observation is “lossy” –

Fong & Spivak^{1}

*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*

- Let \(f(x) = x\); arguments and results are identical so order is preserved.
- Is not order preserving
- \(f(x) = 1/x\)

- Is order preserving
*metric-preserving*\(\iff |x-y| = |f(x)-f(y)|\)- Is order preserving
- \(f(x) = x\)
*Trivial*

- \(f(x) = x\)
- Is not order preserving
- \(f(x) = 1/x\)

- Is order preserving
*addition-preserving*\(\iff f(x+y) = f(x) + f(y)\)- Is addition preserving
- \(f(x) = x\)
*Trivial*

- \(f(x) = x\)
- Is not addition preserving
- \(f(x) = 1/x\)

- Is addition preserving

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

Fong, B., & Spivak, D. (2019). An Invitation to Applied Category Theory: Seven Sketches in Compositionality. Cambridge: Cambridge University Press.↩︎