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Stream: theory: category theory

Topic: Softening of a reflector by a factorization system

Arshak Aivazian (Mar 17 2025 at 19:29):

Can I just duplicate my question from MathOverflow here? https://mathoverflow.net/questions/489579/when-does-the-softening-of-a-reflector-by-a-factorization-system-determine-a-ref

Kevin Carlson (Mar 17 2025 at 20:25):

Arshak wrote on MO:

Let 𝐶 be a category equipped with an idempotent monad 𝑀 (synonym: a reflective subcategory) and an orthogonal factorization system 𝐹=(𝐿,𝑅). The 𝐹-factorization of the unit of 𝑀 defines a pointed endofunctor 𝑀𝐹:𝐶→𝐶 (returning the object that arises in the factorization).

Under what natural and most general assumptions on 𝑀, 𝐹 (and possibly 𝐶) is 𝑀𝐹an idempotent monad?

First and foremost, I am interested in the (∞,1)-version of this question, but of course any information about the (1,1) case will be useful and appreciated.

Is it sufficient to check that the arising natural transformation 𝑀𝐹→𝑀𝐹𝑀𝐹 is an isomorphism? Is there any literature discussing such constructions?

The closest related discussions I have found on MathOverflow are:

• Conditions so that a FS induces a Frobenius adjunction
• The reflection of a reflective factorization system

as well as the paper Rosický-Tholen, "Factorization, Fibration, and Torsion", mentioned in the second link. However, in these sources:

1. A very special case is discussed: 𝑀 is a reflector onto a subcategory 1 (terminal object).
2. Moreover an unrealistic (for me) condition is imposed on the class 𝐿: the full two-out-of-three property. This does not hold even in the best cases I care about (such as effective epimorphisms (surjections/quotients) in algebraic categories or toposes).

Arshak Aivazian (Mar 17 2025 at 23:03):

I suppose I was overthinking it. Instead of checking the structure of an idempotent monad, I can simply check the adjunction. So (unless I'm missing something) the answer is: always.

image.png

Arshak Aivazian (Mar 17 2025 at 23:03):

And I apologize for the confusion in my question: the condition in point 2 that I mentioned is not required for reflectivity, but for some additional property. The authors clearly note that this subcategory is always reflective (see the last line of page 6). I fixed it on MO.

Arshak Aivazian (Mar 17 2025 at 23:35):

Is it possible to see this from general considerations, without resorting to diagramm checks?