Equational Theory
In every cartesian closed category (using exponential notation), (XY)Z and (XZ)Y are isomorphic for all objects X, Y and Z. We write this as the "equation"
- (xy)z = (xz)y.
One may ask what other such equations are valid in all cartesian closed categories. It turns out that all of them follow logically from the following axioms:
- x×(y×z) = (x×y)×z
- x×y = y×x
- x×1 = x (here 1 denotes the terminal object of C)
- 1x = 1
- x1 = x
- (x×y)z = xz×yz
- (xy)z = x(y×z)
Bicartesian closed categories extend cartesian closed categories with binary coproducts and an initial object, with products distributing over coproducts. Their equational theory is extended with the following axioms:
- x + y = y + x
- (x + y) + z = x + (y + z)
- x(y + z) = xy + xz
- x(y + z) = xyxz
- 0 + x = x
- x×0 = 0
- x0 = 1
Read more about this topic: Cartesian Closed Category
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