (a logical biconditional).

The definition of balanced category in ncatlab is an exanple of this usage

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Would the downvoter mind to explain his/her downvote? The OP wrote: if every bimorphism is an iso we have a balanced category, if we have a balanced category , is it true that every bimorphism is an iso? I simply said : yes it is true. What's wrong with it? –  magma Aug 6 '13 at 19:28
    
I wasn't the down-voter, but you should have another look at the original question to see why you were down-voted. "Bimorphism" and "bijective morphism" are distinct notions, so the question has some mathematical content. –  Rob Arthan Aug 6 '13 at 22:24

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