Revisions to Does $\mathbf{Cat}$ have the Cantor–Schröder–Bernstein property?

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edited Aug 19, 2023 at 6:10

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I am wondering if the category of small categories $\mathbf{Cat}$ is known to (not) have the Cantor–Schröder–Bernstein property? That is, for any two categories $\mathcal{C}$ and $\mathcal{D}$, does the statement that there exist embedding functors (monomorphisms in $\mathbf{Cat}$) $\mathcal{F}: \mathcal{C} \rightarrow \mathcal{D}$ and $\mathcal{G}: \mathcal{D} \rightarrow \mathcal{C}$ imply that $\mathcal{C} \cong \mathcal{D}$?

If not, what can we say about the validity of the following weaker statement?

For any two categories $\mathcal{C}$ and $\mathcal{D}$, the statement that there exist embedding functors $\mathcal{F}_1: \mathcal{C} \rightarrow \mathcal{D}$ and $\mathcal{F}_2: \mathcal{D} \rightarrow \mathcal{C}$ implies that there exists bimorphisms (epic embedding functors) in $\mathbf{Cat}$ $\mathcal{G}_1: \mathcal{C} \rightarrow \mathcal{D}$ and $\mathcal{G}_2: \mathcal{D} \rightarrow \mathcal{C}$.

Thank you in advance for your help!

I am wondering if the category of small categories $\mathbf{Cat}$ is known to (not) have the Cantor–Schröder–Bernstein property? That is, for any two categories $\mathcal{C}$ and $\mathcal{D}$, does the statement that there exist embedding functors (monomorphisms in $\mathbf{Cat}$) $\mathcal{F}: \mathcal{C} \rightarrow \mathcal{D}$ and $\mathcal{G}: \mathcal{D} \rightarrow \mathcal{C}$ imply that $\mathcal{C} \cong \mathcal{D}$?

If not, what can we say about the validity of the following weaker statement?

For any two categories $\mathcal{C}$ and $\mathcal{D}$, the statement that there exist embedding functors $\mathcal{F}_1: \mathcal{C} \rightarrow \mathcal{D}$ and $\mathcal{F}_2: \mathcal{D} \rightarrow \mathcal{C}$ implies that there exists bimorphisms (epic embedding functors) in $\mathbf{Cat}$ $\mathcal{G}_1: \mathcal{C} \rightarrow \mathcal{D}$ and $\mathcal{G}_2: \mathcal{D} \rightarrow \mathcal{C}$.

Thank you in advance for your help

I am wondering if the category of small categories $\mathbf{Cat}$ is known to (not) have the Cantor–Schröder–Bernstein property? That is, for any two categories $\mathcal{C}$ and $\mathcal{D}$, does the statement that there exist embedding functors (monomorphisms in $\mathbf{Cat}$) $\mathcal{F}: \mathcal{C} \rightarrow \mathcal{D}$ and $\mathcal{G}: \mathcal{D} \rightarrow \mathcal{C}$ imply that $\mathcal{C} \cong \mathcal{D}$?

If not, what can we say about the validity of the following weaker statement?

For any two categories $\mathcal{C}$ and $\mathcal{D}$, the statement that there exist embedding functors $\mathcal{F}_1: \mathcal{C} \rightarrow \mathcal{D}$ and $\mathcal{F}_2: \mathcal{D} \rightarrow \mathcal{C}$ implies that there exists bimorphisms (epic embedding functors) in $\mathbf{Cat}$ $\mathcal{G}_1: \mathcal{C} \rightarrow \mathcal{D}$ and $\mathcal{G}_2: \mathcal{D} \rightarrow \mathcal{C}$.

Thank you in advance for your help!

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edited Aug 18, 2023 at 18:47

Tian Vlašić

405
1
18

I am wondering if the category of small categories $\mathbf{Cat}$ is known to (not) have the Cantor–Schröder–Bernstein property? That is, for any two categories $\mathcal{C}$ and $\mathcal{D}$, does the statement that there exist embedding functors (monomorphisms in $\mathbf{Cat}$) $\mathcal{F}: \mathcal{C} \rightarrow \mathcal{D}$ and $\mathcal{G}: \mathcal{D} \rightarrow \mathcal{C}$ imply that $\mathcal{C} \cong \mathcal{D}$?

If not, what can we say about the validity of the following weaker statement?

For any two categories $\mathcal{C}$ and $\mathcal{D}$, the statement that there exist embedding functors $\mathcal{F}_1: \mathcal{C} \rightarrow \mathcal{D}$ and $\mathcal{F}_2: \mathcal{D} \rightarrow \mathcal{C}$ implies that there exists bimorphisms (epic embedding functors) in $\mathbf{Cat}$ $\mathcal{G}_1: \mathcal{C} \rightarrow \mathcal{D}$ and $\mathcal{G}_2: \mathcal{D} \rightarrow \mathcal{C}$.

Thank you in advance for your help!

I am wondering if the category of small categories $\mathbf{Cat}$ is known to (not) have the Cantor–Schröder–Bernstein property? That is, for any two categories $\mathcal{C}$ and $\mathcal{D}$, does the statement that there exist embedding functors (monomorphisms in $\mathbf{Cat}$) $\mathcal{F}: \mathcal{C} \rightarrow \mathcal{D}$ and $\mathcal{G}: \mathcal{D} \rightarrow \mathcal{C}$ imply that $\mathcal{C} \cong \mathcal{D}$?

If not, what can we say about the validity of the following weaker statement?

For any two categories $\mathcal{C}$ and $\mathcal{D}$, the statement that there exist embedding functors $\mathcal{F}_1: \mathcal{C} \rightarrow \mathcal{D}$ and $\mathcal{F}_2: \mathcal{D} \rightarrow \mathcal{C}$ implies that there exists bimorphisms (epic embedding functors) in $\mathbf{Cat}$ $\mathcal{G}_1: \mathcal{C} \rightarrow \mathcal{D}$ and $\mathcal{G}_2: \mathcal{D} \rightarrow \mathcal{C}$.

Thank you in advance for your help!

I am wondering if the category of small categories $\mathbf{Cat}$ is known to (not) have the Cantor–Schröder–Bernstein property? That is, for any two categories $\mathcal{C}$ and $\mathcal{D}$, does the statement that there exist embedding functors (monomorphisms in $\mathbf{Cat}$) $\mathcal{F}: \mathcal{C} \rightarrow \mathcal{D}$ and $\mathcal{G}: \mathcal{D} \rightarrow \mathcal{C}$ imply that $\mathcal{C} \cong \mathcal{D}$?

If not, what can we say about the validity of the following weaker statement?

For any two categories $\mathcal{C}$ and $\mathcal{D}$, the statement that there exist embedding functors $\mathcal{F}_1: \mathcal{C} \rightarrow \mathcal{D}$ and $\mathcal{F}_2: \mathcal{D} \rightarrow \mathcal{C}$ implies that there exists bimorphisms (epic embedding functors) in $\mathbf{Cat}$ $\mathcal{G}_1: \mathcal{C} \rightarrow \mathcal{D}$ and $\mathcal{G}_2: \mathcal{D} \rightarrow \mathcal{C}$.

Thank you in advance for your help