Supplies in a Monoidal Category

A supply in a symmetric monoidal category equips every object with a chosen algebraic structure coherent with . That structure is a commutative monoid, or a Frobenius or bimonoid structure. Coherence means the structure on is built from those on and , and each map respects them. So the structure is not a property of one object but a uniform assignment across all of them. The terminology is due to Baez–Coya–Rebro (“Props in Network Theory”) and Fong–Spivak (Seven Sketches). Presented syntactically, such a supply is organised as a layered PROP.

Definition. A supply of monoids

A supply of commutative monoids on is a choice of commutative-monoid structure on each object , natural in and monoidal in the sense that and are the evident composites. A supply object is then not extra data attached to one object but structure carried uniformly by the whole category.

Example. Supplying an order, supplying a vector

Ordered vector spaces read as two supplies interacting, where an order can be embedded so that becomes closure under a positive combination , with positive cones sent to positive cones. One says supplies a vector object while supplies an order; a morphism must then be simultaneously monotone and linear, i.e. respect both supplied structures, which is exactly a positive linear map.