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We characterize modalities $\Box$ on $(\infty,1)$ 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 $\Box$ a modality of

(1) type (Cl1) if $Fact(\Box)$ is orthogonal.

(2a) type (Cl2a) if $Fact(\Box)$ is reflective.

(2b) type (Cl2b) if $Fact(\Box)$ is coreflective.

(3) type (Cl3) if $Fact(\Box)$ is stable.

A factorization system arising in this way from $\Box$ is necessarily orthogonal because of the universal property of the pullback.

Definition and Lemma

Let $\Box$ be an indempotent modality on $C$ . (further assumptions). Let $f:X\to Y$ be a morphism in $C$ .

(1) $\overline{f^\Box}$ in the pullback square

\array{ Y\times_{\Box Y} \Box X&\to &\Box X \\ \downarrow^\overline{f^\Box}&&\downarrow^{\Box f} \\ Y&\to&\Box Y }

is called $\Box$ -closure of $f$ . We denote the class of $\Box$ -closed morphisms in $C$ by $\overline{\Box}$ .

(2) $\overline{f^\Box}$ is $\Box$ -closed.

(3) $f$ is called to be a $\Box$ -equivalence if $\Box f$ is an equivalence. We denote the class of $\Box$ -equivalences in $C$ by $\tilde{\Box}$ .

Proposition (Cl1)

Let $C$ be an $(\infty,1)$ -category in which pullbacks are universal, (further assumptions), let $\Box$ be an indempotent modality on $C$ which commutes with pullbacks.

Then $(\tilde{\Box},\overline{\Box})$ is an orthogonal factorization system on $C$ .

Proof

By naturality of the unit of the monad and the universality of the pullback, $f:X\to Y$ factors as $f=\tilde{f^\Box};\overline{f^\Box}$ .

\array{ X&\stackrel{\tilde{f^\Box}}{\to}&Y\times_{\Box Y} \Box X&\to &\Box X \\ &\searrow^{f}&\downarrow^\overline{f^\Box}&&\downarrow^{\Box f} \\ &&Y&\to&\Box Y }

$\overline{f^\Box}$ is $\Box$ -closed by the previous Lemma. Since $\Box$ preserves by assumption this pullback and since $\Box$ is idempotent, $\Box(X\to \Box X)$ is an equivalence, and $\Box X$ is also a pullback of the $\Box$ -image of the pullback square it follows that $\Box \tilde{f^\Box}$ is an equivalence and hence $\tilde{f^\Box}$ is a $\Box$ -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.

Lemma (some closure properties)

(1) Let $(L,R)$ be an a ~~weak~~ orthogonal factorization system (also called just factorization system) in an $C$ be an $(\infty,1)$ -category. Then: ~~$R$~~ ~~is closed under domain retracts,~~ ~~$L$~~ ~~is closed under codomain retracts.~~

~~(2) Let~~ $L$ ~~$(L,R)$~~ and ~~be a orthogonal factorization system (also called just factorization system) in an~~ $R$ ~~$C$~~ contain all isomorphisms and are closed under composition, retracts, and are replete subcategories of the arrow category ~~be an~~ $C^I$ ~~$(\infty,1)$~~ of ~~-category. Then:~~ $C$ .

$L$ is moreover closed under base change and products. ~~$R$~~ ~~contain all isomorphisms and are closed under composition, retracts, and are replete subcategories of the arrow category~~ ~~$C^I$~~ of ~~$C$~~ .

$L R$ L R is moreover closed under ~~base~~ cobase change and ~~products.~~ coproducts.

~~$R$~~ ((2) As a side remark which is not needed here: Let ~~is moreover closed under cobase change and coproducts.~~ $(L,R)$ be a weak factorization system in an $C$ be an $(\infty,1)$ -category. Then: $R$ is closed under domain retracts, $L$ is closed under codomain retracts.)

Corollary

$\tilde{\Box}$ is closed under:

$\overline{\Box}$ is closed under:

Definition

An orthogonal factorization system $(E,M)$ in $C$ is called to be a reflective factorization system if $M/*\hookrightarrow C$ is a reflective sub $(\infty,1)$ -category where $M/*$ denotes the sub $(\infty,1)$ -category on those objects $X$ for which $X\to *$ is in $M$ .

Proposition (Cl2)

Let $C$ be an $(\infty,1)$ -category in which pullbacks are universal, (further assumptions), let $\Box$ be an indempotent modality on $C$ which commutes with colimits.

(1) Proposition (Cl1) applies.

(2) $(\tilde{\Box},\overline{\Box})$ is a reflective factorization system on $C$ .

(3)

Definition

A reflective factorization system is called to be a stable factorization system if its corresponding reflector preserves finite limits.

Proposition (Cl3)

(…) $(\tilde{\Box},\overline{\Box})$ is a stable factorization system on $C$ .

References

look up in

Lurie, HTT, prop. 5.2.6.8 (7), (8))

Lurie, HTT, lemma 5.2.8.19)

Revision on December 13, 2012 at 18:20:08 by Stephan Alexander Spahn?. See the history of this page for a list of all contributions to it.