$\newcommand{\G}{\mathcal G}$I have been playing around with the category of groups, and seeing if it can be defined in a purely category-theoretic way (without appeal to binary operations and inner set-theoretic structure). My idea was to consider it to be a category in which each object has a morphism corresponding to multiplication.
I then stumbled across the following realisation. Denote the category of groups by $\G$, and the identity functor $I: \G \to \G$. Define the product functor $P: \G \to \G$ by taking any object $G$ to the direct product $G \times G$, and any morphism $\phi: G \to H$ to the product morphism $\phi \times \phi: G\times G \to H \times H$ (set-theoretically, $(\phi \times \phi)(g_1, g_2) = (\phi(g_1), \phi(g_2))$). Now, if we define multiplication on an object $G$ as a map $\mu_G: G \times G \to G$, one can show that for any objects $G, H$ and any morphism $\phi: G \to H$ the following diagram commutes:
As such, it is ever so tantalising to say that group multiplication is a natural transformation from the product functor to the identity functor. However, there's a big problem: generally speaking, $\mu_G: G \times G \to G$ is not a group homomorphism! We can only consider it to be a morphism in our category if we think of our groups as sets, which is what I was attempting to avoid in the first place. It appears that my idea was wrong, but it looks so close to right.
I am therefore wondering what one can do from here! It seems that multiplication is playing an interesting role here, but I haven't been able to find any discussion of this elsewhere (if I'm being honest I don't even know what keywords to look for). To be precise, these are the things I'm curious about:
- Is it possible to define a related category in which multiplication maps are morphisms, without bringing back all of the set structure?
- What role is multiplication playing here, if not that of a natural transformation?
- Are there any analogous examples of this structure in other parts of math?