Let $RVar$ be the category which has complex algebraic verieties as objects and rational maps as morphisms [Edit: for $RVar$ to be a category, the rational maps have to be dominant]. Let's consider a group object $G$ in this category, i.e. a tuple $(G,\mu,e,\iota)$ where $\mu:G\times G \rightarrow G$ , $e:* \rightarrow G$, and $\iota: G \rightarrow G$ are rational maps satisfying the usual commutative diagrams defining a group structure.

Just out of curiosity, two natural questions:

• Is such a $G$ necessarily an algebraic group? That is: is it the case that for any $(G,\mu,e,\iota) \in Grp(RVar)$, there exists an algebraic group $(G',\mu',e',\iota')$ and a birational map $\varphi: G\rightarrow G'$ such that "$\varphi$ intertwines the operations of $G$ and $G'$"?

• Analogous question in the holomorphic/meromorphic setting.