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Stream: theory: category theory

Topic: Definition of streams

Madeleine Birchfield (Dec 24 2023 at 20:14):

On the nLab there is currently a number of discussions over whether the set of streams of a set $A$ is defined as the final coalgebra of the endofunctor $X \mapsto A \times X$ or the endofunctor $X \mapsto 1 + A \times X$:

https://nforum.ncatlab.org/discussion/17554/list-of-notable-initial-algebras-and-terminal-coalgebras/#Item_0

https://nforum.ncatlab.org/discussion/17572/stream/#Item_0

Nathaniel Virgo (Dec 25 2023 at 08:05):

I'd say both deserve to be called "streams", but personally I'd assume $X\mapsto A\times X$ unless you said otherwise. There's lots of work that uses that definition, e.g. Rutten's work on "stream calculus".

fosco (Dec 25 2023 at 11:14):

I'm far from being an expert, but now that I think about it, I have seen both definitions; usually I intend $A\times-$, but Jacobs' coalgebra book (iirc) uses instead $1+A\times-$...

Matteo Capucci (he/him) (Dec 27 2023 at 14:05):

In Glasgow the first are called streams and the latter colists

Oscar Cunningham (Dec 28 2023 at 14:40):

Would it be right to say that costreams are well defined, but that there aren't any?

Morgan Rogers (he/him) (Dec 29 2023 at 09:20):

@Oscar Cunningham what would be your intuitive definition of costream and why do you expect there not to be any?