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Stream: learning: questions

Topic: Terminal coalgebras and topology

Madeleine Birchfield (Jun 17 2025 at 20:08):

A lot of structures from topology can be defined as some kind of terminal coalgebra. For example, the generic convergent sequence (i.e. "extended natural numbers") is the terminal coalgebra of the functor $X \mapsto X + 1$, any sequence space $A^\mathbb{N}$ is the terminal coalgebra of the functor $X \mapsto X \times A$, and the unit interval was defined by Freyd as the terminal coalgebra of the cospan composition functor $X \mapsto X \vee X$ on sets with two elements $0$ and $1$ which are separated in that $x \neq 0$ or $x \neq 1$ for all elements $x$.

I wonder if there are any general characterisation of which terminal coalgebras are topological in nature, and which topological structures are coalgebraic in nature.

fosco (Jun 17 2025 at 20:21):

Very much related to this is the well known fact that some inverse limits / pro-objects acquire a topology. A paper that I find enlightening and one of the best pieces of "applied category theory", whatever it means, is Leinster's "A general theory of self-similarity", where it is shown that [some] Julia sets are terminal coalgebras for endofunctors of Top or the analogue functor with sets / discrete spaces --and in this second case the topology is induced in the aforementioned way, iirc.

fosco (Jun 17 2025 at 20:22):

this is a piece of category theory that is beautiful, elegant and natural, and I stumbled upon Leinster's paper because I was scratching the surface of a similar problem, rediscovering its motivation