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
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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 18-Jan-2002 08:34:03 -0400,897;000000000000-00000000