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Stream: theory: category theory

Topic: strict factorization systems as strict algebras

Matteo Capucci (he/him) (Jul 04 2024 at 15:18):

Koronstenski & Tholen and Grandis showed that various flavours of factorization systems (FS) can be presented as various flavours of algebras of the 2-monad $(-)^\downarrow$ on $\bf Cat$, endowed with the obvious structure.
I wrote up Grandis' argument on the nLab. He says a strict algebra of $(-)^\downarrow$ is a strict FS. When unpacking this statement, however, is clear that an algebra $(X,t)$ for the pointed endofunctor $((-)^\downarrow, \eta)$ is enough to recover a strict FS: $t$ provides factorizations and unitality ($t.\eta = 1$) guarantees these are indeed factorizations. Uniqueness is trivial from the way we built the factorization.
K&T treat a similar thing in §2, calling it a weak FS. More precisely, they call WFS a pseudoalgebra of $((-)^\downarrow, \eta)$ and then claim one can always strictify such, so they work with strict algebras from there on.
Ultimately, this is because associativity can be proven from unitality as I sketched in the nLab write up.

But I must be getting something wrong: both K&T and Grandis prove an OFS is a strictly unital (i.e. normal) pseudoalgebra for the 2-monad $(-)^\downarrow$, but then what role does the associator play? We already know that carrier + strict unitality implies a strict factorization, why isn't every normal pseudoalgebra strict then?

Matteo Capucci (he/him) (Jul 04 2024 at 15:19):

This is going to be one of those cases where I'm drowning in an inch of water but here we are