The present paper is the first of four which will develop a universal Existence Theorem for orbifolds. Given a category C, ultimately we find a categorical hypothesis under which C embeds into a larger category which contains quotients by finite group actions.To define a fundamental group for algebraic geometry, Grothendieck reformulated the topological invariant in terms of algebra and small cat­egories. Different concerns require the present author to revise this ap­proach, and to give a universal format for Galois covers. A complete theory of Galois correspondences and profinite fundamental groups is de­veloped for any category with a suitable notion of connectedness. The new formulation avoids base points and fiber functors.