Abstract Nonsense

Crushing one theorem at a time

Functors (Pt. II)


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

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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 \mathcal{C} and \mathcal{D} a contravariant functor is a mapping F:\mathcal{C}\to\mathcal{D} which is the same as a functor, but which reverses the direction of the arrows. Said differently, if A\xrightarrow{f}B is in \mathcal{C} then one has F(B)\xrightarrow{F(f)}F(A) in \mathcal{D}. Moreover, we see that F is going to (preferably) have to also reverse composition, so that F(g\circ f)=F(f)\circ F(g). It’s not hard to see that contravariant functors \mathcal{C}\to\mathcal{D} are nothing more (secretly) than just covariant functors \mathcal{C}^{\text{op}}\to\mathcal{D} where \mathcal{C} is, as always, the opposite category. Let’s take a look at some examples of contravariant functors.

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Let \mathcal{C} be a category and A\in\text{obj}(\mathcal{C}) some fixed object. Similar to the covariant Hom functor is the contravariant Hom functor \text{Hom}(\bullet,A):\mathcal{C}\to\mathbf{Set} which takes B to \text{Hom}(B,A) and B\xrightarrow{f}C to \text{Hom}(C,A)\xrightarrow{f^\ast}\text{Hom}(B,A) where f^\ast(g)=g\circ f.

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

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For example, the contravariant Hom functor on R\textbf{-}\mathbf{Mod}, as we have seen, actually takes values in \mathbf{Ab}, and if R is commutative it takes values in R\text{-}\mathbf{Mod}.

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As a particular example of the contravariant Hom functor, we can consider the dual functor ^\vee:R\text{-}\textbf{Mod}\to\textbf{Ab} defined by taking a mdoule M to its dual module M^\vee. Even more specifically, we can fix a field k we can look at the dual space functor \mathbf{Vect}_k\to\mathbf{Vect}_k.

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We can create a contravariant functor \mathbf{Top}\to\mathbf{Ring} by taking X to the ring C(X,\mathbb{R}) of continuous maps X\to\mathbb{R}, this is just the contrarvariant Hom functor with \mathbb{R} fixed, only realizing that the image of the functor sits nicely inside of \textbf{Ring} instead of \mathbf{Set}.

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Types of Functors

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Similar to the case of different types of morphisms there are different types of functors. We fix two categories \mathcal{C} and \mathcal{D} and a functor F:\mathcal{C}\to\mathcal{D}. Then, we make the following definitions.

The functor F is known as an endofunctor if \mathcal{C}=\mathcal{D}.

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The functor F:\mathcal{C}\to\mathcal{D} is full if for each A,B\in\text{obj}(\mathcal{C}) the induced Hom set maps F_{A,B}:\text{Hom}_\mathcal{C}(A,B)\to\text{Hom}_\mathcal{D}(F(A),F(B)) are surjective.

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The functor F is faithful if for each A,B\in\text{obj}(\mathcal{C}) the induced Hom set maps F_{A,B}:\text{Hom}_\mathcal{C}(A,B)\to\text{Hom}_\mathcal{D}(F(A),F(B)) are injective.

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Note that we can compose functors, and we call a functor F:\mathcal{C}\to\mathcal{D} an isomorphism  if there exists another functor G:\mathcal{D}\to\mathcal{C} such that F\circ G=1_\mathcal{D} and G\circ F=1_\mathcal{C}. We call two such categories isomorphic and denote this by \mathcal{C}\cong\mathcal{D}. It’s clear that the inverse functor for any isomorphism is unique, and so we may unambiguously denote it by F^{-1}. Moreover, it’s clear that “is isomorphic to” is an equivalence relation on the “class of all categories”.

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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 \mathcal{S} of \mathcal{C} then the inclusion functor I:\mathcal{S}\hookrightarrow\mathcal{C} is going to always be faithful. Moreover, it’s clear by definition (and the name should be a hint) that I is going to be a full functor if and only if \mathcal{S} is a full subcategory of \mathcal{C}.

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Let’s now look at some examples of categorical equiavalences. For example, it’s not hard to see that the obvious functor \textbf{Ab}\to\mathbb{Z}\text{-}\mathbf{Mod} 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”.

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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

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Theorem: Let \mathcal{C} be any category and \mathcal{X},\mathcal{Y} two skeleta of \mathcal{C}. Then, \mathcal{X}\cong\mathcal{Y}.

Proof: We know that for each A\in\text{obj}(\mathcal{X}) there exists a unique F(A)\in\text{obj}(\mathcal{Y}) of the same isomorphism type, we define F on \text{obj}(\mathcal{X}) accordingly–note that this is obviously a bijection. Now, suppose that A,B\in\text{obj}(\mathcal{X}) and let A\xrightarrow{f}F(A) and B\xrightarrow{g}F(B) be the guaranteed isomorphisms. We then define, for A\xrightarrow{j}B the map F(A)\xrightarrow{F(j)}F(B) by g\circ j\circ f^{-1} where f^{-1} is the inverse of f–note that this is obviously a bijection on the Hom sets. So, now the fact that F is an isomorphism follows from the theorem immediately succeeding this theorem. \blacksquare

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Theorem:  Let \mathcal{C} and \mathcal{D} be two categories. Then, a functor F:\mathcal{C}\to\mathcal{D} is an isomorphism if and only if its full, faithful, and bijective on objects.

Proof: Obvious. \blacksquare

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The Main Property of Functors

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

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Theorem: Let \mathcal{C} and \mathcal{D} be functors and F:\mathcal{C}\to\mathcal{D} a functor. Then, if A\xrightarrow{f}B is an isomorphism so is F(A)\xrightarrow{F(f)}F(B).

Proof: Let f^{-1}:B\to A be the inverse of f, then we have that F(B)\xrightarrow{F(f^{-1})}F(A) and by using the functional property of functors we have F(f)\circ F(f^{-1})=F(f\circ f^{-1})=F(1_B)=1_{F(B)} and F(f^{-1})\circ F(f)=F(f^{-1}\circ f)=F(1_A)=1_{F(A)} so that F(f^{-1}) is an inverse for F(f), and so F(f) is an isomorphism as desired. \blacksquare

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Think about what this says. This says that we can tell \mathcal{C}-information entirely by \mathcal{D}-information–this was the founding idea of algebraic topology. For example, it’s near impossible to prove that as topological spaces the 2-sphere \mathbb{S}^2 and the2– torus \mathbb{T}^2 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 \mathbb{S}^2 and \mathbb{T}^2 agree on all of these topological invariants. But, if we recall that the fundamental group is a functor, we can quickly see that \mathbb{S}^2 being homeomorphic to \mathbb{T}^2 would imply, by the above, that \pi_1(\mathbb{S}^2)\cong\pi_1(\mathbb{T}^2) as groups, but this just simply isn’t true \pi_1(\mathbb{S}^2)\cong0 and \pi_1(\mathbb{T}^2)\cong\mathbb{Z}^2. This is where one should step back, and appreciate the power of functors.

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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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December 27, 2011 - Posted by | Algebra, Category Theory | , , ,

7 Comments »

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