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

**Point of Post: **This is a continuation of this post.

*Functors and Functor Categories out of Preordered Sets and *

Perhaps one of the most useful things thinking about preordered sets as categories gives us is how we can interpret functors from them into other, general categories. Indeed, suppose that we have a preordered set and some other category . What does a functor look like? Well, we can think about the object mapping as just being an indexed set of -objects. What about the arrow mapping induced by ? Well, we see that given any in we have an arrow in which then produces an arrow in . Now, since there is a UNIQUE arrow there is no harm in denoting as since there is no other arrow that is the image of an arrow under . Thus, we see that as an arrow mapping is just an indexed set of maps . But, not any old indexed set of maps will do, this indexed set of maps has to satisfy the functor properties. Indeed, the fact that is equivalent to and the fact that if that is equivalent to . So, all in all, we may provocatively identity functors with orderered pairs which satisfy the following two axioms:

But, this should look astonishingly familiar! Indeed, if is this is just the definition of a directed system of modules over ! So, we can basically identify directed systems of modules over some preordered set as being functors . Of course, this allows us to readily generalized and define a *directed system in over *to be a functor where we obviously make the identifications between the actual functor and the ordered pair with conditions on the morphisms as listed above.

Now, since we have identified functors as being just directed systems, it’s seems natural to ask if whether the functor category of all functors is the same as the category of directed systems in over . Basically this entails showing that the morphisms in are really just natural transformations between the associated functors in the sense just described. Indeed, by definition a morphism between and is a set of -maps such that

whenever . But, since there are only arrows if and only if , and the corresponding arrows are . From this the fact that is a natural transformation between and follows. Thus, the identification of and is justified. We then extend this to define the *category of directed systems in over *to be the functor category with the functors identified with the ordered pairs as before.

Ok, fine so an interesting (and I think, if one looks enough, fairly motivated question) is what do functors (i.e. contravariant functors look like? Well, just as the case of covariant functors we see that contravariant functors can be described as ordered pairs of indexed sets. Only, this time it’s easy to see that the ordered pairs . Moreover, just checking the axioms of the contravariant functor it’s easy to see that this indexed set of arrows satisfies:

If we’d call this an inverse system of modules, and so we define an *inverse system in over *to be a functor where we naturally identify the functor with the associated ordered pair of indexed sets of objects/arrows. It’s not then hard to check that the category of inverse systems of modules over is the same thing as . We can thus define, in general, the *category of inverse systems in over *to be the category where the contravariant functors therein are identified with the ordered pair of indexed sets as mentioned above.

**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.

[…] system of right -bimodules over some directed set where, considering our recent discussion of directed systems as functors, we can consider this as a functor . Suppose we then had some -bimodule . Then, post-composing […]

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