Fun with Functors

We again begin this post with citations. The interested reader can refer to:

• Saunders Mac Lane, Categories for the Working Mathematician Graduate Texts in Mathematics (vol 5) Springer-Verlag, Second Edition (1998);
• John Baez, Some Definitions Everyone Should Know from Fall 2004 seminar.
• Saunders Mac Lane and Ieke Moerdijk, Sheaves in Geometry and in Logic: A First Introduction to Topos Theory Springer-Verlag (1992).

We'll then proceed to the notion of functors. We introduced the morphism as a structure-preserving mapping between mathematical objects. What if we take those objects to be categories? That's precisely what a functor is! More formally:

Definition 1 Given categories $C,D$ a "functor" $F:C\to{D}$ consists of:

• a function $F:Ob(C)\to Ob(D)$;
• for any pair of objects $x,y\in Ob(C)$, a function $F:\hom(x,y)\to\hom(F(x),F(y))$

such that

• $F$ preserves identities: for any object $x\in C$, $F(1_{x})=1_{F(x)}$;
• $F$ preserves composition: for any pair of morphisms $f:x\to{y}$, $g:y\to{z}$ in $C$, we have $F(fg)=F(f)F(g)$.

It's straightforward to introduce the notion of composition of functors and identity functors. They are done in "the obvious way" (i.e. composition of functors results in a functor whose object function consists of the composed object functions, and morphism function consists of the composed morphism function, etc.).

So...what do functors do?! Really in a nutshell, they assign information to each object and information-preserving-morphism to each morphism. This is still kinda abstract, so lets break it down to a catch phrase:

*** functors are mathematical procedures. ***

For example, consider the functor $\mathcal{P}:\mathbf{Set}\to\mathbf{Set}$ the powerset functor. It does what you think it does, it eats in a set $X$ and it spits out its power set $\mathcal{P}(X)$; for each function $f:X\to{Y}$ it eats in, it spits out a function between the corresponding powersets $\mathcal{P}:\mathcal{P}(X)\to\mathcal{P}(Y)$.

There is also a broad class of functors that can "forget" aspects of a mathematical object, we can forget stuff, structure, and properties...or we can forget structure, and properties...or we can forget properties. In general, whenever we use the term "forgetful functor", we mean a functor $U:C\to\mathbf{Set}$ which gives us the set underlying a mathematical object.

Definition 2. A "Concrete Category" $(C,U)$ consists of

• a category $C$;
• a (faithful) functor $U:C\to\mathbf{Set}$

We can think of objects of $(C,U)$ as being a set "with some structure". Some examples of concrete categories would be Mon, Rng, Grp, and so on.

Sheaves as Generalization of Vector Fields

On the other hand, we have a notion of having a topological space as a category. We have, from basic vector calculus, the notion of assigning a vector to each point in space. We can generalize this notion by assigning some information to each open subset of a topological space, i.e. each object of the category $O(X)$ of open subsets in the topology $\mathcal{T}$ of $X$. (Note the morphisms of $O(X)$ are inclusion mappings.)

So now the question is: how oh how do we assign information to each object of $O(X)$? It's trivial, we consider the functor $F:O(X)\to(C,U)$ where $(C,U)$ is a concrete category. Since we haven't introduced it yet, lets take a moment to consider $(C,U)$ as a concrete category. It consists of a category $C$ and a "forgetful functor" $U:C\to\mathbf{Set}$ which basically tells us that the objects of $C$ are sets equipped with some structure which is "forgotten" when we consider $U(C)$.

So to recap: $(C,U)$ as a concrete category consists of "sets equipped with structure" and a functor $U$ which gives us the underlying set. We assign information to the open subsets of a topological space $(X,\mathcal{T})$ by considering the topological space as a category $O(X)$ and then we use a functor $F:O(X)\to(C,U)$ to assign to each open subset some object of $C$ in a "nice way".

