Recent work on the formal analysis of kinship category systems turns upon whether the proper modelling of a terminology must, need not, or even cannot adequately be built upon a universal genealogical base. I shall show that the non-genealogical algebraic structures of the sort used to model particular kinship terminologies by Read and others are formally interpretable in terms of category theoretical operations upon a properly structured universal space of genealogical relations if, but only if the latter structure is not limited to that of the left-right concatenation of lexical terms; that is, when these operations take as their domain the inherent dimensionalities of lines of ascent, descent, collaterally and the like and computations on these. I argue that, in spite of flawed arguments by D. M. Schneider and others, genealogical structure is universal in that every terminological system is at least closed under a mapping of genealogical entities onto it even if this map does not define the kin terms in the first instance. I then explore the formal properties of some category theoretic operation as deforming genealogical space into the non-genealogical graph-theoretical structures of Read and others. I also show that some terminological system, in particular Oceanic generational ones, have only the structure of a proper subset of the dimensionalities of genealogical space, and that such systems are not the consequence deforming morphisms from genealogical space.