## Limits, Colimits, and Representable Functors (Pt. III)

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

Well, call (in general) a functor *representable by *if is an object of such that –in other words, is secretly just the covariant Hom functor associated to . In this case we say that * represents* . Call *representable *if it is represented by some object of . Similarly, if is a contravariant functor we say it’s representable if there exists such that –the rest of the terminology also translates.

So, back to thinking about our functor . Fix an object in and denote the functor given by and is mapped to with . Since this is a functor into from we can take its limit to get the set . To be explicit it is the following set

Let now be the contravariant functor which on objects takes and which is harder to define on arrows. Ok, so let be a map. We then define a map by which extends uniquely to a map .

Ok, so we now have this somewhat crazy contravariant functor . What we claim though is that if exists then it represents . Indeed, we define the natural transformation with given by the unique extension of the map given by where are the cone maps. We first claim that each is a bijection. To see that is an injection we merely note that if then for all and so by definition of a limit we must have that . To see that is surjective we take an arbitrary . We know then that is a morphism and so by the definition of there exists a morphism such that and thus we see that as desired.

Ok, now that we know the ‘s are bijections the only thing left to check is that the ‘s satisfy the naturality condition, namely that for each arrow in the following diagram commutes:

To do this let be arbitrary. We see then that

whose coordinate is . Now,

which has -coordinate . Thus, we may conclude that the diagram we want commutes and thus really is a natural isomorphism.

While I won’t go through all of it one can verify that we had instead started with looking at defined on objects by (where the functor is defined in the obvious way) and on arrows in the obvious way that (if it exists) becomes a representing element of the covariant functor .

Of course, we can dualize the property above to instead recover the definition of limit and colimit from these functors. Namely, it’s not hard to see that a representing element of and is necessarily going to be a limit and colimit respectively. How? Well, clearly the representing object of , for example, will be the object and so we are just left trying to figure out what we should take the cone arrows to be. But, if there is any justice in the world (and there is–well, at least in math) we should be able to use the above construction to actually figure them out. In particular, we know that on the coordinate is a map given by . Now, if we wanted to recover it seems pretty sensible to take to be the identity map (so that ) but this only makes sense if we take . Thus, by tinkering, we discover that we should be able to recover the cone arrows by . And, while I won’t prove this, it’s true (check it yourself).

The moral of the story is that we can think about limits/colimits as either their cone definition or as representing elements of certain functors–both ways can be useful at different points.

**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] Mitchell, Barry. *Theory of Categories.* New York: Academic, 1965. Print.

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

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