Schröder–Bernstein theorem in category.

category-theorytopos-theory

In set theory, Schröder–Bernstein theorem assert for every set $A$ and $B$ if there exists injections from $A$ into $B$, from $B$ into $A$. Then there exists a bijection from $A$ onto $B$.

I want to know does this assertion holds in category?

Let $C$ be a category with terminal object and pull backs. For two objects $A$ and $B$ if there exists injections from $A$ into $B$ and from $B$ into $A$, then there exists an arrow from $A$ into $B$ which is injective and surjective.
If necessary, we assume the well-pointed property for category.

The following is some definitons;

  1. An arrow $f:A\rightarrow B$ is injective if for every parallel arrow $h_1:1\rightarrow A$ and $h_2:1\rightarrow A$, $fh_1=fh_2$ implies $h_1=h_2$. where $1$ is the terminal object.
  2. An arrow $f:A\rightarrow B$ is surjective if for an arrow $y:1\rightarrow B$ there exists some arrow $x:1\rightarrow A$ such that $fx=y$ holds.
  3. A category is well-pointed if the terminal object is a separator. An object $S$ is a separator if for any parallel arrows $x,y:A\rightarrow B$ if $x\not= y$, there exists $f:S\rightarrow A$ such that $xf\not= yf$

Best Answer

This fails in the category of topological spaces.

The terminal object is the one-element space, and it is easy to check that this is a separator, and that being "injective" or "surjective" is the same as being injective or surjective as a function.

But if $A=(0,1)$ and $B=[0,1]$ are an open and a closed interval, then there are continuous injective maps in both directions, but no continuous bijection between the two.

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