I have recently acquired a .cat domain and I’m long time owner of a .pro domain. If you wonder, .pro is designed to be used by professionals, and .cat is the domain for catalunya. But for me (and me only, probably):
.PrO is also the category of Pre-Orders. Objects in PrO are preoders, and arrows are homomorphism between preorders.
.Cat is the category of (small) categories. Objects are categories, and arrows are functors between categories.
This post is to tell you that I have migrated my blog from a .pro domain to a .cat domain. What a good Fun occasion to remember that there is a functor between these categories. (There is also the opposite functor, but that’s for another post. :p )
Theorem: There is a functor $i : PrO \to Cat$ such that:
For each $x,y \in Ob(\chi)$, $ | Hom_\chi(x,y) | \leq 1$ |
Moreover, every category with property 2 is in the image of the functor $i$.
Proof:
It’s possible to see that preorders moprhism are transitive (they form a preorder, the preorder of preorder morphism, lol) and since Cat is a category, if $|Hom_{Cat}(\chi, \mathbb{Y})| > 1$ and $|Hom_{Cat}(\mathbb{Y}, \mathbb{Z})| > 1$, then $|Hom_{Cat}(\chi, \mathbb{Z})| > 1$, and this guaratee us to satisfy the law of morphism composition in Cat. In PrO, the identity morphism is the identity (it preservs order relations in a preorder), we see that each $id_{Pro}$ is sent into the trivial functor in Cat.
I really like the book, but since I’m a noob in cat and I want things to be as verbose and pedantic as possible, I rewrote the proof there in a more schematic way.
As slogan 4.2.1.18[1] say: A preorder is a category in which every hom-set has either 0 elements or 1 element. A preorder morphism is just a functor between such categories.
Daje
[1] Spivak, David I. Category theory for the sciences. MIT Press, 2014.