A De Morgan topos $\mathcal{E}$ is a middle thing between a Boolean topos with a classical internal logic and a general topos with an intuitionistic internal logic, in that the lattices of subobjects in $\mathcal{E}$ obey a weak form of the law of excluded middle which defines a De Morgan Heyting algebra.
The de Morgan property has many equivalent formulations and many facets making the class of de Morgan toposes an important one in topos theory. From the perspective on toposes as generalized spaces they correspond to extremally disconnected spaces in topology.
A topos $\mathcal{E}$ is called a De Morgan topos if its subobject classifier $\Omega$ is an internal De Morgan Heyting algebra.
Every Boolean topos is a De Morgan topos.
A simple example of a topos that is De Morgan but not Boolean is the Sierpinski topos $Set^{\to}$ the arrow category of $Set$: the diagram category $\to$ is easily seen to satify the Ore condition (cf. below).
The topos $Sh(X)$ of sheaves on a topological space $X$ is De Morgan precisely iff $X$ is extremally disconnected, i.e. the closure of every open subset is open. (Since $Set^{\to}$ is equivalently the topos of sheaves on Sierpinski space it instantiates this case as well.)
Injective Grothendieck toposes are De Morgan (cf. Johnstone 2002, p.739).
The importance of the concept of a De Morgan topos stems partly from the fact that for many statements whose classical (Boolean) validity is lost in intuitionistic logic the De Morgan property delineates precisely the range of toposes where the statement is valid. An illustration of this is e.g. the implication that maximal ideals are prime in commutative rings.
Though such statements are in fact equivalent to the De Morgan property they are better viewed as higher-order properties of De Morgan toposes.
In contrast, there exists also a long list, mostly due to Johnstone (1979), of statements equivalent to the De Morgan property, that are logically of the same complexity and are often more convenient to work with than the original definition. Some of the most important of these appear in the following:
Let $\mathcal{E}$ be a topos. The following are equivalent:
$\mathcal{E}$ is a De Morgan topos.
For all formulae $\varphi,\psi$ : $\mathcal{E}\models \neg (\varphi\wedge\psi)\Leftrightarrow(\neg\varphi\vee\neg\psi)$ .
For every formula $\varphi$ : $\mathcal{E}\models \neg\varphi\vee\neg\neg\varphi$.
For every subobject $X\rightarrowtail Y$: $\neg X\vee\neg\neg X=Y$.
$Sub(X)$ is a De Morgan Heyting algebra for every $X\in\mathcal{E}$.
The canonical monomorphism $(\top,\bot):1\coprod 1\rightarrowtail\Omega_{\neg\neg}$ to the subobject classifier of $Sh_{\neg\neg}(\mathcal{E})$ is an isomorphism.
$\Omega_{\neg\neg}$ is decidable.
$1\coprod 1$ is a retract of $\Omega_{\neg\neg}$.
$1\coprod 1$ is injective.
Every $\neg\neg$-sheaf is decidable.
$\bot:1\to\Omega$ has a complement.
The first few equivalences are based mainly on the validity of the “law of excluded middle” for elements of the form $\neg a$ in any De Morgan Heyting algebra. For the rest see e.g. Johnstone (2002, pp.999-1000).
The following shows that ‘being De Morgan’ is a local property i.e. stable under slicing. This is immediate from the fact that the functor $\mathcal{E}\to\mathcal{E}/A$ sending $X$ to the projection $p_A: X\times A\to A$ is logical hence preserves the various equivalent formulations of De Morgan’s law.
Let $A$ be an object of the De Morgan topos $\mathcal{E}$. Then the slice topos $\mathcal{E}/A$ is De Morgan as well.
Let $\mathcal{C}$ be a small category. The presheaf topos $Set^{\mathcal{C}^{op}}$ is De Morgan precisely if $\mathcal{C}$ satifies the Ore condition.
This result due to Peter Johnstone appears e.g. in Johnstone (1979). Compare also the generalizations in Kock-Reyes (1994) and Caramello (2012).
Since the dense topology coincides with the atomic topology on $\mathcal{C}$ satisfying the Ore condition the preceding proposition implies that the double negation subtopos of a De Morgan presheaf topos $Set^{\mathcal{C}^{op}}$ is always atomic.
