Nothing to do with differential geometry but there was a question on the haskell-cafe mailing list asking for “a proof that initial algebras & final coalgebras of a CPO coincide”. I presume that means a CPO-category.

A category is a CPO-category iff

- There is a terminator, 1.
- Each hom-set, , is a CPO with a bottom.
- For any three objects, arrow composition is strict continuous.
- Idempotents split, that is, if then and .

The following lemma answers the poster’s question.

**Lemma**

If is a covariant endofunctor on a CPO-category, then is an initial -algebra with respect to the sub-category of strict maps if and only if is a final -co-algebra.

**Proof**

See Freyd.

We can do a bit more. We borrow some notation from Meijer, Fokkinga and Paterson. If, for a given functor, the initial algebra exists and is an algebra then we denote the unique arrow from the initial algebra to this algebra as . Dually, if for a given functor, the final co-algebra exists and is a co-algebra then we denote the unique arrow to the final co-algebra as .

**Theorem**

In a CPO-category, if is an isomorphism then and .

**Proof**

Let be the initial -algebra and let be the final -co-alegebra. Then by the above lemma, we have that and . We also have

Since there is only one arrow from the initial algebra to itself then we must have .

For the other way, there is always a morphism from the algebra to the initial algebra. Therefore, must be isomorphic to the initial algebra.

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