8 research outputs found
The Category of Matroids
The structure of the category of matroids and strong maps is investigated: it
has coproducts and equalizers, but not products or coequalizers; there are
functors from the categories of graphs and vector spaces, the latter being
faithful; there is a functor to the category of geometric lattices, that is
nearly full; there are various adjunctions and free constructions on
subcategories, inducing a simplification monad; there are two orthogonal
factorization systems; some, but not many, combinatorial constructions from
matroid theory are functorial.Comment: 31 pages, 10 diagrams, 28 reference
Monoidal characterisation of groupoids and connectors
We study internal structures in regular categories using monoidal methods. Groupoids in a regular Goursat category can equivalently be described as special dagger Frobenius monoids in its monoidal category of relations. Similarly, connectors can equivalently be described as Frobenius structures with a ternary multiplication. We study such ternary Frobenius structures and the relationship to binary ones, generalising that between connectors and groupoids
Tensor topology
A subunit in a monoidal category is a subobject of the monoidal unit for
which a canonical morphism is invertible. They correspond to open subsets of a
base topological space in categories such as those of sheaves or Hilbert
modules. We show that under mild conditions subunits endow any monoidal
category with a kind of topological intuition: there are well-behaved notions
of restriction, localisation, and support, even though the subunits in general
only form a semilattice. We develop universal constructions completing any
monoidal category to one whose subunits universally form a lattice, preframe,
or frame.Comment: 44 page
Boolean Subalgebras of Orthoalgebras
We develop a direct method to recover an orthoalgebra from its poset of
Boolean subalgebras. For this a new notion of direction is introduced.
Directions are also used to characterize in purely order-theoretic terms those
posets that are isomorphic to the poset of Boolean subalgebras of an
orthoalgebra. These posets are characterized by simple conditions defining
orthodomains and the additional requirement of having enough directions.
Excepting pathologies involving maximal Boolean subalgebras of four elements,
it is shown that there is an equivalence between the category of orthoalgebras
and the category of orthodomains with enough directions with morphisms suitably
defined. Furthermore, we develop a representation of orthodomains with enough
directions, and hence of orthoalgebras, as certain hypergraphs. This hypergraph
approach extends the technique of Greechie diagrams and resembles projective
geometry. Using such hypergraphs, every orthomodular poset can be represented
by a set of points and lines where each line contains exactly three points.Comment: 43 page
Limits in dagger categories
We develop a notion of limit for dagger categories, that we show is suitable
in the following ways: it subsumes special cases known from the literature;
dagger limits are unique up to unitary isomorphism; a wide class of dagger
limits can be built from a small selection of them; dagger limits of a fixed
shape can be phrased as dagger adjoints to a diagonal functor; dagger limits
can be built from ordinary limits in the presence of polar decomposition;
dagger limits commute with dagger colimits in many cases
Monads on Dagger Categories
The theory of monads on categories equipped with a dagger (a contravariant
identity-on-objects involutive endofunctor) works best when everything respects
the dagger: the monad and adjunctions should preserve the dagger, and the monad
and its algebras should satisfy the so-called Frobenius law. Then any monad
resolves as an adjunction, with extremal solutions given by the categories of
Kleisli and Frobenius-Eilenberg-Moore algebras, which again have a dagger. We
characterize the Frobenius law as a coherence property between dagger and
closure, and characterize strong such monads as being induced by Frobenius
monoids.Comment: 28 page