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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

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

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 $\tilde{\Box}$ .

We characterize modalities $\Box$ on $(\infty,1)$ categories by

Proposition (Cl 1)

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 $C$ .

(1) the kind of factorization system they induce.

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.

(2) the kind of subcategory they induce.

Lemma (some closure properties)

(1) Let $(L,R)$ be a factorization system in an $C$ be an $(\infty,1)$ -category. Then:

(2) Let $(L,R)$ be an orthogonal factorization system in an $C$ be an $(\infty,1)$ -category. Then:

(3) the kind of cohesion they induce.

Corollary

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

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

We call $\Box$ a modality of

Proposition (Cl 2)

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 (Cl 1) applies.

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

(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 a factorization system in an $C$ be an $(\infty,1)$ -category. Then:

(2) Let $(L,R)$ be an orthogonal factorization system in an $C$ be an $(\infty,1)$ -category. Then:

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 $C$ .

Revision on December 13, 2012 at 03:41:29 by Stephan Alexander Spahn?. See the history of this page for a list of all contributions to it.