We report the results of the first study of the multi-stream environment of dark matter haloes in cosmological N-body simulations in the Λ cold dark matter cosmology. The full dynamical state of dark matter can be described as a three-dimensional sub-manifold in six-dimensional phase space – the dark matter sheet. In our study we use a Lagrangian sub-manifold (where and are comoving Eulerian and Lagrangian coordinates, respectively), which is dynamically equivalent to the dark matter sheet but is more convenient for numerical analysis. Our major results can be summarized as follows. At the resolution of the simulation, i.e. without additional smoothing, the cosmic web represents a hierarchical structure: each halo is embedded in the filamentary framework of the web predominantly at the filament crossings, and each filament is embedded in the wall-like fabric of the web at the wall crossings …