Consider the Mal’tsev condition on a variety which asserts the existence of an n-ary term p satisfying some equations of the form p(x_1 , . . . , x_n ) = y. Such a Mal’tsev condition can be reformulated within the internal language of an abstract category, using the so-called ‘matrix method’ of Z. Janelidze. The corresponding categorical condition is captured by a matrix of positive integers, and is formally called a closedness property of internal relations, or a ‘matrix property’ for short. Some of these properties follow from (conjunctions of) others, for example, the matrix property corresponding to the equations defining a majority term follows from the matrix property corresponding to a Pixley term. This then is the categorical theorem corresponding the fact that if p(x, y, z) is a Pixley term, then m(x, y, z) = p(x, p(x, y, z), z) is a majority term. The main aim of this talk is to present some recent work together with Z. Janelidze and P.-A. Jacqmin which has produced an algorithm for deciding implications of (conjunctions of) matrix properties in the light context of left-exact categories. Such implications of matrix properties are context sensitive, i.e., they depend on what further conditions the base category satisfies. In particular, we will see that implications of matrix properties in the context of varieties of algebras can be different from the context of left exact categories.
A classification of left exact categories motivated from universal algebra
There are many instances throughout mathematics where the representation theory of algebraic structures becomes prohibitively challenging. In the theory of finite groups, there is a reasonable work-around that is known as a supercharacter theory. The basic idea is to limit attention to modules that are better behaved than irreducible modules, but that still retain many of the nice properties we expect from irreducibles. This talk gives an introduction to this approach with a focus on techniques that might have a chance of extending to other types of algebraic structures. Many such techniques allow the user to tailor a representation theory to specific applications, a process I will illustrate with some examples in combinatorics.
An introduction to approximating representation theories