Biclosed monoidal structures on the categories of digraphs and graphs
Adrien Grenier, Chris Kapulkin
TL;DR
The paper classifies biclosed monoidal structures on two graph categories: directed and undirected reflexive graphs, proving there are exactly two such structures in each case (the box product and the categorical product). The authors provide a conceptual proof using reflective presentations of graphs as finite-limit-preserving functors and Yoneda/density arguments, offering a more streamlined approach than prior combinatorial proofs. They also extend the discussion to graphs with optional loops, conjecturing three biclosed monoidal structures (box product, categorical product, strong product) and outlining methodological gaps that prevent full resolution in that setting. Overall, the work sharpens our understanding of which graph products are compatible with biclosed monoidal structures for naive graph homotopy theories.
Abstract
We show that the categories of directed and undirected reflexive graphs carry exactly two (up to isomorphism) biclosed monoidal structures.