By Daniel Lin
We all have a fair idea of what polynomials are from our high school days. Remember having to “factorise” expressions of the form x^2+3x+2? These are examples of what are called polynomials, where x is just some unknown variable.
However, we can also think about these polynomials as functions. For example, the value of the function f(x)=x^2+3x+2 at x=3 is 20. Notice that when we input a natural number (e.g., 1,2,3 etc.) into the function, the output is also a natural number. Similarly, if we had used a fraction as a our input, then we would get a fraction as our output. Therefore, a question arises. Would it make sense to have a set of objects as our input?
Well, it turns out we can if we think about how polynomials are constructed in the first place (taking as our example the same polynomial x^2+3x+2 ).
- Begin with our variable x.
- Make five copies of x to form a set {x,x,x,x,x}.
- Take two of the x’s above to form x^2. Leave the other x’s as they are. Construct two copies of the number 1.
- Finally, sum everything together to make x^2+x+x+x+1+1=x^2+3x+2.
Now if we want to be even more abstract, we may in fact represent the above process by the following diagram of objects and arrows (the building blocks of a branch of mathematics known as Category Theory).
{x}← {x,x,x,x,x}→{x^2,x,x,x,1,1}→{x^2+3x+2}
A category is just a collection of objects and arrows between these objects, and a function between two categories is called a functor. So in a way, they are similar to sets (but with the addition of arrows). However, we can take this one step further and think about arrows between arrows (bicategories).
Object → Object
⇓
Object → Object
The aim of my project was to define such a new bicategory involving this abstract notion of polynomials.
But why polynomial functors? Well, it turns out that these have very important applications in computer science, particularly in aiding our understanding of mathematical structures called trees.
Daniel Lin was one of the recipients of a 2013/14 AMSI Vacation Research Scholarship.