In the paper, the notion is introduced of outer section of a polyverse (a function representation of a Boolean-valued universe) and a new function representation is obtained of K-spaces and vector lattices as outer sections. In particular, an isomorphism is constructed between an arbitrary vector lattice and an outer subset of the field of reals in the corresponding Boolean-valued universe. Within the new function representation, analogs are discovered of the main motions and facts of the theory of vector lattices. It is also clarified which of the properties of K-spaces under consideration admit “pointwise criteria.”