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Stream: theory: category theory

Topic: "Idempotent" monads

Patrick Nicodemus (Feb 13 2023 at 22:29):

Is this a studied notion?
I do not mean an Idempotent monad in the usual sense, I mean a monad where all operations are idempotent, so that in $T(X)$ if you have some binary operation $m$ then an equation like $m(m(x, y), m(x, y))=m(x, y)$ holds.

Patrick Nicodemus (Feb 13 2023 at 22:31):

More generally for any term $t(x_1.... x_n)$ in $T(X)$, $t(\vec{x}) =t(t(\vec{x}),..., t(\vec{x}))$

Max New (Feb 13 2023 at 22:48):

It's one of the notions considered here in @Ohad Kammar and @Gordon Plotkin 's paper (https://dl.acm.org/doi/pdf/10.1145/2103656.2103698). If a commutative algebraic theory has this property then its category of algebras is a "relevance monoidal category" (https://ncatlab.org/nlab/show/relevance+monoidal+category)

Patrick Nicodemus (Feb 13 2023 at 22:57):

Magnificent. Thank you very much.

Max New (Feb 14 2023 at 01:42):

In fact I think I was talking to Ohad about this at POPL and I think he called such a theory an idempotent theory

Nathanael Arkor (Feb 14 2023 at 04:11):

Theories satisfying this condition (which look to be called relevant in Kammar–Plotkin's paper) were called hyperaffine by Johnstone in Collapsed toposes and cartesian closed varieties.

Nathanael Arkor (Feb 14 2023 at 04:11):

I think "idempotent theory" is a misleading term, because usually "P theories" correspond to "P monads" under the monad–theory correspondence for some property P, and that's not true here.

Ryan Wisnesky (Feb 14 2023 at 07:29):

When monads are used to model collection types in union presentation, idempotency is one of the four axes along which the collection can vary in the so-called "Boom Hierarchy". For example, the set monad is idempotent because set union is but the list monad is not because list append is not. The others are associativity, commutativity, and existence of a unit. Here's a paper: https://citeseerx.ist.psu.edu/document?repid=rep1&type=pdf&doi=dd3ed1c7d523781513f01d25e1be472de4279d72 and blog post https://blog.acolyer.org/2014/11/06/the-boom-hierarchy/