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August 21, 2016

Migrations and functors

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:

  1. each $(X, \leq) \in$ Pr0 is an element of Ob(Cat)
  2. 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:

  1. On object part: we have to send each preorder $(X,\leq)$ into a category $\chi=i(X, \leq_X)$ built in this way:
    1. Objecs: every element x of the preorder $(X,\leq_X)$ is sent into an object of the category $\chi$. $Ob(\chi)=X$
    2. Morphisms: for every $x,y \in X$ such that $x \leq y$ there is a unique morphism in Hom_\chi(x, y). By the way, since we are in Cat, this morphism is a functor between category x and y. For each element $x \in (X,\leq)$, reflexivity becomes the $id_x$ (in this case, the identity functor for each category in Cat) The transitivity property in preorders is expressed as morphism composition (in this case, functor composition) Laws of morphism composition are satisfied automatically, because your codomain is already a category.
  2. On morphism part, we are sending each preorder morphism into a functor (aka a morphism in Cat). The idea is to show that composition of preorder morphisms satisfy the laws of arrows composition. Given a morphism between preorders $f : (X, \leq_x) \to (Y, \leq_y)$ there is a morphism $i(f): \chi \to \mathcal{Y} \in Hom_Cat(\chi,\mathbb{Y})$. This morphism is a functor, such that:
    1. On object: for each $x \in Ob(\chi) = X $ there is an element $y = f(x) \in Ob(\mathbb{Y})=Y$.
    2. On morphism part: if there is an arrow $x \to x’$ in $\chi$, then we know that $x\leq x’$ \in (X, \leq_X). By definition of preorder morphism we know therefore that $f(x) \leq f(x’)$, and so we require a morphism between $f(x) \to f(x’) \in \mathbb{Y}$.

    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.

To recap:

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

References

[1] Spivak, David I. Category theory for the sciences. MIT Press, 2014.