nLab
pointwise order

Contents

Contents

Idea

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.

Definition

Definition

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.

2-cells

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

Created on October 16, 2019 at 23:31:43. See the history of this page for a list of all contributions to it.