Coproducts in dagger categories

Question

I am a beginner with category theory. I have defined a category that seems like it might be useful for me. (The details don't matter here.) It turns out that this category has a product, and it also turns out that this category is a dagger category, in that for every morphism $f:A\to B$ there is a morphism $f:B\to A$, and vice versa.

From these two properties it seems to follow that my category has a coproduct, since I can just reverse all the arrows in the definition of the product to get the definition of a coproduct. However, while the product is nice and easy to interpret the coproduct seems a bit strange and mysterious.

It seems like the dagger property might make the coproduct relate to the product in some simple, convenient way that would make its interpretation obvious. However, I haven't yet found much information about how products and coproducts behave, in general, in dagger categories. Are there general results on this that could help with understanding my specific case?

Answer 1

In a dagger category, coproducts are products: applying $\dagger$ to a limit cone defining the product gives the colimit cone defining the coproduct.

In fact, you can check directly that the product satisfies the formula defining the coproduct: there is a natural bijection

$$ \hom(X \times Y, Z) \cong \hom(X, Z) \times \hom(Y, Z) $$

The forward and backwards directions are

• $f \mapsto (f \pi_0^\dagger, f \pi_1^\dagger) $
• $(g, h) \mapsto \langle g^\dagger, h^\dagger \rangle ^\dagger$

where $\pi_0 : X \times Y \to X$ (et al), and for $u : Z \to X$ and $v : Z \to Y$, I use $\langle u, v \rangle$ to denote the corresponding map $Z \to X \times Y$.