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Stream: learning: questions

Topic: A converse to Lambek theorem

fosco (Feb 02 2021 at 16:41):

A famous theorem by Lambek asserts that if $G$ is an endofunctor of a category $\mathcal C$, and if $\xi : X\to GX$ is its terminal coalgebra, then $\xi$ is an isomorphism.

Until today, I stupidly believed that this was an if and only if! "By the universal property of $\xi$, any other coalgebra $(A,\alpha)$ where $\alpha$ is an isomorphism, is the terminal coalgebra." However, it takes just the time to draw the right diagram to understand that this is false.

So, I have two questions:

• What are the endofunctors $G$ for which it is true that if a coalgebra $(A,\alpha)$ has $\alpha$ invertible, then $(A,\alpha)$ is terminal? I expect there are very few.
• Given G, is there a way to "count" how many $G$-coalgebras $(A,\alpha)$ have $\alpha$ invertible? I expect there are probably "as much as the object $GX$ is big", where $X$ is the carrier of the terminal coalgebra.

Feel free to assume whatever you want on G and on its domain!

Tom Hirschowitz (Feb 02 2021 at 17:27):

Not quite answering your question, but are you aware of work by Stefan Milius and colleagues about characterising various kinds of coalgebras with invertible $\alpha$?

Nathanael Arkor (Feb 02 2021 at 17:34):

However, it takes just the time to draw the right diagram to understand that this is false.

Even more simply, this applies to both terminal coalgebras and initial algebras, so, unless they coincide, this is immediately seen to be false.

Jules Hedges (Feb 02 2021 at 17:34):

You beat me to it by 30 seconds..... I thought I was going crazy thinking that Lambek's theorem also applies to initial algebras...

fosco (Feb 02 2021 at 17:35):

:grinning: I left vague the concept of "the right diagram" on purpose

fosco (Feb 02 2021 at 17:35):

Tom Hirschowitz said:

Not quite answering your question, but are you aware of work by Stefan Milius and colleagues about characterising various kinds of coalgebras with invertible $\alpha$?

No! My question is also meant to get acquainted with more literature on the subject

Tom Hirschowitz (Feb 02 2021 at 21:46):

IIRC, one of their contributions was to characterise a kind of "regular" fixed point. In lots of examples, the initial algebra intuitively consists of finite trees, while the final coalgebra consists of all, potentially wildly infinite trees. And their regular fixed point would catch something like regular trees, in the sense of finite trees with "loops".

Todd Trimble (Feb 03 2021 at 01:47):

The only thing I remember which seems remotely relevant are "algebraically compact functors" where the initial algebra is isomorphic to the terminal algebra. There's some stuff by Barr here.