Abstract:For Floer-Fukaya theory, there is a good theory of functoriality, using the notion of A_\infty-morphism introduced by Stasheff and Fukaya. I will talk about a "complexification" of this notion which we conjecture applies to functoriality in Gromov-Witten theory, namely the behavior of Gromov-Witten invariants under symplectic quotients: I will define "morphism of cohomological field theories" developed by Ma'u and myself, and also investigated by Nguyen, and explain its relationship to work of Gaio-Salamon and Ziltener.