Spahn newpage étale types

We characterize modalities \Box on (,1)(\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()Fact(\Box) is orthogonal.

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

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

(3) type (Cl3) if Fact()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 CC. (further assumptions). Let f:XYf:X\to Y be a morphism in CC.

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

Y× YX X f ¯ f Y Y\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 ff. We denote the class of \Box-closed morphisms in CC by ¯\overline{\Box}.

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

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

Proposition (Cl1)

Let CC be an (,1)(\infty,1)-category, (further assumptions), let \Box be an indempotent modality on CC which commutes with pullbacks.

Then (˜,¯)(\tilde{\Box},\overline{\Box}) is an orthogonal factorization system on CC.

Proof

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

X f ˜ Y× YX X f f ¯ f Y Y\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 }

f ¯\overline{f^\Box} is \Box-closed by the previous Lemma. Since \Box preserves by assumption this pullback and since \Box is idempotent, (XX)\Box(X\to \Box X) is an equivalence, and X\Box X is also a pullback of the \Box-image of the pullback square it follows that f ˜\Box \tilde{f^\Box} is an equivalence and hence f ˜\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)(L,R) be a orthogonal factorization system (also called just factorization system) in an CC be an (,1)(\infty,1)-category. Then:

LL and RR contain all isomorphisms and are closed under composition, retracts, and are replete subcategories of the arrow category C IC^I of CC.

LL is moreover closed under base change and products.

RR is moreover closed under cobase change and coproducts.

((2) As a side remark which is not needed here: Let (L,R)(L,R) be a weak factorization system in an CC be an (,1)(\infty,1)-category. Then: RR is closed under domain retracts, LL is closed under codomain retracts.)

Corollary

˜\tilde{\Box} is closed under:

¯\overline{\Box} is closed under:

Definition

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

Proposition (Cl2)

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

(1) Proposition (Cl1) applies.

(2) (˜,¯)(\tilde{\Box},\overline{\Box}) is a reflective factorization system on CC.

(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 CC.

References

look up in

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

Lurie, HTT, lemma 5.2.8.19)

Last revised on December 13, 2012 at 23:08:18. See the history of this page for a list of all contributions to it.