Is elliptic curves an group object in some category?

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Is elliptic curves an group object in some category?

Group like object, like lie group,formal group, algebraic group,topological group, is sometimes define as group object in some category. What about elliptic curves?

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For a fixed elliptic curve, it is an abelian variety, which is a group scheme, so it is a group object in $(Sch)$.