nLab supply in a monoidal category

Idea

Consider a type $P$ of algebraic structure (such as monoid-structure, comonoid-structure, etc.) that may be put on an object of a symmetric monoidal category. We say that a symmetric monoidal category $C$ supplies $P$ if every object of $C$ can be equipped with such $P$-structure, compatibly with the given tensor product. Importantly, the morphisms of $C$ are not all required to be $P$-homomorphisms.

Definition

Let $P$ be a (single-sorted) prop, and $C$ a symmetric monoidal category. A supply of $P$ in $C$ consists of

1. A $P$-algebra structure on every object of $C$, such that
2. The $P$-algebra structure on any tensor product $x\otimes y$ is that canonically induced from those on $x$ and $y$.

If there is a given supply of $P$ in $C$, we also say that $C$ supplies $P$-algebras.

Remarks

The notion of “supply” does not satisfy the principle of equivalence: if $C\simeq D$ then a supply of $P$ in $C$ does not automatically transfer to a supply of $P$ in $D$. Accordingly, a “symmetric monoidal category that supplies $P$” should be thought of as an atomic categorical structure, not as a category equipped with structure.

Examples

• A hypergraph category is a symmetric monoidal category that supplies special commutative Frobenius algebras.

• A Markov category is a semicartesian symmetric monoidal category that supplies cocommutative comonoids.

• A gs-monoidal category is a symmetric monoidal category that supplies cocommutative comonoids.

References

• Brendan Fong, David Spivak, Supplying bells and whistles in symmetric monoidal categories, [arxiv:1908.02633]