Category theory, first steps

observation is “lossy” – Fong & Spivak¹

compositionality systems or relationships combined to form new systems or relationships

structures and structure-preserving maps \(f: X \rightarrow Y\): which aspects of \(X\) are preserved by \(f\)?

Consider functions in \(\mathbb{R}\): only some preserve order, distance, sum:

A function \(f: \mathbb{R} \Rightarrow \mathbb{R}\) is said to be

• order-preserving \(\iff x \leq y \Rightarrow f(x) \leq f(y), \forall x \in \mathbb{R}\) (aka monotonically increasing)
  • Is order preserving
    • Let \(f(x) = x\); arguments and results are identical so order is preserved. Trivial
  • Is not order preserving
    • \(f(x) = 1/x\)
• metric-preserving \(\iff |x-y| = |f(x)-f(y)|\)
  • Is order preserving
    • \(f(x) = x\) Trivial
  • Is not order preserving
    • \(f(x) = 1/x\)
• addition-preserving \(\iff f(x+y) = f(x) + f(y)\)
  • Is addition preserving
    • \(f(x) = x\) Trivial
  • Is not addition preserving
    • \(f(x) = 1/x\)

Quaere: Is the identity \(f\) always preserving?