Talk:F-coalgebra

Example

Surely that example is wrong. If F(X) = X+1 then the final coalgebra is the natural numbers, not a stream of X's. A stream of X's, with a possible "end of file", would come from F(A) = XxA + 1. Sigfpe 01:42, 14 March 2006 (UTC)

Indeed. I have corrected it. Tillmo 10:15, 2 April 2006 (UTC)

Proposed rehash

Hi, I'm thinking of changing this article considerably. Currently it seems rather sparse and dry, lacking both interesting examples and motivation -- I could add these and bring the article a bit more up to date i.e. post 2001. Any objections? Algebran 11:06, 16 February 2009.

Go ahead, the article needs that and more. — Charles Stewart (talk) 21:01, 4 March 2009 (UTC)

Technical

The article seems difficult to understand without prior knowledge of the subject. —Preceding unsigned comment added by 79.49.169.202 (talk) 14:19, 8 April 2010 (UTC)

I agree, 10 years later 🙂. I added the {{Technical}} box to the article. --Jhertel (talk) 12:29, 26 July 2020 (UTC)

It would be useful for you to explain what it is that is confusing or hard to understand. That way, someone could try to fix at least those parts of it. Myself, I've never heard of F-coalgebras before, but I read it and it seems not only easy to understand, but even kind of boring, because they seem to be kind of trivial. At least the first two examples are "trivial": the first example decrements an unsigned int until it goes negative at which point an exception is thrown. Woot. The second example describes a finite state machine. Wow. What else is new? The third example seems to cover standard object-oriented factorization. Wow! Sooo ... my complaint is that it's missing a non-trivial example, some discussion as to why this is an interesting or useful abstraction. What can I do with F-colagrbas that is new and exciting? 67.198.37.16 (talk) 05:40, 18 March 2021 (UTC)

Proposed Change

An F-coalgebra is not necessarily based on a signature since the functor F is arbitrary. Therefore, the lede is in my opinion incorrect. Any objections? Citogenitor^{[talk needed]} 11:49, 25 February 2015 (UTC)

F-algebras are *not* the dual concept

They are only mostly dual. I haven't got the details around me, but this was said in a lecture 1 year ago and I also verified it myself on paper. For that, just take the definition of F-algebras and swap *all* arrow directions. It turns out that the definition of F-coalgebras is not the result! I think one or two arrows in the commutation requirement for morphisms are not matched.

Also, viewed from a different perspective, if they were dual, there would be less point in exploring both theories in parallel that much. Yes, you can transport results by partial duality, but not all. Interestingly, you can transport the statement of a theorem (whose name I forgot) via partial duality, but not its proof. The proof is different.

I edited the page to reflect that.

Edit I heard it actually is as follows: Coalg F = (Alg F^{op})^{op} =/= (Alg F)^{op}. So yes, they are dual in a specific sense, but not in the obvious one which would be Coalg F = (Alg F)^{op}, which doesn't hold. So what shall we write in the articles on F-algebras and F-coalgebras? — Preceding unsigned comment added by 188.195.0.202 (talk) 09:30, 23 December 2019 (UTC)

Edit 2 I've undone my changes. I still think including a section on the exact duality would be beneficial.

Example: (I) `∀(C: category) ∀(F:C->C) Alg(F) has all limits which C has`. By forall introduction & elimination we can get (II) `∀(C: category) ∀(F^{op}: C^{op}->C^{op}) Alg(F^{op}) has all limits which C^{op} has`. Note that I used C^{op} and F^{op} in the forall elimination. If we now classically dualize (II) to the opposite category, we get (III) `∀(C: category) ∀(F: C->C) Alg(F^{op})^{op} = Coalg(F) has all colimits which C has`. Wait... now I am wondering why we didn't dualize F^{op} to simply F. — Preceding unsigned comment added by 188.195.0.202 (talk) 10:39, 23 December 2019 (UTC)

--188.195.87.79 (talk) 15:59, 22 December 2019 (UTC)