The flavor of Galois theory called categorical Galois theory is a variant of Grothendieck's Galois theory as generalised by George Janelidze.
The main reference is the monograph
G. Janelidze, Magid’s theorem in categories, Bull. Georgian Acad. Sci. 114, 3, 1984, 497-500 (in Russian)
G. Z. Dzhanelidze, The fundamental theorem of Galois theory, Mat. Sb. (N.S.) 136(178) (1988), no. 3, 361–376, 431, pdf; translation in Math. USSR Sbornik 64 (2), 1989, 359-374, MR89k:18008
G. Janelidze, Pure Galois theory in categories, J. Algebra 132 (2), 1990, 270-286, MR91k:18004, doi
G. Janelidze, Precategories and Galois theory, Proc. Como, Springer Lect. Notes in Math. 1488, 1991, 157-173, MR93h:18011, doi
G. Janelidze, D. Schumacher, R. Street, Galois theory in variable categories, Applied Categorical Structures 1, 1993, 103-110