Hi, 1. That's pretty easy. Since [X,_] maps the one-element (final) set onto itself, i.e., it preserves the final object, the final coalgebra of this functor is nothing but the one-element-set itself. Of course one need not use this abstract argument ("the forgetful functor U:Set_F --> Set creates any limit which the functor F:Set->Set preserves") but can give a direct proof. 2. That's much more complicated. I do not know any elementary description. For one possible description as a quotient of an automaton you may refer to H. Peter Gumm, Tobias Schroeder : Coalgebras of bounded type. Mathematical Structures in Computer Science, to appear which you can find Peter Gumm's homepage (http://www.mathematik.uni-marburg.de/~gumm/Papers/publ.html) and the papers quoted there. ... If somebody could offer a really nice description of that final coalgebra this would be quite interesting I assume. Hope that helps a bit Tobias Schroeder -----Original Message----- From: N Ghani To: categories@mta.ca Cc: coalgebras@iti.cs.tu-bs.de Sent: 17.01.02 12:23 Subject: coalgebras: Final Coalgebra Question Can anyone help me with the following final coalgebra questions 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. 2. Consider the functor [[_,2],2]:Set -> Set. This functor doesnt have a final coalgebra for cardinality reasons. However one may define a finitary variant of this functor as follows: First let TX = [[X,2],2] if X is finitely presentable. Thus T:Set_fp -> Set is a functor from the full subcategory of fintely presentable objects of Set into Set. Next define T' to be the left Kan extension of T along the inclusion Set_fp -> Set. In other words T'X is the filtered colimit of all the TX_0 where X_0 is a finitely presentable subobject of X Now, T' is clearly finitary and from general nonsense we know that it has a final coalgebra. But what is it concretely? Thanks for any help you can offer Neil 18-Jan-2002 08:31:46 -0400,3073;000000000001-00000000