Let $a$ be a loop about an object $A$. Suppose that $aa=1_A$. Show that $a=1_A$. I'm pretty sure this is true, but I can't prove it. Help?
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3What is a "loop about an object"? A self-map of $A$? If so, your conjecture is really false. – Randall Feb 12 '18 at 19:27
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@Randall: It almost certainly means an endomorphism, otherwise the composite $aa$ doesn't make sense. – Clive Newstead Feb 12 '18 at 19:28
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Assuming what you mean by a loop about an object $A$ is a morphism $A \to A$, this isn't necessarily true. For example, the group $\mathbb{Z}_2$ considered as a one-object category in the usual way has such a morphism.
Clive Newstead
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What if there is a morphism $a:A\to B$ and $b:B\to A$ and $baba=1_A$. Is it true that $ba=1_A$? – alexanderyaacov Feb 12 '18 at 19:40
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@alexanderyaacov: Again no. There is a counterexample (with $A=B$ and $a=b$) in the group $\mathbb{Z}_4$, considered again as a one-object category. – Clive Newstead Feb 12 '18 at 20:15
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Obviously I meant $A\neq B$ and therefore $a\neq b$. But don't bother to look for a counterexample. Thanks, – alexanderyaacov Feb 13 '18 at 00:55
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