A Categorical Approach to Rough Equality Algebras via Approximation Functors

This paper develops a categorical framework for congruence-based rough set theory on equality algebras. We introduce the category AppEqAlg of equality algebras equipped with a congruence, analyse the induced rough upper and lower approximations on the power set and on the lattice of subalgebras, and characterise exact (θ-definable) subalgebras via the quotient algebra E/θ. On the categorical side we construct the quotient functor U : AppEqAlg → EqAlg and the diagonal embedding G : EqAlg → AppEqAlg, prove the adjunction U ⊣ G, and show that the forgetful functor V : AppEqAlg → EqAlg is topological, so that (co)limits lift from EqAlg with canonical congruences.