Subobject Lattices Research Articles

Semisimplicity and Indecomposable Objects in Interpolating Partition Categories

Abstract We study Karoubian tensor categories, which interpolate representation categories of families of the so-called easy quantum groups in the same sense in which Deligne’s interpolation categories $\ensuremath {\mathop {\textrm {\underline {Rep}}}}(S_t)$ interpolate the representation categories of the symmetric groups. As such categories can be described using a graphical calculus of partitions, we call them interpolating partition categories. They include $\ensuremath {\mathop {\textrm {\underline {Rep}}}}(S_t)$ as a special case and can generally be viewed as subcategories of the latter. Focusing on semisimplicity and descriptions of the indecomposable objects, we prove uniform generalisations of results known for special cases, including $\ensuremath {\mathop {\textrm {\underline {Rep}}}}(S_t)$ or Temperley–Lieb categories. In particular, we identify those values of the interpolation parameter, which correspond to semisimple and non-semisimple categories, respectively, for all the so-called group-theoretical easy quantum groups. A crucial ingredient is an abstract analysis of certain subobject lattices developed by Knop, which we adapt to categories of partitions. We go on to prove a parametrisation of the indecomposable objects in all interpolating partition categories for non-zero interpolation parameters via a system of finite groups, which we associate to any partition category, and which we also use to describe the associated graded rings of the Grothendieck rings of these interpolation categories.

Cartesian closedness in categories with an idempotent closure operator and closed morphisms

Given a subobject-structured category $$\mathcal X$$ , we construct a new category whose objects are the pairs (X, c) where X is an $$\mathcal X$$ -object and c is an idempotent, monotonic and extensive endomap of the subobject lattice of X, and whose morphisms between objects are the closed maps between the corresponding subobject lattices. We give a sufficient condition on $$\mathcal X$$ for the new category to be cartesian closed.

Monomorphisms in categories of firm acts

Abstract We prove that in the category of firm acts over a firm semigroup monomorphisms co-incide with regular monomorphisms and we give an example of a non-injective monomorphism in this category. We also study conditions under which monomorphisms are injective and we prove that the lattice of subobjects of a firm act over a firm semigroup is isomorphic to the lattice unitary subacts of that act.

Closure, interior and neighbourhood in a category

The natural correspondences in topology between closure, interior and neighbourhood no longer hold in an abstract categorical setting where subobject lattices are not necessarily Boolean algebras. We analyse three canonical correspondences between closure, interior and neighbourhood operators in a category endowed with a subobject structure. While these correspondences coincide in general topology, the analysis highlights subtle differences which distinguish different approaches taken in the literature.

An induction principle for consequence in arithmetic universes

Suppose in an arithmetic universe we have two predicates ϕ and ψ for natural numbers, satisfying a base case ϕ(0)→ψ(0) and an induction step that, for generic n, the hypothesis ϕ(n)→ψ(n) allows one to deduce ϕ(n+1)→ψ(n+1). Then it is already true in that arithmetic universe that (∀n)(ϕ(n)→ψ(n)). This is substantially harder than in a topos, where cartesian closedness allows one to form an exponential ϕ(n)→ψ(n).The principle is applied to the question of locatedness of Dedekind sections.The development analyses in some detail a notion of “subspace” of an arithmetic universe, including open or closed subspaces and a Boolean algebra generated by them. There is a lattice of subspaces generated by the open and the closed, and it is isomorphic to the free Boolean algebra over the distributive lattice of subobjects of 1 in the arithmetic universe.

Complementarity in Categorical Quantum Mechanics

We relate notions of complementarity in three layers of quantum mechanics: (i) von Neumann algebras, (ii) Hilbert spaces, and (iii) orthomodular lattices. Taking a more general categorical perspective of which the above are instances, we consider dagger monoidal kernel categories for (ii), so that (i) become (sub)endohomsets and (iii) become subobject lattices. By developing a `point-free' definition of copyability we link (i) commutative von Neumann subalgebras, (ii) classical structures, and (iii) Boolean subalgebras.

Quantum Logic, Dagger Kernel Categories and Inverse Baer*-Categories

This paper investigates dagger kernel categories which are considered first by Crown (J Nat Sci Math 15:11–25, 1975) and used by Heunen and Jacobs (Order 27:177–212, 2010) in their study of quantum logic from the perspective of categorical logic. The inverse Baer*-categories with splitting projections as special dagger kernel categories have a central place in our investigations. The inverse Baer*-categories with splitting and closed projections are Boolean and therefore the subobject lattices of such categories are representing classical logics. Examples are presented at every stage of our investigations.

A lattice-theoretical perspective on adhesive categories

It is a known fact that the subobjects of an object in an adhesive category form a distributive lattice. Building on this observation, in the paper we show how the representation theorem for finite distributive lattices applies to subobject lattices. In particular, we introduce a notion of irreducible object in an adhesive category, and we prove that any finite object of an adhesive category can be obtained as the colimit of its irreducible subobjects. Furthermore we show that every arrow between finite objects in an adhesive category can be interpreted as a lattice homomorphism between subobject lattices and, conversely, we characterize those homomorphisms between subobject lattices which can be seen as arrows.

Neighborhoods with respect to a categorical closure operator

We introduce and study a concept of neighborhoods with respect to a categorical closure operator. The concept, which is based on using pseudocomplements in subobject lattices, naturally generalizes the classical neighborhoods in topological spaces and we show that it behaves accordingly. We investigate also separation and compactness defined in a natural way by the help of the neighboorhoods introduced.

Subobject Transformation Systems

Subobject transformation systems STS are proposed as a novel formal framework for the analysis of derivations of transformation systems based on the algebraic, double-pushout (DPO) approach. They can be considered as a simplified variant of DPO rewriting, acting in the distributive lattice of subobjects of a given object of an adhesive category. This setting allows a direct analysis of all possible notions of dependency between any two productions without requiring an explicit match. In particular, several equivalent characterizations of independence of productions are proposed, as well as a local Church–Rosser theorem in the setting of STS. Finally, we show how any derivation tree in an ordinary DPO grammar leads to an STS via a suitable construction and show that relational reasoning in the resulting STS is sound and complete with respect to the independence in the original derivation tree.

Saturated models of intuitionistic theories

We use the language of categorical logic to construct generic saturated models of intuitionistic theories. Our main technique is the thorough study of the filter construction on categories with finite limits, which is the completion of subobject lattices under filtered meets. When restricted to coherent or Heyting categories, classifying categories of intuitionistic first-order theories, the resulting categories are filtered meet coherent categories, coherent categories with complete subobject lattices such that both finite disjunctions and existential quantification distribute over filtered meets. Such categories naturally embed into Grothendieck toposes which then contain saturated models of the theory we started with.