The classical Sobolev spaces built from the fundamental first-order operators grad, curl, and div fit elegantly into the de Rham complex, a unifying algebraic framework that has transformed the numerical analysis of boundary value problems. For scalar and vector fields, this picture is complete. Yet matrix-valued finite elements, which arise naturally in mechanics and geometry, resist the same de Rham framework. This talk presents a new commuting diagram of Sobolev spaces of scalar, vector, and matrix fields, organized not by a complex, but by a 2-complex, a generalization in which the composition of any three successive morphisms, rather than two, vanishes. The spaces in this diagram are characterized by very weak second-order regularity conditions; for instance, one such space consists of symmetric matrix fields with square-integrable components whose double divergence lies in a standard negative-order Sobolev space. Analogous spaces defined via curl-div and double-curl (incompatibility) operators appear as natural companions. We establish regular decompositions of these spaces, prove duality relationships between such spaces with and without boundary conditions. The 2-complex structure reveals unifying connections across scalar, vector, and matrix Sobolev spaces that point toward a broader synthesis encompassing the disparate families of nn-, nt-, and tt-continuous matrix finite elements.