Nested partial unary operation category ( $C_{n}$ )

An object of

C_{n}

is a set with a finite list of partial unary operations. The first is total; the domain of each later operation consists exactly of the defined fixed points of the immediately preceding operation. Morphisms preserve every defined operation. Take

C_{0}

to be the Category of sets. This tower gives examples of arbitrary finite monadic length even though its long free–forgetful composites induce the same first monad.

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- Free extension of nested partial unary operations Nested partial unary operation category

Free extension of nested partial unary operations ( $F_{n} ⊣ G_{n}$ )

 0  0

For an object of the nested partial unary operation category

C_{n}

, adjoining the next operation freely gives one new value at each defined fixed point of the previous operation. Each new value starts an infinite free chain under the first unary operation; the other old operations are undefined on those new points. The new top operation consequently has no fixed points, so subsequent free extensions add no points. This gives a left adjoint to forgetting the last operation; extensions of maps along the chains are forced by the first target operation.