nLab cartesian object

Definition

A cartesian object in a 2-category with finite 2-products is an object $X$ such that the diagonal morphism $X \to X \times X$ and the unique map $X \to 1$ have right adjoints.

For example, a cartesian object in Cat is precisely a category with finite products.

Properties

Any category with finite products becomes a symmetric monoidal category by making a choice of products. The symmetric monoidal structure is canonical in the sense that for any other choice of products, there is a canonical isomorphism between the two symmetric monoidal categories. Specifically, there is an invertible strong symmetric monoidal functor whose underlying functor is the identity and whose structure maps are given by the universal property of products.

This fact generalizes to cartesian objects:

Proposition

A cartesian object in a 2-category $C$ with finite 2-products is a symmetric pseudomonoid in $C$ in a canonical way.

Proof

The idea of the proof is to reduce to the case of Cat using the 2-categorical Yoneda embedding.

Let $X$ be a cartesian object in $C$, and make a choice of right adjoints to the diagonal and terminal maps. Because 2-functors preserve adjunctions and the 2-Yoneda embedding $C \to [C^{op}, Cat]$, $A \mapsto C(-,A)$ preserves 2-products, $C(-,X)$ is a cartesian object with a choice of right adjoints in the 2-category of 2-presheaves on $C$. In particular, since 2-products in this 2-category are computed pointwise, $C(Y,X)$ is a cartesian object with a choice of right adjoints in Cat, and hence has the structure of a symmetric pseudomonoid in Cat, for each object $Y$ of $C$. Then, by similar reasoning, $C(-,X)$ is a symmetric pseudomonoid in the 2-presheaf 2-category. Finally, because the 2-Yoneda embedding is locally fully faithful and in particular locally reflects isomorphisms, $X$ is a symmetric pseudomonoid in $C$.

For any other choice of rights adjoints for the cartesian object $X$, there are unique invertible 2-cells between them that commute with the units and counits of the adjunctions. Threading these through the 2-Yoneda embedding as before, the 2-cells define the comparison iso-cells of a strong morphism of symmetric pseudomonoids whose underlying morphism between the carrier objects is the identity.

References

• Jacques Penon, Sous-categorie classifiee d’une 2-categorie (1974), CRAS

• Carboni?, Kelly?, Wood?, A 2-categorical approach to change of base and geometric morphisms I (1991) (Numdam), Section 5: Cartesian objects in $F$ and various morphisms between them