Nullital categories

and a new research programme in

arithmetic of structures

Joint work with Michael Hoefnagel

The part of categorical algebra which deals with the study of interaction between limits and colimits can be seen as “arithmetic of structures”, i.e., an elementary number theory of abstract mathematical structures. In elementary number theory, one studies relation between multiplication and addition. But these operations can be seen as products and coproducts in the category of finite sets. In categorical algebra, we are interested in the relation between limits and colimits not in one particular category, but in classes of categories where main examples are categories of mathematical structures such as vector spaces, monoids, groups, rings and many others.

There are categories where 0=1. The category of finite sets is not such, as it would force the 0=1 to be a numerical equality. But the category of vector spaces is such, where 0 and 1 are both the zero dimensional vector space (vector space with only the zero vector in it). Many other categories are such too: vector spaces, monoids, groups, rings without identity element, etc. In such categories we can define a quotient, by considering the colimit of a morphism and a parallel morphism factoring through 0=1. Such categories are called “pointed” categories.

In pointed categories we can investigate further relations between limits and colimits. Some of these can resemble laws of elementary number theory.

In this talk, we will consider the law

A/(BxC) = (A/B)/C.

Remarkably, this and other similar laws have not yet been studied in full generality, except the following, which was the subject of my first paper:

(AxB)/A = B.

In algebraic pointed categories, a sufficient condition for the law A/(BxC) = (A/B)/C is to have a binary operation + such that

x+0=0+x, and,

x+0=0 implies x=0.

This explains why the law above holds for monoids, groups, vector spaces, and many other algebraic structures.

It can be shown that algebras (A,+0) where + has the two properties above do not, in general, fulfil the law (AxB)/A=B. Thus, the second implies the first, but the converse is not true.

Let us conclude by proposing the following research programme in the field of arithmetic of structures:

Provide a complete classification of all laws that can be expressed using multiplication and division.