  We present a general formalism for identifying the caustic structure of an
evolving mass distribution in an arbitrary dimensional space. For the class of
Hamiltonian fluids the identification corresponds to the classification of
singularities in Lagrangian catastrophe theory. Based on this we develop a
theoretical framework for the formation of the cosmic web, and specifically
those aspects that characterize its unique nature: its complex topological
connectivity and multiscale spinal structure of sheetlike membranes, elongated
filaments and compact cluster nodes. The present work represents an extension
of the work by Arnol'd et al., who classified the caustics for the 1- and
2-dimensional Zel'dovich approximation. His seminal work established the role
of emerging singularities in the formation of nonlinear structures in the
universe. At the transition from the linear to nonlinear structure evolution,
the first complex features emerge at locations where different fluid elements
cross to establish multistream regions. The classification and characterization
of these mass element foldings can be encapsulated in caustic conditions on the
eigenvalue and eigenvector fields of the deformation tensor field. We introduce
an alternative and transparent proof for Lagrangian catastrophe theory, and
derive the caustic conditions for general Lagrangian fluids, with arbitrary
dynamics, including dissipative terms and vorticity. The new proof allows us to
describe the full 3-dimensional complexity of the gravitationally evolving
cosmic matter field. One of our key findings is the significance of the
eigenvector field of the deformation field for outlining the spatial structure
of the caustic skeleton. We consider the caustic conditions for the
3-dimensional Zel'dovich approximation, extending earlier work on those for 1-
and 2-dimensional fluids towards the full spatial richness of the cosmic web.
