Structure and Power: an Emerging Landscape

In this paper, we give an overview of some recent work on applying tools from category theory in finite model theory, descriptive complexity, constraint satisfaction, and combinatorics. The motivations for come Computer Science, but there may also be something interest theorists other logicians. basic setting involves studying the relational structures via a resource-indexed family adjunctions with process - which unfolds into tree-like forms, allowing natural resource parameters to assigned these unfoldings. One instance scheme allows us recover, purely structural, syntax-free way: Ehrenfeucht-Fraïssé game; quantifier rank fragments first-order logic; equivalences induced by (i) fragments, (ii) restriction fragment existential positive part, (iii) extension counting quantifiers; combinatorial parameter tree-depth (Nesetril Ossona de Mendez). Another recovers k-pebble game, finite-variable corresponding equivalences, treewidth. Other instances cover modal, guarded hybrid generalized quantifiers, wide range parameters. This whole has been axiomatized very general setting, arboreal categories covers. Beyond level, landscape is beginning emerge, structural features categories, comonads are reflected degrees logical computational tractability languages. Examples include semantic characterisation preservation theorems, Lovász-type results homomorphisms.