In mathematics, a category is distributive if it has finite products and finite coproducts and such that for every choice of objects, the canonical map
is an isomorphism, and for all objects , the canonical map is an isomorphism (where 0 denotes the initial object). Equivalently, if for every object the endofunctor defined by preserves coproducts up to isomorphisms .[1] It follows that and aforementioned canonical maps are equal for each choice of objects.
The categories Grp and Ab are not distributive, even though they have both products and coproducts.
An even simpler category that has both products and coproducts but is not distributive is the category of pointed sets.[2]
Referencesedit
^Taylor, Paul (1999). Practical Foundations of Mathematics. Cambridge University Press. p. 275.
^F. W. Lawvere; Stephen Hoel Schanuel (2009). Conceptual Mathematics: A First Introduction to Categories (2nd ed.). Cambridge University Press. pp. 296–298. ISBN 978-0-521-89485-2.
Further readingedit
Cockett, J. R. B. (1993). "Introduction to distributive categories". Mathematical Structures in Computer Science. 3 (3): 277–307. doi:10.1017/S0960129500000232.
Carboni, Aurelio (1993). "Introduction to extensive and distributive categories". Journal of Pure and Applied Algebra. 84 (2): 145–158. doi:10.1016/0022-4049(93)90035-R.