We characterize modalities on categories by
(1) the kind of factorization system they induce.
(2) the kind of subcategory they induce.
(3) the kind of cohesion they induce.
We call a modality of
(1) type (Cl1) if is orthogonal.
(2a) type (Cl2a) if is reflective.
(2b) type (Cl2b) if is coreflective.
(3) type (Cl3) if is stable.
A factorization system arising in this way from is necessarily orthogonal because of the universal property of the pullback.
Let be an indempotent modality on . (further assumptions). Let be a morphism in .
(1) in the pullback square
is called -closure of . We denote the class of -closed morphisms in by .
(2) is -closed.
(3) is called to be a -equivalence if is an equivalence. We denote the class of -equivalences in by .
Let be an -category in which pullbacks are universal, (further assumptions), let be an indempotent modality on which commutes with pullbacks.
Then is an orthogonal factorization system on .
By naturality of the unit of the monad and the universality of the pullback, factors as .
is -closed by the previous Lemma. Since preserves by assumption this pullback and since is idempotent, is an equivalence, and is also a pullback of the -image of the pullback square it follows that is an equivalence and hence is a -equivalence.
That the factorization system is orthogonal follows from the definition, naturality of the modality unit, the pullback pasting lemma, and the universal property of the pullback giving finally the unicity of the lift.
(1) Let be an weak factorization system in an be an -category. Then: is closed under domain retracts, is closed under codomain retracts.
(2) Let be a orthogonal factorization system (also called just factorization system) in an be an -category. Then:
and contain all isomorphisms and are closed under composition, retracts, and are replete subcategories of the arrow category of .
is moreover closed under base change and products.
is moreover closed under cobase change and coproducts.
is closed under:
is closed under:
An orthogonal factorization system in is called to be a reflective factorization system if is a reflective sub -category where denotes the sub -category on those objects for which is in .
Let be an -category in which pullbacks are universal, (further assumptions), let be an indempotent modality on which commutes with colimits.
(1) Proposition (Cl1) applies.
(2) is a reflective factorization system on .
(3)
A reflective factorization system is called to be a stable factorization system if its corresponding reflector preserves finite limits.
(…) is a stable factorization system on .
look up in
Lurie, HTT, prop. 5.2.6.8 (7), (8))
Lurie, HTT, lemma 5.2.8.19)