Abstracting the genealogical aspect of a given society yields a mathematical structure within which aunts, cousins, etc. can be precisely defined. Other objects, in the same category of such structures, which seem very different from actual societies, are nonetheless shown to be important tools in a societys conceptualizing about itself, so that for example gender and moiety become labelling morphisms within that category. Topological operations, such as contracting a connected subspace to a point, are shown to permit rationally neglecting the remote past. But such operations also lead to the qualitative transformation of a topos of pure particular Becoming into a topos of pure general Being; the latter two kinds of mathematical toposes are distinguished from each other by precise conditions. The mythology of a primal couple is thus shown to be a naturally-arising didactic tool. The logic of genealogy is not at all 2-valued nor Boolean, because the truth-value space naturally associated with the ancestor concept has a rich lattice structure. A finitary approximation to this theory is also considered and the corresponding category of structures is fully analyzed.