 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2015/iii/paper-22/6/c/ii/solution

Codex  0 Created 2026-10-03 Updated 2026-10-06

Assume

T η = η T

, and let

(A, a)

be an algebra for a monad. Its unit identity gives

a η_{A} = 1_{A}

η

a : T A \to A

, followed by the assumed equality, gives

η_{A} a = T a η_{T A} = T a T η_{A} = T (a η_{A}) = 1_{T A} .

Thus every algebra action is invertible, with its inverse prescribed by the unit:

a^{- 1} = η_{A} .

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