Igor Bakovic HomePage

On this page I would like to describe the project which I grandiosely entitled

Higher Galois Theory

whose more appropriate name would be Bicategorical Galois Theory since it is a weakest possible generalization of Categorical Galois Theory developed by Janelidze, Schumacher and Street in their

• G. Janelidze, D. Schumacher, R. Street, Galois theory in variable categories, Applied Categorical Structures 1, 1993, 103-110

to the immediate next level of dimension.

Motivation

Categorical Galois theory was developed by Janelidze, Schumacher and Street who studied Galois structures in a context of a 2-topos

of $\mathbf{B}$-indexed categories $Cat^{\mathbf{B}^{op}}$ for some bicategory $\mathbf{B}$, whose objects I also called 2-presheaves in my preprint Bicategorical Yoneda Lemma following the philosophy of categorification. The yoga of categorification than naturally leads us to investigate a 3-category

of $\mathbf{B}$-indexed bicategories which I defined in Fibrations of Bicategories as (contravariant) trihomomorphisms in a sense of Gordon, Power and Street to the tricategory $Bicat$ of bicategories, their homomorphisms, pseudonatural transformations and modifications. One of important aspects would be to describe factorizations systems in the tricategory $Bicat$ analogous to bicategorical essentially surjective on objects - fully faithful factorization system on the 2-category $Cat$, and to transfer it pointwise to the 3-category $Bicat^{\mathbf{B}^{op}}$.