More precisely, the "nice way" is such that upon restrictions of open subsets of $X$ we get consistent information. So when we consider the subset $v\subseteq u\subseteq X$, we demand a morphism $res_{v,u}:F(u)\to F(v)$ in the category $C$. This morphism $res_{V,U}$ is called the restriction morphisms and they are such that:

1. For every open set $u$ of $O(X)$, the restriction morphism $res_{u,u}:F(u)\to{F(u)}$ is precisely the identity morphism;
2. if we have three open sets $w\subseteq v\subseteq u$, then $res_{w,v}\circ{}res_{v,u}=res_{w,u}$.

When this happens, we have a bit of a surprise: we have $F$ be called a "presheaf". We can formally define it in our favorite algorithmic way:

Definition 3. A "presheaf on $X$ with values in $C$" consists of

• a functor $F:O(X)^{op}\to(C,U)$ from the dual of $O(X)$ (i.e. $O(X)$ with the arrows reversed) to a concrete category $(C,U)$ which assigns
1. for each open set $u\subseteq{X}$, it assigns an object $F(u)$ of $C$;
2. for each inclusion of open sets $v\subseteq{u}$, it assigns a morphism $res_{v,u}:F(u)\to{F(v)}$ in $C$ called the "restriction morphism"

such that

• for each open set $u\subseteq{X}$, the restriction morphism $res_{u,u}:F(u)\to{F(u)}$ is the identity $1_{F(u)}$;
• if $w\subseteq{v}\subseteq{u}$ are open subsets of $X$, then $res_{w,v}\circ{}res_{v,u}=res_{w,u}$.

Notice: we a mild specification of "consistency on overlapping subsets". When we impose a stronger demand and some more properties, we'll turn a presheaf into a sheaf. We won't get to this, however, since it is far beyond the scope of this entry.

A presheaf assigns information in a way that's basically a generalization of vector fields, or should be intuitively considered as such. It gives us the intuition of the functor as assigning information to each object of a category in a way that's "consistent on overlaps".

Functors are my Hom-Boy

Recall that a category C has defined for any pair of objects $x,y\in\mathbf{C}$ a set $\hom(x,y)$ of morphisms from $x$ to $y$. So...what happens if we choose some fixed object $x_{0}\in\mathbf{C}$, and consider $\hom(x_{0},-)$ as a morphism?

Well it has to eat in objects from C. It spits out a set of morphisms. How does it behave on morphisms? Let $f:y\to{y^{\prime}}$ be some morphism in C, we expect our $\hom(x_{0},-)$ "morphism" to take the domain and the codomain to sets, it would seem logical that the morphism becomes a function of these sets. That is, $\hom(x_{0},f):\hom(x_{0},y)\to\hom(x_{0},y^{\prime})$ is a function.

Or, pithily, hom induces a functor $\hom(x_{0},-):\mathbf{C}\to\mathbf{Set}$(!)

Is this wild allegation true?! Lets see, if it satisfies the properties of a natural transformation, then it must be one.

Does it preserve the identity? Well, $\hom(x_{0},1_{y}):\hom(x_{0},y)\to\hom(x_{0},y)$, so that's a check on that.

Does it preserve composition of morphisms? If $f:y\to{z}$ and $g:z\to{w}$, then $g\circ{f}:y\to{w}$. We see that $\hom(x_{0},g\circ{f}):\hom(x_{0},y)\to\hom(x_{0},w)$ in the "obvious way", i.e. it's $\hom(x_{0},g)\circ\hom(x_{0},f)$. So that's check.

That means that $\hom(x_{0},-):\mathbf{C}\to\mathbf{Set}$ is indeed a functor.

The observant reader might ask "But what if we don't fix the first slot, so we have hom(-,-); what is that?" We can view it as a sort of generalization of the inner product for categories. It's actually a "bifunctor" from $\mathbf{C}^{op}\times\mathbf{C}\to\mathbf{Set}$ where C^op is the dual category to C. The reader knowledgeable in linear algebra should immediately see a parallel to the current situation, and we can think of functors as "operators" (later on we will see that we can define "adjoint functors" this way.) There is one exception: instead of "scalars" we have objects in Set, which is weird!