nLab supply in a monoidal category


Consider a type PP 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 CC supplies PP if every object of CC can be equipped with such PP-structure, compatibly with the given tensor product. Importantly, the morphisms of CC are not all required to be PP-homomorphisms.


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

  1. A PP-algebra structure on every object of CC, such that
  2. The PP-algebra structure on any tensor product xyx\otimes y is that canonically induced from those on xx and yy.

If there is a given supply of PP in CC, we also say that CC supplies PP-algebras.


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



Last revised on February 2, 2024 at 09:58:13. See the history of this page for a list of all contributions to it.