  A complete mathematical framework for coalgebraic formulation of
supergeometry and its infinite-dimensional extension is proposed. Within this
approach a supermanifold is defined as a graded coalgebra endowed with a smooth
structure. The category of such coalgebras is constructed and analysed. It is
shown that it contains as its full subcategories both the category of smooth
Fr\'{e}chet manifolds and the category of finite-dimensional
Berezin-Leites-Kostant supermanifolds.
