pointwise order



The pointwise order is the canonical order on the set of functions into a poset or preorder.

Using negative thinking, this can be seen as the (0,1)-decategorification of the concept of category of functors. Just as well, it can be seen as the enriched functor category for categories enriched in truth values.



Let XX be a set, and let YY be a preordered set. Given f,g:XYf,g:X\to Y, we say that fgf\le g in the pointwise order or product order if and only of for every xXx\in X we have f(x)g(x)f(x)\le g(x).

The terminology “product order” comes from the fact that the order defined above can be seen as the one of the object

xXY \prod_{x\in X} Y

in the category of preorders (which has products). The underlying set of the object above is indeed naturally isomorphic to the set of functions XYX\to Y.


Just as Cat is naturally a 2-category, with 2-cells given by natural transformations (i.e. morphisms of functors), many categories of preorders (such as Pos) are naturally locally posetal 2-categories, with the 2-cells given by the pointwise order. That is, given f,g:XYf,g:X\to Y, we draw a 2-cell fgf\Rightarrow g if and only if fgf\le g in the pointwise order. If the morphisms are chosen to be monotone, this choice of 2-cells gives indeed the structure of a 2-category.

See also