2021/01/25 by Hadzihasanovic, Amar
#18M85 #18N20 #18N30 #18N70 #Algebraic Topology (math.AT) #Category Theory (math.CT) #FOS: Mathematics
paper · doi:10.48550/arxiv.2101.10361
The tensor product of props was defined by Hackney and Robertson as an extension of the Boardman-Vogt product of operads to more general monoidal theories. Theories that factor as tensor products include the theory of commutative monoids and the theory of bialgebras. We give a topological interpretation (and vast generalisation) of this construction as a low-dimensional projection of a "smash product of pointed directed spaces". Here directed spaces are embodied by combinatorial structures called diagrammatic sets, while Gray products replace cartesian products. The correspondence is mediated by a web of adjunctions relating diagrammatic sets, pros, probs, props, and Gray-categories. The smash product applies to presentations of higher-dimensional theories and systematically produces higher-dimensional coherence cells.