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Functor

A functor is a mapping from one category to another.

In particular, a functor \(F: C \to D\) for categories \(C, D\) must map objects \(c \in C\) to objects \(Fc \in D\), and morphisms \(f \in C\) to morphisms \(Ff \in D\), preserving sources, targets, identities, and composition. This last fact, equivalent to the statement that for all morphisms \(f, g \in C\) , \(Ff \circ Fg = F(f \circ g)\), is known as the functoriality axiom.

Examples of Functors

There are many, many examples of functors. Some simple examples include:

  1. The "forgetful functor" \(U: \text{Grp} \to \text{Set}\) which maps a group to its underlying set and group homomorphisms to their respective functions. Likewise, there is an analogous functor from \(\text{Top}\) to \(\text{Set}\).
  2. The fundamental group of a topological space defines a functor \(\pi_1: \text{Top} \to \text{Grp}\), and the \(n\)-cycles \(Z_n\), \(n\)-boundaries \(B_n\), and \(n\)-th homology \(H_n\) all define functors from the category of chain complexes to the category of graded \(R\)-modules.
  3. The there a free functor from the category of sets to the category of groups which maps a set \(X\) to the free group on \(X\).
  4. There is an opposite functor \((-)^{\text{op}}: \text{Cat} \to \text{Cat}\) which maps a category to its opposite category.

Examples of the Functoriality Axiom

There are many seemingly unrelated corollaries of the functoriality axiom.

The Chain Rule

The derivative can be thought of as an endofunctor on \(\text{Diff}\), mapping manifolds to their tangent bundles and functions to their derivatives.

Suppose \(A, B, C\) are smooth manifolds, and \(f: A \to B\) and \(g: B \to C\) are differentiable functions. Then \(f \circ g: A \to C\) is a smooth function, and functoriality implies that \(d(f \circ g) = d(f) \circ d(g)\). However, this is simply the statement that \(f'(g(x)) = f'(x) g'(x)\), which is the ordinary chain rule from introductory calculus.

Brouwer's Fixed Point Theorem

Consider the unit disk \(D^2\). Suppose that there exists a continuous function \(f: D^2 \to D^2\) that does not admit any fixed points. Then there must exist a function \(r: D^2 \to S^1\) which maps a point \(x \in D^2\) to the point on the boundary circle \(S^1\) intersected by a ray from \(f(x)\) through \(x\). The points on the unit circles are fixed points of \(r\), and so if we compose with the inclusion mapping \(i: S^1 \to S^2\), \(r \circ i\) must be the identity morphism on \(S^1\). Functoriality of the fundamental group functor then implies \(\pi_1(r) \circ \pi_1(i) = \pi_1(r \circ i) = \pi_1(id_{S^1}) = id_{\pi_1(S^1)}\), which is simply the identity homomorphism.

However, the homotopy groups of both the circle and the disk are well known: \(\pi_1(D^2) = 0\) and \(\pi_1(S^1) = Z\). Therefore, because \(i: S^1 \to D^2\), \(\pi_1(i): Z \to 0\) can only be the trivial/zero homomorphism, in turn implying \(\pi_1(i) \circ \pi_1(r)\) is the trivial homomorphism.

\(\pi_1(r) \circ \pi(i)\) cannot be both the identity homomorphism and the trivial homomorphism, and so there is a contradiction. Therefore, no such function can exists. The statement that every continuous endomorphism on the disk must admit a fixed point is precisely Brouwer's Fixed Point Theorem.

Natural Transformations

A natural transformation is a mapping from one functor to another, sharing the same domain and codomain, subject to certain naturality constraints. The original goal of Category Theory was to study natural transformations, and categories and functors were a means by which to do so.