nLab
De Morganization

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Context

Category theory

Topos Theory

topos theory

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Internal Logic

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Extra stuff, structure, properties

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In higher category theory

Theorems

Contents

Idea

The De Morganization of a topos \mathcal{E} is a universal way to turn \mathcal{E} into a de Morgan topos Sh m()Sh_m(\mathcal{E}) with the use of a certain Lawvere-Tierney topology mm, called the De Morgan topology on \mathcal{E}.

This can be viewed as an analogue to the Booleanization Sh ¬¬()Sh_{\neg\neg}(\mathcal{E}) of \mathcal{E} with the help of the double negation topology ¬¬\neg\neg .

Definition

Let \mathcal{E} be a topos. The De Morgan topology mm on \mathcal{E} is defined as the smallest Lawvere-Tierney topology jj such that the canonical monomorphism (,):11Ω ¬¬(\top,\bot): 1\coprod 1\rightarrowtail \Omega_{\neg\neg} is jj-dense. The De Morganization of \mathcal{E} is the associated topos Sh m()Sh_m(\mathcal{E}) of mm-sheaves.

Remark

Here Ω ¬¬\Omega_{\neg\neg} denotes the subobject classifier for Sh ¬¬()Sh_{\neg\neg}(\mathcal{E}) with ¬¬\neg\neg the double negation topology on \mathcal{E}. The De Morgan topology mm is well-defined due to Joyal’s lemma (cf. Johnstone 1977, p.99; or Johnstone 2002, p.215). Compare its definition to this proposition about Sh ¬¬()Sh_{\neg\neg}(\mathcal{E}) .

Example

Proposition

The De Morganization of the classifying topos for the theory of fields is the classifying topos for the geometric theory of fields of finite characteristic, in which every element is algebraic over the prime field.

This is proposition 2.3 in Caramello-Johnstone (2009).

Properties

Proposition

The De Morgan topology mm is the smallest dense topology jj on \mathcal{E} , i.e. j¬¬j\leq \neg\neg , such that Sh j()Sh_j(\mathcal{E}) is a De Morgan topos.

This appears as theorem 1 in Caramello (2009). In other words, Sh m()Sh_{m}(\mathcal{E}) is the largest dense De Morgan subtopos of \mathcal{E}

Proposition

The De Morgan topology mm is the smallest topology jj on \mathcal{E} such that all monomorphisms of the form ¬A¬¬AE\neg A\vee\neg\neg A\rightarrowtail E for subobjects AEA\rightarrowtail E in \mathcal{E} are jj-dense.

This appears as proposition 6.2 in Caramello (2012a).

Proposition

Let \mathcal{E} be a topos and mm be the De Morgan topology on it.

  • Sh m()=Sh_m(\mathcal{E})=\mathcal{E} iff \mathcal{E} is a De Morgan topos.

  • For any dense topology jj on \mathcal{E} , Sh m(Sh j())=Sh mj()Sh_m(Sh_j(\mathcal{E}))=Sh_{m\vee j}(\mathcal{E}).

Caramello (2009), prop.1.5. In fact, in the second statement it suffices to demand that jj is a weakly open topology i.e. the associated sheaf functor a j:Sh j()a_j:\mathcal{E}\to Sh_j(\mathcal{E}) preserves the pseudo-complementation operator in the lattices of subobjects (cf. Caramello (2012, prop.4.5)).

Notice that Sh mj()=Sh m()Sh j()Sh_{m\vee j}(\mathcal{E})=Sh_{m}(\mathcal{E})\cap Sh_{ j}(\mathcal{E}) and, accordingly, for a dense or, more generally a weakly open subtopos De Morganization simply amounts to intersection with Sh m()Sh_m(\mathcal{E}).

Geometric morphisms preserving De Morganizations

Notice that in analogy to Sh ¬¬()Sh_{\neg\neg}(\mathcal{E}) and the class of skeletal geometric morphism, the universality of the De Morganization affords to define a class of m-skeletal geometric morphisms as those geometric morphisms f:f:\mathcal{F}\to\mathcal{E} that restrict to geometric morphisms f| m:Sh m()Sh m()f|_m:Sh_m(\mathcal{F})\to Sh_m(\mathcal{E}) .

Due to a result in Johnstone (2002, p.194), this is equivalent to the preservation of mm-dense monomorphisms by f *f^\ast.

By the above proposition, Sh m(Sh j())Sh m()Sh_m(Sh_{j}(\mathcal{E}))\hookrightarrow Sh_{m}(\mathcal{E}) for jj dense. Accordingly, dense inclusions Sh j()Sh_{j}(\mathcal{E})\hookrightarrow\mathcal{E} are m-skeletal !

The characterization of Boolean toposes by skeletal morphisms carries over to m-skeletal morphisms and De Morgan toposes:

Proposition

A topos \mathcal{E} is De Morgan iff every geometric morphism \mathcal{F}\to\mathcal{E} is m-skeletal.

Proof: Assume \mathcal{E} is De Morgan, then it coincides with Sh m()Sh_m(\mathcal{E}) and mm-sheaves of \mathcal{F} necessarily have to land there.

Conversely, assume all \mathcal{F}\to\mathcal{E} are m-skeletal. Then the surjective morphism f:γf:\gamma\mathcal{E}\to\mathcal{E} from the Gleason cover γ\gamma\mathcal{E} is m-skeletal. But γ\gamma\mathcal{E} is De Morgan and, therefore, so is im(f)=im(f)=\mathcal{E}. \qed

References

Last revised on December 16, 2017 at 06:08:17. See the history of this page for a list of all contributions to it.