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Stream: theory: category theory

Topic: factorization system from arrows to objects

Matteo Capucci (he/him) (Feb 02 2023 at 12:17):

Suppose I have a factorization system on $\cal E^\downarrow$ (the arrow category on $\cal E$). Is it always valid to transport it to $\cal E$ along the adjunction $\rm dom \vdash id$?

Matteo Capucci (he/him) (Feb 02 2023 at 12:19):

The references I consulted are a bit hard to parse bc I'm not very comfortable with FS literature yet, but for example here seems to suggest you can do this as long as $\rm id$ is left exact, which it is

Ivan Di Liberti (Feb 02 2023 at 15:12):

A factorisation system on the arrow category of C is the same of a torsion theory on C, these correspond to special families of factorisation systems on C. See "From torsion theories to closure operators and factorization systems" by Grandis and Janelidze.

fosco (Feb 02 2023 at 19:08):

Another place can be "Factorization, fibration and torsion", although I would bet on the literature cited rather than the paper itself, most of these questions are better answered in terms of factorizations induced by a fibration!

Matteo Capucci (he/him) (Feb 03 2023 at 11:35):

Grazie mille!!