## Preordered Sets as Categories, and their Functor Categories (Pt. I)

**Point of Post: **In this post we discuss the way in which preordered sets are, in spirit, the same as categories whose Hom sets never exceed one in cardinality–we also discuss then the interpretation of already-discussed categorical concepts into the language of preordered sets.

*Motivation*

Having a plethora of categorical examples is invaluable in two ways. Firstly, having a wide-array of examples allows us to test our knowledge of categorical concepts, get our “hands dirty” which of course gives us better insight into what the concepts “really mean”. Secondly, it allows us to create even more categorical concepts by combining previously defined ideas.

*Preordered Sets as Categories*

Let be a preordered set, i.e. is a relation on which is reflexive and transitive (but not necessarily symmetric or antisymmetric, thus generalizing both posets and ordered sets). There is a particularly simple way to make into a category. Namely, let be the “category” whose object class is and such that given one has that

define composition in by stating that whenever this composition is admissible (i.e. whenever ). It’s easy to see that this is a category, the only perhaps unclear thing is why is the identity arrow of . To see this, all we have to note is that, when defined, we have the following and so that really is an identity arrow for . Thus, we see that, with these definitions, really is a category. We call this the category *induced by .*

There is actually a way of going backwards, to create a poset from a category with Hom sets which are either empty or singletons. Indeed, suppose conversely that we have a (small) category such that for all objects in . If we define to be the object set of we claim that there is a natural preorder on from the category structure on . Indeed, define on by declaring if and only if is non-empty. To see that really is a preorder we merely note that reflexivity follows from the existence of an identity arrow for each , and transitivity follows from the composibility of an arrow in and an arrow in to get an arrow in .

It’s easy to see that there is a kind of “duality” between these two constructions. Indeed, if we start with a preordered set , create the corresponding category , and then from this category create the corresponding preordered set, we just get again (i.e. we have a “bijection” between preordered sets and categories whose Hom sets have at most one element). Similarly, if we start with a category whose Hom sets have at most one element, create the associated preordered set, and from this preordered set create the associated category we end up with the original category we started with. In this way we often forego the distinction between preordered sets and their associated categories–this is in the same vein that we may forego the distinction between monoids and their associated categories.

*Remark: *There was, as mentioned at one point but not enforced, a blanket assumption the smallness of all categories in the above discussion. If one is willing to accept the notion of relations on classes (possibly proper classes) then the above can be generalized to general categories.

Ok, so we have a new general example of a category to play with, let’s see if we can figure out what some of our previous categorical constructions look like on such categories. For example, what do functors between two preordered sets look like? Well, let’s say we have to preordered sets thought of as categories, call them and . Then, a functor induces a set function , but note that this set function has the property that if then . Indeed, if then there is an arrow and so there exists a corresponding arrow so that . Conversely, it’s easy to see that any set map such that implies induces a functor . Thus, we see that functors are nothing more than monotone functions .

What about the product of two preordered sets? How does this relate to the product of categories? Well, letting and be as above we see that the object set of is , so how about the morphisms? Well, we see that given a pair of objects and in one has, by definition, that an arrow is a pair of arrows and . Thus, we see that there is an arrow if and only if and . Moreover, it’s clear that if an arrow exists then it is unique, for two different arrows between must induce either two different arrows or two different arrows both of which are impossible. From this we can conclude that is indeed a category coming from a preordered set, and moreover it’s the category coming from the usual product preordered set .

Ok, so given a preordered set what does the opposite category look like? Since the arrows in are just the arrows of just reversed, it’s easy to see that is a category whose Hom sets have at most one object, and thus arises as the category of a preordered set, but which? I think it’s fairly easy to see that the preordered set from which is none other than the preordered set where is the usual *opposite order *defined by if and only if .

**References:
**

[1] Mac, Lane Saunders. *Categories for the Working Mathematician*. New York: Springer-Verlag, 1994. Print.

[2] Adámek, Jirí, Horst Herrlich, and George E. Strecker. *Abstract and Concrete Categories: the Joy of Cats*. New York: John Wiley & Sons, 1990. Print.

[3] Berrick, A. J., and M. E. Keating. *Categories and Modules with K-theory in View*. Cambridge, UK: Cambridge UP, 2000. Print.

[4] Freyd, Peter J. *Abelian Categories.* New York: Harper & Row, 1964. Print.

[5] Rotman, Joseph J. *Introduction to Homological Algebra*. Springer-Verlag. Print.

[6] Herrlich, Horst, and George E. Strecker. *Category Theory: An Introduction*. Lemgo: Heldermann, 2007. Print.

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