categories: Re: Final Coalgebra Question

[Dusko Pavlovic <dusko@kestrel.edu>]

> 1. Let X be a fixed Set. What is the final coalgebra of the functor
> [X,_]:Set -> Set. If you wish to make X finite then that's fine by me.

you mean exponentiation? why isn't 1 --->[X,1] final?

> 2. Consider the functor [[_,2],2]:Set -> Set. This functor doesnt have
> a final coalgebra for cardinality reasons. However one may define a

[snip]

> Now, T' is clearly finitary and from general nonsense we know that it
> has a final coalgebra. But what is it concretely?

depends on how we represent the final coalgebra for the finite powerset
functor [-,2]_fin. if you like to view its elements as finitely branching
apgs, then the final coalgebra for [[-,2],2]_fin presumably consists of
*bipartite* finitely branching apgs: the root is blue, its successors are
red, the successors of successors are blue again, and so on. (given a
coalgebra X-->[[X,2],2], write Y = [X,2]. this is the set of the red
nodes of this apg; X is the set of the blue nodes. each of the red nodes
the char function of its blue successors. and the structure map X-->[Y,2]
tells the red successors of each blue node. so each elt of X induces, as
its trace through the coalgebra, a bipartite apg with a blue root. this
gives the final coalgebra homomorphism from X to the blue-rooted
bipartite apgs. unless i am wrong.)

-- dusko