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Improved formula markup
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edit approved Jan 25, 2015 at 16:04
Jake
edit approved Jan 25, 2015 at 16:04
Jake
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Can anybody give me an example of a "naturally-occurring" algebraic category C$C$ in which:

  1. C$C$ has two non-isomorphic objects A$A$ and B$B$ which are bi-embeddable via monic maps; but

  2. C$C$ does NOT have any infinite collection A_1$A_{1}$, A_2$A_{2}$, ... of objects which are pairwise bi-embeddable (via monic maps) and pairwise nonisomorphic?

Alternatively: can anybody give a reason why, under some reasonable hypothesis about the category C$C$, property 1 should imply that 2 fails ("there is an infinite collection of pairwise bi-embeddable, pairwise nonisomorphic objects")?

Can anybody give me an example of a "naturally-occurring" algebraic category C in which:

  1. C has two non-isomorphic objects A and B which are bi-embeddable via monic maps; but

  2. C does NOT have any infinite collection A_1, A_2, ... of objects which are pairwise bi-embeddable (via monic maps) and pairwise nonisomorphic?

Alternatively: can anybody give a reason why, under some reasonable hypothesis about the category C, property 1 should imply that 2 fails ("there is an infinite collection of pairwise bi-embeddable, pairwise nonisomorphic objects")?

Can anybody give me an example of a "naturally-occurring" algebraic category $C$ in which:

  1. $C$ has two non-isomorphic objects $A$ and $B$ which are bi-embeddable via monic maps; but

  2. $C$ does NOT have any infinite collection $A_{1}$, $A_{2}$, ... of objects which are pairwise bi-embeddable (via monic maps) and pairwise nonisomorphic?

Alternatively: can anybody give a reason why, under some reasonable hypothesis about the category $C$, property 1 should imply that 2 fails ("there is an infinite collection of pairwise bi-embeddable, pairwise nonisomorphic objects")?

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John Goodrick
John Goodrick
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Bi-embeddability vs. isomorphism

Can anybody give me an example of a "naturally-occurring" algebraic category C in which:

  1. C has two non-isomorphic objects A and B which are bi-embeddable via monic maps; but

  2. C does NOT have any infinite collection A_1, A_2, ... of objects which are pairwise bi-embeddable (via monic maps) and pairwise nonisomorphic?

Alternatively: can anybody give a reason why, under some reasonable hypothesis about the category C, property 1 should imply that 2 fails ("there is an infinite collection of pairwise bi-embeddable, pairwise nonisomorphic objects")?