nLab cartesian object


A cartesian object in a 2-category with finite 2-products is an object XX such that the diagonal morphism XX×XX \to X \times X and the unique map X1X \to 1 have right adjoints.

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


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:


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


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

Let XX be a cartesian object in CC, and make a choice of right adjoints to the diagonal and terminal maps. Because 2-functors preserve adjunctions and the 2-Yoneda embedding C[C op,Cat]C \to [C^{op}, Cat], AC(,A)A \mapsto C(-,A) preserves 2-products, C(,X)C(-,X) is a cartesian object with a choice of right adjoints in the 2-category of 2-presheaves on CC. In particular, since 2-products in this 2-category are computed pointwise, C(Y,X)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 YY of CC. Then, by similar reasoning, C(,X)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, XX is a symmetric pseudomonoid in CC.

For any other choice of rights adjoints for the cartesian object XX, 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.


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

Last revised on April 1, 2023 at 00:16:16. See the history of this page for a list of all contributions to it.