Category theory MOC

Cartesian category

A cartesian category is a monoidal category whose tensor product is a categorical product and whose tensor unit is a terminal object. cat Specifically, for any 𝐴,𝐵 ∈𝖢 there exist natural transformations

𝜆𝐵:1⊗𝐵⇒1:𝖢→𝖢𝜌𝐴:𝐴⊗1⇒1:𝖢→𝖢

so that for any 𝐴,𝐵 ∈𝖢 the data

(𝐴⊗𝐵,𝜆𝐵𝐴,𝜌𝐴𝐵)

satisfy the universal property of the product.

See also

• The dual concept is Cocartesian category.
• A Bicartesian category is both of these simultaneously.

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