He was a close collaborator of Grothendieck, notably contributing to SGA 4 his theory of hypercovers and anticipating the later development of étale homotopy by Michael Artin and Barry Mazur, following a suggestion he attributed to Pierre Cartier.

Saul Lubkin's related theory of rigid hypercovers was later taken up by Eric Friedlander in his definition of the étale topological type.

For many years he directed a joint seminar at the École Normale Supérieure with Adrien Douady.

In 1976 Verdier developed a useful regularity condition on stratified sets that the Chinese-Australian mathematician Tzee-Char Kuo had previously shown implied the Whitney conditions for subanalytic sets (such as real or complex analytic varieties).

Work of Bernard Teissier, aided by Jean-Pierre Henry and Michel Merle at the École Polytechnique, led to the 1982 result that Verdier's condition (w) is equivalent to the Whitney conditions for complex analytic stratifications.