nLab De Morgan Heyting category




A De Morgan Heyting category is a Heyting category CC (such as a topos or pretopos) in which for every object XOb(C)X \in \mathrm{Ob}(C) the pseudocomplement of the intersection of two subobjects ASub(X)A \in \mathrm{Sub}(X) and BSub(X)B \in \mathrm{Sub}(X) is the union of the pseudocomplements of AA and BB, ((AB))=(A)(B)((A \cap B) \implies \emptyset) = (A \implies \emptyset) \cup (B \implies \emptyset). Equivalently, it is a Heyting category CC in which for every object XOb(C)X \in \mathrm{Ob}(C) the union of the pseudocomplement of every subobject ASub(X)A \in \mathrm{Sub}(X) and the double pseudocomplement of AA is an improper subobject, ASub(X)A \in \mathrm{Sub}(X), (A)((A))=(X)(A \implies \emptyset) \cup ((A \implies \emptyset) \implies \emptyset) = \Im(X).

Therefore, the subobject poset Sub(X)Sub(X) of any object XOb(C)X \in \mathrm{Ob}(C) is a De Morgan Heyting algebra. De Morgan's law and weak excluded middle hold in the internal logic of CC.

In addition, every De Morgan Heyting category CC is a first order De Morgan Heyting hyperdoctrine given by the subobject poset functor Sub:𝒞 opDeMorgHeytAlg\mathrm{Sub}:\mathcal{C}^{op} \to DeMorgHeytAlg.


 See also

Last revised on November 16, 2022 at 20:58:16. See the history of this page for a list of all contributions to it.