A mathematical preprint has identified a precise three-way link in a class of abstract categories. For essentially small rigid tensor-triangulated categories, the Baer property, the second tensor-triangular De Morgan law and extremal disconnectedness of the Hochster-dual Balmer spectrum are equivalent. In this setting, establishing any one of the three establishes the other two.
The study seeks an intrinsic way to recognize the second De Morgan law from the category itself, as well as a construction that produces categories with the corresponding property. Its formal objects are thick tensor ideals within essentially small tensor-triangulated categories.
A law with two versions
The paper defines the two laws through an operation called a pseudocomplement, a complement-like construction in the lattice of ideals. The first law uses the pseudocomplement of a generated ideal; the second uses the pseudocomplement of an intersection. That difference is central to the paper’s effort to distinguish the two structural conditions.
The first test is universal. Every essentially small tensor-triangulated category satisfies the first tensor-triangular De Morgan law. For the second law, the paper establishes an inclusion that always goes in one direction, leaving the full condition to be characterized more carefully.
That characterization is topological. For an essentially small tensor-triangulated category without nilpotent objects, the second law holds exactly when its Hochster-dual Balmer spectrum is extremally disconnected. In ordinary language, this means that the closure of every open set is open again. The paper uses the supports of tensor-triangular ideals as the open sets of that Hochster-dual spectrum.
Building a category where it works
The paper then turns the characterization into a construction. For an input category K, it defines the Baerification B(K) as a filtered colimit of a constructed family of categories and supplies an inclusion functor from K. A filtered colimit is an organized way to assemble a directed collection of compatible stages into one category.
For every essentially small rigid tensor-triangulated category K, the construction produces a Baer category B(K). It also supplies an annihilator-compatible functor from K and a universal factorization property: when K is sent in a compatible way into a Baer category, that map factors through B(K) in one and only one way.
The categorical construction has a topological counterpart. The Hochster-dual Balmer spectrum of B(K) is homeomorphic to the absolute of the Hochster-dual Balmer spectrum of K. In other words, the paper identifies the space associated with the constructed category through a specific topological companion of the original spectrum.
The result has clear boundaries
The paper’s clearest warning comes from Dperf(Z). That category does not satisfy the second tensor-triangular De Morgan law, is not a Baer tensor-triangulated category, and has a Hochster-dual Balmer spectrum that is not extremally disconnected. The example shows why the second law cannot be treated as automatic across all essentially small tensor-triangulated categories.
Rigidity is another explicit boundary. A non-rigid example shows that the idempotent-triangle assumption used in the paper’s definition of a Baer category need not hold outside the rigid setting. The main three-way equivalence is therefore stated for essentially small rigid categories, while the topological characterization carries its own no-nilpotent-objects condition.
What the construction preserves
The paper also records two consequences. For essentially small tensor-triangulated categories without nilpotent objects, the tensor-triangular annihilator ideals form a complete Boolean algebra, giving them all the operations required of that structure. Baerification also preserves the Nerves of Steel conjecture when the input category already satisfies it.
The supplied document is an arXiv preprint, version 1, dated 26 August 2026. Its contribution is a set of formal equivalences and a universal construction for tensor-triangulated categories: it sets out when the second law aligns with the Baer property and a topological condition, and how to build B(K) with that property from a rigid input.
Paper data and sources
Original title: De Morgan's Laws in Tensor-Triangular Geometry
Authors: Mark Lyttle
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-26
DOI: Not available
Original paper · Full text