Spahn newpage étale types (Rev #1)

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

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 (Cl 1)

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

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

Corollary

˜\tilde{\Box} is closed under:

¯\overline{\Box} is closed under:

Proposition (Cl 2)

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 (Cl 1) applies.

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

Revision on December 13, 2012 at 02:40:05 by Stephan Alexander Spahn?. See the history of this page for a list of all contributions to it.