nLab De Morganization

Idea

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

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

Definition

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

Remark

Here $\Omega_{\neg\neg}$ denotes the subobject classifier for $Sh_{\neg\neg}(\mathcal{E})$ with $\neg\neg$ the double negation topology on $\mathcal{E}$. The De Morgan topology $m$ 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 $Sh_{\neg\neg}(\mathcal{E})$ .

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 $m$ is the smallest dense topology $j$ on $\mathcal{E}$ , i.e. $j\leq \neg\neg$ , such that $Sh_j(\mathcal{E})$ is a De Morgan topos.

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

Proposition

The De Morgan topology $m$ is the smallest topology $j$ on $\mathcal{E}$ such that all monomorphisms of the form $\neg A\vee\neg\neg A\rightarrowtail E$ for subobjects $A\rightarrowtail E$ in $\mathcal{E}$ are $j$-dense.

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

Proposition

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

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

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

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

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

Geometric morphisms preserving De Morganizations

Notice that in analogy to $Sh_{\neg\neg}(\mathcal{E})$ 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 $f:\mathcal{F}\to\mathcal{E}$ that restrict to geometric morphisms $f|_m:Sh_m(\mathcal{F})\to Sh_m(\mathcal{E})$ .

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

By the above proposition, $Sh_m(Sh_{j}(\mathcal{E}))\hookrightarrow Sh_{m}(\mathcal{E})$ for $j$ dense. Accordingly, dense inclusions $Sh_{j}(\mathcal{E})\hookrightarrow\mathcal{E}$ 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 $Sh_m(\mathcal{E})$ and $m$-sheaves of $\mathcal{F}$ necessarily have to land there.

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

Related entries

• De Morgan topos

• De Morgan Heyting algebra

• double negation

• dense subtopos

• dense topology

• skeletal geometric morphism

References

• Igor Arrieta, The DeMorganization of a Locale , arXiv:2406.12486 (2024). (abstract)

• Olivia Caramello, De Morgan classifying toposes , Adv. in Math. 222 (2009) pp.2117-2144. (arXiv:0808.1519)

• Olivia Caramello, Universal models and definability , Math. Proc. Cam. Phil. Soc. (2012) pp.279-302. (arXiv:0906.3061)

• Olivia Caramello, Topologies for intermediate logics , arXiv:1205.2547 (2012). (arXiv:1205.2547)

• Olivia Caramello, Peter Johnstone, De Morgan’s law and the Theory of Fields , Adv. in Math. 222 (2009) pp.2145-2152. (arXiv:0808.1572)

• Peter Johnstone, Topos Theory , Academic Press New York 1977. (Dover reprint New York 2014)

• Peter Johnstone, Sketches of an Elephant vol. I, Oxford UP 2002.