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Problem.

Let $\mathbf{A}$ and $\mathbf{B}$ be two categories and $F:\mathbf{A}\to\mathbf{B}$, $G:\mathbf{B}\to\mathbf{A}$ be two functors that are embeddings (i.e., injective on objects and on $\mathsf{Hom}$-set restrictions). Are $\mathbf{A}$ and $\mathbf{B}$ isomorphic?

It is clear that the object part (respectively, the morphism) part of $\mathbf{A}$ and the object part (respectively, the morphism) part of $\mathbf{B}$ are in bijective correspondence. But whether the bijection on $\mathbf{A}$-morphisms satisfies the functorial properties is not clear to me and a problem which I haven't been able to solve yet.

The reason for giving the title as I have given is due to the "natural" similarity between this problem to the Schröder–Bernstein theorem. Hence I was wondering whether the following is true,

In the metacategory of all categories $\mathbf{CAT}$, is it true that the embeddings are the monomorphisms?

If this is so (I couldn't prove it either) then the only the difference between the Schröder–Bernstein theorem and this problem would be that the Schröder–Bernstein theorem applies to the morphisms of the category $\mathbf{Set}$ whereas here in this problem we are talking about the same question for the $\mathbf{CAT}$-morphisms.

2 Answers2

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The analogue of Schröder–Bernstein is false for groups, as shown here. Let $f:G\to H$, $k:H\to G$ be such an example. We can translate this example to the case of categories as follows.

Let $\mathbf BG$ be the category with one object, called $\bullet$, and with $\mathrm{Hom}(\bullet,\bullet)=G$. Composition of morphisms is by multiplication in G, and $\mathrm{id}_\bullet$ is just the identity of $G$. Define $\mathbf BH$ analogously.

Now define a functor $\mathbf Bf:\mathbf BG\to \mathbf BH$ by sending $\bullet_G$ to $\bullet_H$ and acting on morphisms by $f$. Likewise define $\mathbf Bk$.

Then $\mathbf Bf$ and $\mathbf Bk$ are injective on objects and morphisms, but $\mathbf BG$ and $\mathbf BH$ aren't isomorphic because $G$ and $H$ aren't.

Oscar Cunningham
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    What if we assume full-and-faithful? – Oscar Cunningham Sep 27 '19 at 14:45
  • I am not saying that the analogue of Schröder–Bernstein theorem is true for every category. I am only asking whether it is true for $\mathbf{CAT}$. I am not sure how your post answers my question. –  Sep 27 '19 at 15:00
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    @user170039 You can see every group as a one-object category, and group morphisms as functors, so the counterexample in groups is also a counterexample in categories. – Arnaud D. Sep 27 '19 at 15:01
  • @user170039 I added some explanation. – Oscar Cunningham Sep 27 '19 at 15:08
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The Bernstein-Schröder theorem also fails for posets : for example, take $A$ to be the poset of natural numbers and $B=A\sqcup A$, with no order relation between the two copies of $A$. $A$ can be embedded into $A$ as either copy, $B$ can be embedded into $A$ by identifying one copy with the odd numbers and the other with the even numbers. The two posets are not isomorphic since $A$ is a total order and not $B$. Note that this implies that as categories, they are not even equivalent.

This MO answer gives another counterexample : the posets $[0,1]$ and $[0,1)$ can be embedded into each other, but are not isomorphic. Here the two orders are total, and the embeddings are initial segments. This means that as functors between categories, they are actually fully faithful; yet the two corresponding categories are, again, not even equivalent since only one has a terminal object.

Arnaud D.
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