Cohomology and homotopy
In higher category theory
In logic, double negation is the operation that takes to , where is negation. In other words, double negation is the composite of negation with itself. This is a closure operator/modality.
In classical logic, the double negation of any truth value or proposition is itself. More abstractly, double negation is the identity function on any boolean algebra.
In intuitionistic logic, double negation is weaker than the identity. That is, we have but not conversely. In paraconsistent logic, it is the other way around. More abstractly, this holds in any Heyting algebra (intuitionistic) or its dual (paraconsistent).
In linear logic, double negation is the identity again, although linear logic also has notions of intuitionistic negation and paraconsistent negation which act as above.
In locale theory
Even in classical mathematics, a frame is a Heyting algebra but not a boolean algebra. Accordingly, double negation is usually not the identity on a frame. However, the operation of double negation is a nucleus on any frame.
Thus, every locale has a sublocale given by that nucleus, called the double negation sublocale and denoted . This sublocale is dense, and in fact it is the smallest dense sublocale of , the intersection of all dense sublocales.
Classically, we have if and only if is the discrete locale on some set of points. In constructive mathematics, must also have decidable equality.
In topos theory
The notion of double negation sublocale may be categorified from locales to toposes.
If is a topos with subobject classifier , there is a negation operator , defined by virtue of the fact that is an internal Heyting algebra.
The double negation morphism
constitutes a Lawvere-Tierney topology on .
This is called the double negation topology.
The topology axioms can be formulated in purely equational form, i.e., as equations between operations of the form . By the Yoneda lemma, it suffices to verify the corresponding equations between transformations , which boils the problem down to checking the equations for ordinary Heyting algebras in . For ordinary Heyting algebras, proofs may be found here.
This appears as MacLaneMoerdijk, theorem VI 3. The topos of double-negation sheaves is in fact the maximal Boolean subtopos contained in .
For every presheaf topos the double negation topology coincides with the dense topology?.
This appears as MacLaneMoerdijk, corollary VI 5.
Let be a poset. Then the double negation sheaf topos
satisfies the axiom of choice.
This appears as MacLaneMoerdijk, corollary VI 9.
In a presheaf topos (on a small category) the notions of -topology and that of dense topology? coincide.
In higher topos theory
Classically the double negation modality is equivalent to the n-truncation modality for (the bracket type). In general, it's still true that double negation takes any type (object in the higher topos) to a -type, but the bracket type only entails the double negation :
there is a canonical function
and this is a 1-epimorphism precisely if the law of excluded middle holds.
Section VI of