# December 2020

# The Fibonacci Sequence as a Functor

Over the years, the articles on this blog have spanned a wide range of audiences, from fun facts (Multiplying Non-Numbers), to undergraduate level (The First Isomorphism Theorem, Intuitively), to graduate level (What is an Operad?), to research level. Today's article is more on the fun-fact side of things, along with—like most articles here—an eye towards category theory.

So here's a fun fact about greatest common divisors (GCDs) and the **Fibonacci sequence **$F_1,F_2,F_3,\ldots$, where $F_1=F_2=1$ and $F_n:=F_{n-1} + F_{n-2}$ for $n>1$. For all $n,m\geq 1$,

In words, the greatest common divisor of the $n$th and $m$th Fibonacci numbers is the Fibonacci number whose index is the greatest common divisor of $n$ and $m$. (Here's a proof.) Upon seeing this, your "spidey senses" might be tingling. *Surely* there's some structure-preserving map $F$ lurking in the background, and this identity means it has a certain nice property. But what *is* that map? And what structure does it preserve? And what's the formal way to describe the nice property it has?

The short answer is that the natural numbers $\mathbb{N}=\{1,2,3,\ldots\}$ form a partially ordered set (poset) under division, and the function $F\colon \mathbb{N}\to\mathbb{N}$ defined by $n\mapsto F_n:=F(n)$ preserves *meets*: $F_n\wedge F_m = F(n\wedge m)$.