Relaxations of cartesian categories

The ubiquitous notion of cartesian category can be weakened, or relaxed, in many useful ways. We also consider weakenings of cocartesian and bicartesian categories.

Weakening the structure

No copying

  • Semicartesian category : monoidal category whose monoidal unit is terminal
    • A.k.a., “monoidal category with projections” (nCat Cafe )
    • “Tensor category with projections” (or inclusions) also defined in Franz, 2002: What is stochastic independence? (doi, arxiv, MathOverflow )
  • Semicocartesian category: monoidal category whose monoidal unit is initial
    • Terminology is not common but is sometimes used, e.g., in John Wiegley’s Coq library for category theory
  • Pointed monoidal category: monoidal category whose monoidal unit is both initial and terminal (i.e., is a zero object )

No deleting

  • Relevance monoidal category : symmetric monoidal category with duplication
    • Equipped with natural family of cocommutative co-semigroup objects, instead of the usual cocommutative comonoid objects

Relaxing the axioms

TODO: Hierarchy of cartesian monoidal, monoidal with diagonals, symmetric premonoidal