Define an exponential ring to be a set $X$ together with a collection of subsets $X_n$ for $n \in \mathbb{Z}$, elements $0_n \in X_n$, $1_n \in X_n$ and maps $+_n: X_n \times X_n \to X_n$ and $\cdot_n: X_n \times X_n \to X_n$ such that:
$(X_n, 0_n, 1_n, +_n, \cdot_n)$ is a commutative ring
$X_{...
@coiso This is an answer to the question, not just a comment. Please notice that comments should only be used to clarify, not answer the question. See How do comments work for more information. Therefore, I suggest to post your answer as an answer. This has many advantages, for example this brings extra visibility to the answer und puts the question off the unanswered list.
Actually, it's now interesting to me what the initial object of this category looks like. But I don't want to move the goalpost, so I might ask this separately.
@SeverinSchraven I don't see how. For example, what would be a morphism to the example given above with every $X_n \cong \mathbb{F}_2 \times \mathbb{R}$? A homomorphism is at the very least a levelwise ring homomorphism, but there is no ring homomorphism $0 \to \mathbb{F}_2 \times \mathbb{R}$.
@SeverinSchraven You mean a ring $R$ with $(R, 0,+) \cong \left( R^\times, 1, \cdot \right)$? Well, $R = 0$ does the job, and as said above we can also use $R = \mathbb{F}_2 \times \mathbb{R}$. I think the latter example is rather interesting.
@SeverinSchraven Or do you mean a ring $R$ with $(R^{\times}, \cdot) \cong (\mathbb{Z}, +)$?
@SeverinSchraven There is an example given here: mathoverflow.net/q/75192/155881 . This is essentially the group ring $\mathbb{F}_2[\mathbb{Z}]$. I don't think this gives the initial object, though.
I think math.stackexchange.com/questions/93409/… is highly relevant here. The ticked answer and comments thereunder show how to construct groups that could not serve as $X_0^\times$.
Discussion on question by Smiley1000:…
Discussion on question by Smiley1000:…