Stuff, Structure, Property

Baez’s stuff / structure / property trichotomy classifies what a forgetful functor throws away, graded by how faithful, full, and essentially surjective it is. Read as a design principle, adding a -property filters whereas adding -structure equips, and the two are forgotten with very different ease.

A forgetful functor discards, in increasing severity, property, structure, and stuff, and the grade is read off three conditions. forgets at most property iff it is full and faithful, since the extra data was a mere condition and no morphisms are added or lost. It forgets at most structure iff it is faithful, because a morphism is still determined by its underlying map yet there were choices. It forgets stuff once it fails to be faithful, since the underlying map then loses actual elements.

Proposition. Trichotomy as functor conditions

Take a forgetful . Then forgets at most property iff full and faithful, at most structure iff faithful, and forgets stuff exactly when it is not faithful. Essential surjectivity controls whether the forgotten thing was present to begin with, so a full, faithful, essentially surjective , that is an equivalence, forgets nothing.

Remark. filters, equips

Adding a property is a -condition, such as commutativity of a monoid, so it restricts to a full subcategory whose inclusion is full and faithful. Adding structure or stuff is an -choice, such as an inverse or a basis, so it extends and forgetting it is no longer faithful. Forgetting a property is therefore cheap, since one only widens the class back, whereas forgetting stuff destroys information.

Example. Viewer = Editor + read-only

Roles make the grading concrete. A Viewer is an Editor carrying the extra -property “issues no writes”. Because that is a property, Viewers form a full subcategory of Editors and the role is cheap to forget by granting write access. An Editor’s actual edit capabilities are structure / stuff and cannot be recovered once dropped.

References

• John Baez, Michael Shulman, “Lectures on n-Categories and Cohomology” (2006)