The primordial velocity dispersion of dark matter is small compared to the velocities attained during structure formation. The initial density distribution is close to uniform, and it occupies an initial sheet in phase space that is single valued in velocity space. Because of gravitational forces, this 3D manifold evolves in phase space without ever tearing, conserving phase-space volume and preserving the connectivity of nearby points. N-body simulations already follow the motion of this sheet in phase space. This fact can be used to extract full fine-grained phase-space structure information from existing cosmological N-body simulations. Particles are considered as the vertices of an unstructured 3D mesh moving in 6D phase space. On this mesh, mass density and momentum are uniquely defined. We show how to obtain the space density of the fluid, detect caustics and count the number of streams as well as their individual contributions to any point in configuration space. We calculate the bulk velocity, local velocity dispersions and densities from the sheet – all without averaging over control volumes. This gives a wealth of new information about dark matter fluid flow which had previously been thought of as inaccessible to N-body simulations. We outline how this mapping may be used to create new accurate collisionless fluid simulation codes that may be able to overcome the sparse sampling and unphysical two-body effects that plague current N-body techniques.