Triads

Triads are four-termed complexes with end terms connected by a translation functor, similar to triangles in a triangular category. We use triads to give an adequate theory of L-functors, introduced in [W. Rump, The category of lattices over a lattice-finite ring, Algebras and Representation Theory, in press] to investigate the global structure of categories with almost split sequences. Roughly speaking, L-functors extend the Auslander–Reiten translate to morphisms and thereby make it functorial.