Stellenbosch University (Professor of Mathematics)

National Institute for Theoretical and Computational Sciences (PI of the Mathematical Structures and Modelling Research Group)

A. Razmadze Mathematical Institute (Associated Member)

South African Mathematical Society (President)

Zurab Janelidze

Joint work with Michael Hoefnagel, Pierre-Alain Jacqmin, and Emil van der Walt

What does Category Theory do?

Category theory mainly studies “hyperstructures”: mathematical structures whose constituents can themselves be mathematical structures, such as vector spaces, groups, rings, lattices, etc. The purpose of category theory is to understand unifying conceptual principles across mathematics and its applications.

Categories

Most standard categories have “finite limits”. Limits combine several mathematical structures into a single structure. Examples of limits are Cartesian product of mathematical structures. Examples of categories having finite limits include:

• The categories of sets, of natural numbers
  
• The categories of semigroups, of monoids, of groups, of abelian groups
  
• The categories of modules over a ring, of vector spaces, of matrices
  
• The categories of topological spaces, of graphs
  
• The categories of algebras in a variety of universal algebras, of lattices, of Boolean algebras
  
• The category of spaces of pure quantum states (Hilbert spaces)
  

having finite limits

Matrix properties

The study of matrix properties of categories is inspired by Mal’tsev conditions in universal algebra. A matrix property of a category describes the combinatorial geometry of the structure of its internal n-dimensional “subspaces” (equivalently, of internal n-ray relations), encoded by integer matrices. Matrix properties are in some sense most basic type of properties that finite limits may have.

Examples of Matrix properties

Some of the well-studied varieties/categories in universal/categorical algebra are defined by matrix properties. For varieties, matrix properties are equivalent to a particular type of linear Mal’tsev conditions

The poset of matrix properties

Define a matrix property to be less than another one, when every category with finite limits having that property also has the other one. We then get an (infinite) poset.

It turns out that such algorithm for establishing implications of matrix properties can establish all implications — the algorithm is not only sound, but also complete (a fundamental result in the theory). There is a geometric and an algebraic way of describing the algorithm.

Fragments of the poset of matrix properties

Implementation of the algorithm on a computer allows to capture various finite fragments of the infinite poset of matrix properties.

The picture to the left displays the fragment consisting of matrix properties presentable using matrices with three rows and two entires: 0 and 1 (binary matrices). Can you identify arithmetical, majority and Mal’tsev matrices?

Already for binary matrices having four rows, the poset is too large to visualise.

On the left: computer-generated image of the fragment of the poset of matrix properties given by matrices having up to four rows and up to five columns

What have we been able to establish thus far about the structure of the poset of matrix properties?

Observation of these computer-generated images of fragments of the infinite poset of matrix properties suggests:

• The down-set of the Mail’tsev matrix and the up-set of the majority matrix cover the entire poset (proven TRUE for binary matrices)
  
• The down-set of the Mal’tsev matrix is smaller than the up-set of the majority matrix (proven FALSE — both are infinite)
• The arithmetical matrix is the bottom element of the poset (proven TRUE for binary matrices)
• The poset of matrix properties does not have a top element, if we do not count the matrix property that holds in every category (proven TRUE, there is an infinitely increasing unbounded sequence)
• Conjunction of the Mal’tsev matrix property and the majority property gives the airthmetical matrix property (known to be TRUE)
  

Diagonal

matrices

Mal’tsev, majority and arithmetical matrices are diagonal matrices. We were able to fully determine the structure of the poset of diagonal matrix properties.

The poset of diagonal matrix properties has above the majority matrix, an infinite ray, and between the Mal’tsev and the arithmetical matrices, an infinite quarter plane where ordering is by divisibility in the second coordinate and the usual “less than” in the first coordinate.

Mathematical richness of the study of the poset of matrix properties

Theory of matrix properties brings together:

• Category theory
  
• Universal algebra
• Finite incidence geometry
• Combinatorics
• Lattice theory
• Various mathematical structures, including the group of integers
  

Some References