An example of a non-distributive category
An example of a category that is not distributive is 1/S, the category of pointed sets. An object of this category is a set X with a chosen base point, or distinguished point, 1 ->(x0)-> X. A map that preserves the structure is a map of sets that takes the base point of the domain into the base point of the codomain. A map in 1/S from a set X with base point 1 ->(x0)-> X to a set Y with base point 1 ->(y0)-> Y is any map of sets X ->(f)-> Y such that fx0 = y0. The base point of the domain is mapped to the base point of the codomain, while the other points can be mapped to any points of the codomain, including the base point.
The terminal object is a set with one element, and the one element is the base point. A set with only one point is initial. Thus in this category 0 = 1. And the empty set is not an object of this category. Because it doesn’t have a point to be chosen as base point.
In this category the unique map 0 -> 1 is an isomorphism, and this category has zero maps.