It was discovered by Olivia Caramello that every topos $\mathcal{E}$ can be associated with a De Morgan topos $Sh_m(\mathcal{E})$, its De Morganization, in a universal way:
Given a topos $\mathcal{E}$. There exists a unique largest dense subtopos $Sh_m(\mathcal{E})$ of $\mathcal{E}$ that is a De Morgan topos.
For a proof see Caramello (2009). The $m$ that occurs here, refers to the De Morgan topology , cf. De Morganization.
As $Sh_{\neg\neg}(\mathcal{E})$, the sheaf subtopos for the double negation topology $\neg\neg$ , is the smallest dense De Morgan subtopos, we see that all dense De Morgan subtoposes lie in the interval between $Sh_{\neg\neg}(\mathcal{E})$ and $Sh_{m}(\mathcal{E})$.
In analogy with skeletal geometric morphisms that preserve $\neg\neg$-sheaves, geometric morphisms $\mathcal{F}\to\mathcal{E}$ that map $m$-sheaves in $\mathcal{F}$ to $m$-sheaves in $\mathcal{E}$ are called $m$-skeletal. They occur in the following characterization of De Morgan toposes (cf. De Morganization):
A topos $\mathcal{E}$ is De Morgan iff every geometric morphism $\mathcal{F}\to\mathcal{E}$ is $m$-skeletal. $\qed$
Another way to associate a De Morgan topos to an arbitrary topos $\mathcal{E}$ was proposed by Peter Johnstone in the late 70s (Johnstone 1979b, 1980). The so called Gleason cover $\gamma\mathcal{E}$ of $\mathcal{E}$ generalizes a construction in topology that covers an arbitrary space by an extremally disconnected topological space.
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Given that the above conditions concern $1\coprod 1$ and the contractability of $\Omega_{\neg\neg}$ it comes as no surprise that the De Morgan property interacts interestingly with Lawvere's axiomatic approach to cohesion and, in particular, with the part of it that concerns the connectedness of the subobject classifier in a cohesive topos of spaces. A first indication of this is the following:
Let $Set^{\mathcal{C}^{op}}$ be a presheaf topos. Then its subobject classifier is connected iff $1\in Set^{\mathcal{C}^{op}}$ is a connected object and $Set^{\mathcal{C}^{op}}$ is not de Morgan iff there exists a connected object $X\in Set^{\mathcal{C}^{op}}$ such that $1\coprod 1\rightarrowtail X$ .
This result appears in La Palme-Reyes-Reyes-Zolfaghari (2004, p.220) where it is attributed to Lawvere.
The following result due to Mielke (1984) shows that the De Morgan property coincides with the (local) absence of a non-trivial interval objects in a topos.
Here by an interval object we mean an internal linearly ordered object $I$ with disjoint least and greatest elements $m:1\to I$ and $M:1\to I$, respectively, i.e. roughly the sort of thing that is classified by sSet.
An interval object is trivial if $I\simeq I_1\coprod I_2$ and $m,M$ factor through $I_1,I_2$, respectively. A topos $\mathcal{E}$ is said to be homotopically trivial if every interval object in $\mathcal{E}$ is trivial, $\mathcal{E}$ is said to locally homotopically trivial if the slice topos $\mathcal{E}/X$ is homotopically trivial for all $X\in\mathcal{E}$.
A topos $\mathcal{E}$ is locally homotopically trivial iff $\mathcal{E}$ is a De Morgan topos. If subobjects of $1$ generate, then $\mathcal{E}$ is homotopically trivial iff $\mathcal{E}$ is a De Morgan topos.
For situations where subobjects of $1$ generate see e.g. Johnstone (1977, sec.5.3 pp.145ff). For a Grothendieck topos $\mathcal{E}$, this happens precisely when $\mathcal{E}$ is equivalent to $Sh(L)$ for some locale $L$. From e.g. Borceux (1994, p.443) it follows then that $Sh(L)$ is homotopically trivial precisely when $L$ is a De Morgan locale.
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Let $\mathcal{T}$ be a coherent theory. Then $\mathcal{T}$ admits a model companion? and $Mod(\mathcal{T})$ has the amalgamation property iff for each $B\in\mathcal{T}$ , the lattice of subobjects of $B$ admits a negation satisfying $\neg A\vee\neg\neg A = B$.
This result appears without proof in Harun (1976, p.73) where it is attributed to a preprint of André Joyal and Gonzalo E. Reyes.
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