## Functors (Pt. II)

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

Now that we have seen ample examples of functors, it’s important to note that covariant functors are often times not what comes up naturally but contravariant functors. Indeed, given categories and a *contravariant functor* is a mapping which is the same as a functor, but which reverses the direction of the arrows. Said differently, if is in then one has in . Moreover, we see that is going to (preferably) have to also reverse composition, so that . It’s not hard to see that contravariant functors are nothing more (secretly) than just covariant functors where is, as always, the opposite category. Let’s take a look at some examples of contravariant functors.

Let be a category and some fixed object. Similar to the covariant Hom functor is the *contravariant Hom functor *which takes to and to where .

More generally, “functors” which take objects to mapping objects “tend” to be contravariant, because it’s usually the target space that’s fixed.

For example, the contravariant Hom functor on , as we have seen, actually takes values in , and if is commutative it takes values in .

As a particular example of the contravariant Hom functor, we can consider the *dual functor* defined by taking a mdoule to its dual module . Even more specifically, we can fix a field we can look at the *dual space *functor .

We can create a contravariant functor by taking to the ring of continuous maps , this is just the contrarvariant Hom functor with fixed, only realizing that the image of the functor sits nicely inside of instead of .

*Types of Functors*

Similar to the case of different types of morphisms there are different types of functors. We fix two categories and and a functor . Then, we make the following definitions.

The functor is known as an *endofunctor *if .

The functor is *full* if for each the induced Hom set maps are surjective.

The functor is *faithful *if for each the induced Hom set maps are injective.

Note that we can compose functors, and we call a functor an *isomorphism ** *if there exists another functor such that and . We call two such categories *isomorphic* and denote this by . It’s clear that the inverse functor for any isomorphism is unique, and so we may unambiguously denote it by . Moreover, it’s clear that “is isomorphic to” is an equivalence relation on the “class of all categories”.

Now, let’s see if we can get our heads around some of these definitions. It’s easy to see that if we have some subcategory of then the inclusion functor is going to always be faithful. Moreover, it’s clear by definition (and the name should be a hint) that is going to be a full functor if and only if is a full subcategory of .

Let’s now look at some examples of categorical equiavalences. For example, it’s not hard to see that the obvious functor is going to be an isomorphism. Now, I feel it is necessary to pause for a second and examine this statement. Those who have had any study in module theory might scoff at this statement “of course they are ‘isomorphic’–they are equal!”. Ah, but therein lies the rub. They are certainly not equal, are they? Their defining qualities are not the same, or as my father (a computer programmer) would say “they are not the same ‘type'”. But, as one may protest, they are just the same thing in different light–in one light we are looking at aspect X of them and in the other we are looking at aspect Y, but if we could illuminate the room properly we’d realize we are looking at the same objects! But there is the beauty. This is one of startlingly powerful (at least in terms of gaining intuition) uses of functors, they allow us to rigorously make sense of the statement “they are the same thing, just thought about in different ways”.

Now, backing away from the more philosophical side of things, now that we have defined the different type of functors we can now prove some interesting theorems relating them to some previously defined notions. For example

**Theorem: ***Let be any category and two skeleta of . Then, .*

**Proof: **We know that for each there exists a unique of the same isomorphism type, we define on accordingly–note that this is obviously a bijection. Now, suppose that and let and be the guaranteed isomorphisms. We then define, for the map by where is the inverse of –note that this is obviously a bijection on the Hom sets. So, now the fact that is an isomorphism follows from the theorem immediately succeeding this theorem.

**Theorem: ** *Let and be two categories. Then, a functor is an isomorphism if and only if its full, faithful, and bijective on objects.*

**Proof: **Obvious.

*The Main Property of Functors*

We’d now like to, lastly, discuss far and by large, one of the most important aspects of functors.

**Theorem: ***Let and be functors and a functor. Then, if is an isomorphism so is .*

**Proof: **Let be the inverse of , then we have that and by using the functional property of functors we have and so that is an inverse for , and so is an isomorphism as desired.

Think about what this says. This says that we can tell -information entirely by -information–this was the founding idea of algebraic topology. For example, it’s near impossible to prove that as topological spaces the -sphere and the– torus aren’t isomorphic if one only knows the basic topological invariants from point-set topology: connectedness, local connectedness, Hausdorffness, compactness, metrizability, completeness, etc. One can quickly check that and agree on all of these topological invariants. But, if we recall that the fundamental group is a functor, we can quickly see that being homeomorphic to would imply, by the above, that as groups, but this just simply isn’t true and . This is where one should step back, and appreciate the power of functors.

**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.

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