Convex Biproducts, Stochastic Matrices and Tape Diagrams

Categories with finite biproducts play a central role in category theory, providing an abstract setting in which additive and linear structures can be studied uniformly. In this paper, we introduce categories with \emph{convex} biproducts, which intuitively restrict the linear structures to convex ones. We show that, whereas categories with finite biproducts give rise to a matrix calculus based on arbitrary linear combinations, convex biproduct categories instead induce a matrix calculus based on stochastic (more generally, substochastic) matrices. This perspective yields a refined algebraic and compositional framework tailored to probabilistic settings. We exploit this connection to establish an isomorphism that underpins probabilistic tape diagrams, a graphical formalism for bimonoidal (also known as rig) categories, and we demonstrate its effectiveness by providing a complete axiomatisation of probabilistic Boolean circuits.