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I have started an article well-founded coalgebra, where I’m trying to put together some things I’ve learned while reading Paul Taylor’s work. All comments welcome.
Ha ha, “well-illustrated”
Got the link to ’coalgebra for an endofunctor’ working by changing ’of’ to ’for’. At that page we seem to switch between ’for’ and ’over’. Trivial stuff, but is there a best choice out of ’coalgebra of/for/over an endofunctor’?
For making the links work, it’s most important to put the redirects at the page to which we are linking (which I have now done).
In this case I like “of” best. For “over”, I would want something that feels more like a (co)algebra to be over, such as a comonad. I don’t understand why people want to use “for”.
I have added some more hyperlinks, a TOC and a floating Context. Also I added a link to the entry from coalgebra, such that it can be found by people who don’t yet know that it exists.
You mean that you added a link from coalgebra for an endofunctor.
Yes, right, sorry.
I added a bit more to well-founded coalgebra, to help create an opening for further development.
In particular I added two observations that, while trivial, I hadn’t seen mentioned elsewhere and which casts a suggestive light I think. Recall that for an endofunctor $T: E \to E$, a $T$-algebra is Peano if every $T$-subalgebra inclusion is an isomorphism (the classical case being the endofunctor $T: X \mapsto 1 + X$ on a topos, where a Peano $T$-algebra is called a Peano natural numbers object). Now, if $(X, \theta: X \to T X)$ is a $T$-coalgebra, then we can form an endofunctor on the slice $E/X$ as an evident composite
$E/X \stackrel{T_\ast}{\longrightarrow} E/T X \stackrel{\theta^\ast}{\longrightarrow} E/X$and of course the terminal object $1_X: X \to X$ is automatically a $\theta^\ast T_\ast$-algebra. The first observation is that a subobject $i: U \to X$ is an inductive subobject of the $T$-coalgebra $X$ iff $i$ is a $\theta^\ast T_\ast$-subalgebra of $1_X$. The second observation is that the coalgebra $X$ is well-founded iff $1_X$ is a Peano $\theta^\ast T_\ast$-algebra.
Is being a Peano $T$-algebra the same as being a surjective image of the initial $T$-algebra?
No; for example there are many surjective images of the natural numbers object $\mathbb{N}$ in a topos as $T$-algebra over $T(X) = 1 + X$, but in a topos a Peano NNO must be the usual initial algebra type of NNO.
I need to think more on what (if any) is the real significance of the observation in #8.
Hmm, it seems to me that the terminal set with its unique $T$-algebra structure (for $T(X)=1+X$) is Peano, since its only proper subset is empty and that is not a $T$-subalgebra. What am I missing?
(Redacted.)
Oh, duh. I was thinking Peano postulates for N in a topos when I said that, but forgot about half of the postulates. :-P
I don’t know off-hand the answer to your question in #9. But in the meantime I should at least name this concept something else besides “Peano”. Just to have a placeholder until somethin better comes along, I’ve revised the article to name this subalgebra property “semi-Peano”.
It’s kind of like being simple…
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