Etale cohomology, whose definition and study would be inconceivable without the use of categorical methods, underlies the construction of many interesting Galois representations as well as the entire context for Deligne's work on the Riemann Hypothesis for varieties over finite fields and its various generalizations, and work of Taylor et al. on Sato-Tate.

The concept of moduli schemes and the study of their non-trivial properties (and reasons for their existence) would likewise be impossible with the systematic use of categorical reasoning, and these underlie Faltings' work on the Mordell conjecture, the work of Drinfeld/Lafforgue on global Langlands correspondence for function fields, the work of Wiles et al. on modularity of Galois representations, the work of Mazur/Merel on torsion in elliptic curves over number fields, the use of Heegner points by Gross-Zagier...and the role of Galois cohomology in all of these matters (informed through homological reasoning) is utterly pervasive.

There's so much more which could be said, but I think the above is quite sufficient to make it clear that categorical and homological methods are ubiquitous throughout the deepest parts of modern algebraic number theory.

Etale cohomology, whose definition and study would be inconceivable without the use of categorical methods, underlies the construction of many interesting Galois representations as well as the entire context for Deligne's work on the Riemann Hypothesis for varieties over finite fields and its various generalizations, and work of Taylor et al. on Sato-Tate.

The concept of moduli schemes and the study of their non-trivial properties (and reasons for their existence) would likewise be impossible without the systematic use of categorical reasoning, and these underlie Faltings' work on the Mordell conjecture, the work of Drinfeld/Lafforgue on global Langlands correspondence for function fields, the work of Wiles et al. on modularity of Galois representations, the work of Mazur/Merel on torsion in elliptic curves over number fields, the use of Heegner points by Gross-Zagier...and the role of Galois cohomology in all of these matters (informed through homological reasoning) is utterly pervasive.

There's so much more which could be said, but I think the above is quite sufficient to make it clear that categorical and homological methods are ubiquitous throughout the deepest parts of modern algebraic number theory.