The De Morganization of a topos is a universal way to turn into a de Morgan topos with the use of a certain Lawvere-Tierney topology , called the De Morgan topology on .
This can be viewed as an analogue to the Booleanization of with the help of the double negation topology .
Let be a topos. The De Morgan topology on is defined as the smallest Lawvere-Tierney topology such that the canonical monomorphism is -dense. The De Morganization of is the associated topos of -sheaves.
Here denotes the subobject classifier for with the double negation topology on . The De Morgan topology 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 .
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).
The De Morgan topology is the smallest dense topology on , i.e. , such that is a De Morgan topos.
This appears as theorem 1 in Caramello (2009). In other words, is the largest dense De Morgan subtopos of
The De Morgan topology is the smallest topology on such that all monomorphisms of the form for subobjects in are -dense.
This appears as proposition 6.2 in Caramello (2012a).
Let be a topos and be the De Morgan topology on it.
iff is a De Morgan topos.
For any dense topology on , .
Caramello (2009), prop.1.5. In fact, in the second statement it suffices to demand that is a weakly open topology i.e. the associated sheaf functor preserves the pseudo-complementation operator in the lattices of subobjects (cf. Caramello (2012, prop.4.5)).
Notice that and, accordingly, for a dense or, more generally a weakly open subtopos De Morganization simply amounts to intersection with .
Notice that in analogy to 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 that restrict to geometric morphisms .
Due to a result in Johnstone (2002, p.194), this is equivalent to the preservation of -dense monomorphisms by .
By the above proposition, for dense. Accordingly, dense inclusions are m-skeletal !
The characterization of Boolean toposes by skeletal morphisms carries over to m-skeletal morphisms and De Morgan toposes:
A topos is De Morgan iff every geometric morphism is m-skeletal.
Proof: Assume is De Morgan, then it coincides with and -sheaves of necessarily have to land there.
Conversely, assume all are m-skeletal. Then the surjective morphism from the Gleason cover is m-skeletal. But is De Morgan and, therefore, so is .
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.
Last revised on June 19, 2024 at 09:36:54. See the history of this page for a list of all contributions to it.