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(retitled the question as per the comment below)
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Jakob
Jakob
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Monomorphisms with retracts When are all split monomorphisms complemented?

In a category $\mathcal C$, let $f: X \rightarrow Y$ and $g: Y \rightarrow X$ such that $g \circ f = id_X$. Is there a general criterion on $\mathcal C$ such that the following holds: there is an object $Z$ such that $Y = X \sqcup Z$ (the coproduct of $X$ and $Z$)?

Thanks for any answer.

(retitled the question as per the comment below)

Monomorphisms with retracts split?

In a category $\mathcal C$, let $f: X \rightarrow Y$ and $g: Y \rightarrow X$ such that $g \circ f = id_X$. Is there a general criterion on $\mathcal C$ such that the following holds: there is an object $Z$ such that $Y = X \sqcup Z$ (the coproduct of $X$ and $Z$)?

Thanks for any answer.

When are all split monomorphisms complemented?

In a category $\mathcal C$, let $f: X \rightarrow Y$ and $g: Y \rightarrow X$ such that $g \circ f = id_X$. Is there a general criterion on $\mathcal C$ such that the following holds: there is an object $Z$ such that $Y = X \sqcup Z$ (the coproduct of $X$ and $Z$)?

Thanks for any answer.

(retitled the question as per the comment below)

Source Link
Jakob
Jakob
  • 31
  • 2

Monomorphisms with retracts split?

In a category $\mathcal C$, let $f: X \rightarrow Y$ and $g: Y \rightarrow X$ such that $g \circ f = id_X$. Is there a general criterion on $\mathcal C$ such that the following holds: there is an object $Z$ such that $Y = X \sqcup Z$ (the coproduct of $X$ and $Z$)?

Thanks for any answer.