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.
Let $P$ be a (single-sorted) prop, and $C$ a symmetric monoidal category. A supply of $P$ in $C$ consists of
If there is a given supply of $P$ in $C$, we also say that $C$ supplies $P$-algebras.
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.
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 commutative comonoids.
Last revised on April 11, 2023 at 12:55:32. See the history of this page for a list of all contributions to it.