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I have tidied up the entry initial algebra and then made sure that it is cross-linked with inductive type (which it wasn’t!).
We really need to rename this entry to initial algebra for an endofunctor. But since I would have to fight the cache bug if I did it now, I decide not to be responsible for that at the moment.
at initial algebra the archetypical example of the natural number object was missing. So I have now added it.
What’s the good notion of “the category of initial $F$-algebras” as $F$ varies?
Maybe: what is a good notion of a natural transformation $F \Rightarrow G$ between endofunctors to qualify as a morphism of the respective initial $F$-algebras.
If you think this is not a good question to ask, do you see a roughly similar question that would be good to ask?
Somehow from a category-theoretic viewpoint the notion of an algebra over an endofunctor is a funny thing, with so little structure there. Of course initiality makes the difference. But in order to understand what a bunch of objects means, here: initial algebras over endofunctors, I should try to understand the category they form. Therefore the question: what’s a decent category of initial algebras of endofunctors on a given category?
I don’t actually see any roughly similar question that I think would be good to ask. (-:
Here is something that might be helpful, though. Rather than initial algebras for endofunctors, you can think about (algebraically-)free monads generated by endofunctors. The initial algebra of an endofunctor $F$ is the same as $T(\emptyset)$ where $T$ is the algebraically-free monad generated by $F$. Conversely, $T(A)$ is the initial algebra of the endofunctor $X\mapsto F(X) + A$. Thus being able to construct initial algebras for endofunctors is more or less equivalent to being able to construct algebraically-free monads on endofunctors.
Okay, yes, it’s that apparent absence of a monad structure from the definition of an “algebra over an endofunctor” that makes it feel a funny concept. But I see, of course one can just freely add that.
Maybe I can tell you what I am really trying to figure out: in the context of the discussion at the $n$Café, I started wondering if somewhere in the higher geometry that I am looking at I can identify a naturally occuring (higher) inductive type. I am not sure where and why, but I thought maybe I can say “Look, this $\infty$-local system may be thought of as the interpretation of a higher inductive type” or the like.
But the notion of an initial algebra over an endofunctor seems to be a bit alien to the constructions that I would naturally consider in higher geometry. Maybe I am wrong about that. But I started wondering if from the category-theoretic side there is maybe some more natural perspective on these beasts.
I replied to your Cafe comment with some thoughts that might be helpful.
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