### In Search of Terminal Coalgebras

#### Posted by David Corfield

Tom Leinster has put up the slides for his joint talk – Terminal coalgebras via modules – with Apostolos Matzaris at PSSL 88.

It’s all about establishing the existence of, and constructing, terminal coalgebras in certain situations. I realise though looking through the slides that I never fully got on top of the *flatness* idea, and *n*Lab is a little reluctant to help at the moment (except for flat module).

So perhaps someone could help me understand the scope of the result, maybe via an example. Say I take the polynomial endofunctor

$\Phi(X) = 1 + X + X^2.$

Given that terminal coalgebras can be said to have cardinality $i$, in which categories will I find such a thing?

In $Set$ we have that the initial algebra for $\Phi$ is the set of Motzkin trees. I guess the terminal coalgebra is the set of extended such trees, just as the initial algebra for $\Psi(X) = 1 + X$ is the natural numbers and the terminal coalgebra the extended natural numbers.

## Re: In Search of Terminal Coalgebras

Of course, I should read again Tom’s introductory paper, and the longer two it introduces.