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1. • CommentRowNumber1.
   • CommentAuthorTodd_Trimble
   • CommentTimeSep 4th 2011
   • PermaLink
   Author: Todd_Trimble
   Format: MarkdownItexI 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.

   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.

2. • CommentRowNumber2.
   • CommentAuthorTobyBartels
   • CommentTimeSep 4th 2011
   • PermaLink
   Author: TobyBartels
   Format: MarkdownItexHa ha, "well-illustrated"

   Ha ha, “well-illustrated”

3. • CommentRowNumber3.
   • CommentAuthorDavid_Corfield
   • CommentTimeSep 5th 2011
   • PermaLink
   Author: David_Corfield
   Format: MarkdownItexGot 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'?

   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’?

4. • CommentRowNumber4.
   • CommentAuthorTobyBartels
   • CommentTimeSep 5th 2011
   • PermaLink
   Author: TobyBartels
   Format: MarkdownItexFor 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)[[nonassociative algebra|algebra]] to be over, such as a comonad. I don't understand why people want to use "for".

   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”.

5. • CommentRowNumber5.
   • CommentAuthorUrs
   • CommentTimeSep 5th 2011
   • (edited Sep 5th 2011)
   • PermaLink
   Author: Urs
   Format: MarkdownItexI 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.

   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.

6. • CommentRowNumber6.
   • CommentAuthorTobyBartels
   • CommentTimeSep 5th 2011
   • PermaLink
   Author: TobyBartels
   Format: MarkdownItexYou mean that you added a link from [[coalgebra for an endofunctor]].

   You mean that you added a link from coalgebra for an endofunctor.

7. • CommentRowNumber7.
   • CommentAuthorUrs
   • CommentTimeSep 5th 2011
   • PermaLink
   Author: Urs
   Format: MarkdownItexYes, right, sorry.

   Yes, right, sorry.

8. • CommentRowNumber8.
   • CommentAuthorTodd_Trimble
   • CommentTimeJan 30th 2016
   • PermaLink
   Author: Todd_Trimble
   Format: MarkdownItexI 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.

   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.

9. • CommentRowNumber9.
   • CommentAuthorMike Shulman
   • CommentTimeJan 31st 2016
   • PermaLink
   Author: Mike Shulman
   Format: MarkdownItexIs being a Peano $T$-algebra the same as being a surjective image of the initial $T$-algebra?

   Is being a Peano $T$-algebra the same as being a surjective image of the initial $T$-algebra?

10. • CommentRowNumber10.
    • CommentAuthorTodd_Trimble
    • CommentTimeJan 31st 2016
    • PermaLink
    Author: Todd_Trimble
    Format: MarkdownItexNo; 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.

    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.

11. • CommentRowNumber11.
    • CommentAuthorMike Shulman
    • CommentTimeJan 31st 2016
    • PermaLink
    Author: Mike Shulman
    Format: MarkdownItexHmm, 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?

    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?

12. • CommentRowNumber12.
    • CommentAuthorTodd_Trimble
    • CommentTimeJan 31st 2016
    • (edited Feb 1st 2016)
    • PermaLink
    Author: Todd_Trimble
    Format: MarkdownItex(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".

    (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”.

13. • CommentRowNumber13.
    • CommentAuthorMike Shulman
    • CommentTimeFeb 1st 2016
    • PermaLink
    Author: Mike Shulman
    Format: MarkdownItexIt's *kind* of like being [[simple object|simple]]...

    It’s kind of like being simple…