(Extra)natural transformation

Let $𝐹 : 𝖠 \times 𝖡 \times 𝖡 𝐨 𝐩 \to 𝖣$ and $𝐺 : 𝖠 \times 𝖢 \times 𝖢 𝐨 𝐩 \to 𝖣$ be functors. A transformation (family of morphisms) with components

𝛼 𝑎, 𝑏, 𝑐 : 𝐹 (𝑎, 𝑏, 𝑏) \to 𝐺 (𝑎, 𝑐, 𝑐)

can of course never be natural in $𝑏$ or $𝑐$ , but it can be extraordinary-natural, or extranatural in $𝑏$ and $𝑐$ . Extranaturality in an argument appearing in the domain is given by the diagram on left, whille extranaturality an argument appearing in the codomain is given by the diagram on the right.¹ #m/def/cat

A transformation is said to be (extra)natural iff it is natural in arguments appearing in both its domain and codomain, and extranatural in any arguments appearing both covariantly and contravariantly in either its domain or codomain only.

#state/tidy | #lang/en | #SemBr

1966. A generalization of the functorial calculus, p. 367 ↩

(Extra)natural transformation

Footnotes