20th century logic has founded mathematics on the binary relation of set membership, formalized with the axioms of Zermelo and Fraenkel. At mid-century algebra reacted by proposing instead to found mathematics on the associative binary operation of function composition, formalized as category theory. But category theory then fell into the trap of making categories themselves algebraic by introducing functors and natural transformations, which led to 2-categories and thence to n-categories and omega-categories. In this talk we avoid this trap while simultaneously accounting for the compatibility of topology and algebra in terms of the arbitrarily selected free and co-free objects of any category whatsoever. Applications include the topoalgebraic specification of the notion of an acyclic graph, an extensional conception of the concept of property or attribute, and a demonstration that C.I. Lewis's 1927 quasi-psychological notion of qualia is at least mathematically consistent.
