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. [Edit: in the light of the comments, e.g. the observation that rational maps have better be dominant, the identity $e:*\rightarrow G$ doesn't seem to make much sense; same for the diagram for the inverse, then; what people, among whom Weil, actually considered were "rational group chunks" in which there's only an associative rational $\mu$, and you can ask the same question(s)].
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.