In 2006, Katsura discovered optimal conditions for morphisms of topological graphs to contravariantly induce morphisms between their C*-algebras. We use these morphisms in the special case of discrete topology (usual directed graphs) to extend the well studied theory of unions of graphs to one-injective pushouts of graphs. Our main result is a pushout-to-pullback theorem that yields a one-surjective gauge equivariant pullback of graphs C*-algebras, which suits the Mayer-Vietoris six-term exact sequence in K-theory and applications to noncommutative topology. To exemplify our theorem, first we vastly generalize the old concept of a derived graph to obtain locally derived graphs. Then we use locally derived graphs, which provide us with a very rich class of non-injective graph homomorphisms satisfying Katsura's conditions, to construct pushouts satisfying the assumptions of our pushout-to-pullback theorem. (Based on joint work with Mariusz Tobolski.)

LOCALLY DERIVED GRAPHS AND GAUGE EQUIVARIANT PULLBACKS OF GRAPH C*-ALGEBRAS