I am trying to figure out how much one can figure out about an object using category theory. Ideally, any property that is well defined up to isomorphism should be computable using only category theory. Let us say that we are trying to figure out how many elements are in a group? For a set, we could "simply" count the morphisms to it from the terminal object. Obviously, this wouldn't work. Is there a "categorical" method to find the cardinality of a group? If one wants a rigorous definition of what I mean by "categorically", here is your compass and straightedge:
- You have an abstract symbol for the object in question.
- For any two abstract symbols for objects $G$ and $H$, you can get abstract symbols for $Hom(G,H)$
- For any abstract symbol for an object $G$, you can get the abstract symbol for $id_G$.
- For any two abstract symbols for morphisms $f$ and $g$ such that $f \circ g$ is defined, you can get the abstract symbol of $f \circ g$
- In terms of the objects and morphisms for abstract symbols you already have, you can get the abstract symbol of any object that uniquely (up to isomorphism) satisfies a given universal property (if needed, I can clarify this.)
Using a finite number of steps, I am trying to find the cardinality of the group. Note for example, you can't take the forgetful functor from $Grp$ to $Set$ (since you can't compute a functor on an abstract symbol.)
I am thinking one method would be to count how many automorphisms there are, but am I not sure how to get the cardinality of the group from this.
If these set of "rules" are too restrictive, it would be interesting to see how they could be lightened to make it possible.