This thesis discusses two ideas, multiversal algebra and algebraic enrichment, and one potential application for the latter, sequential scheduling. Multiversal algebra is a proposal for the reconsideration of semigroupoid and category theory within a framework that extends the approach of universal algebra. The idea is to introduce the notion of algebraic operation relative to a given binary relation, as an alternative to the notion of operation on a carrier class. It is shown that for a particular class of relations the derived notion of category coincides with that of standard category theory. Algebraic enrichment is the name given to a series of similar constructions translating between external and internal algebraic structure, which are studied as a first step towards generalizing the seminal results of Eckmann and Hilton, and for the application to sequential scheduling. This well-known combinatorial engine of game semantics is shown to form part of a double semigroupoid, and this new algebraic perspective on scheduling oﬀers a new direction for the study of game models and their innocence condition.