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(0,1)-category

(0,1)-topos

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

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

$\prod_{x\in X} Y$

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

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 $f,g:X\to Y$, we draw a 2-cell $f\Rightarrow g$ if and only if $f\le g$ 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.