Intrinsic centrality and associated classifying properties
by
Dominique Bourn
Université du Littoral, Calais Cedex France

The notion of unital category is a special case of pointed category, which allows to characterize Mal'cev categories via the fibration \pi of pointed objects. The paradigmatic example of unital category is the category Mag of unitary magmas. Indeed given a pair (X, Y) of objects in Mag, the identity (x, y)=(x, 1).(1, y) in the product X×Y implies that any subobject Z of X×Y containing X×1 and 1×Y is actually equal to X×Y. In other words, for each pair (X, Y) of objects in Mag, the pair of canonical inclusions (l_X, r_Y) is jointly strongly epic:

X lX -->   X ×Y rY <--   Y

The major examples of unital categories are the categories Mon, CoM, Gp, Ab, Rg of respectively monoids, commutative monoids, groups, abelian groups, rings, and the categories Mag(E), Mon(E), CoM(E), Gp(E), Ab(E), Rg(E) of respectively internal unitary magmas, monoids, commutative monoids, groups, abelian groups, rings in a left exact category E.

The aim of this work is to show that, inside any unital category C, there is an additive core which has a powerful classifying potential. More precisely there is a right ideal Z(C), namely the right ideal of central morphisms, such that the class Z(X, Y) of central morphisms between X and Y forms a commutative monoid which has a canonical action on C(X, Y). From this ideal, we can extract the right ideal \Sigma(C) of the symmetrizable morphisms of Z(C), such that the class \Sigma(X, Y) of symmetrizable morphisms between X and Y is of course an abelian group. A first aspect of the discriminatory power of this additive core is given by the following table, where \Omega(C) is the ideal of null maps, and where the intersection of the line L and the column D defines the class of categories which satisfies the property L = D :

= C \Omega(C) Z(C) Linear categories Antilinear categories \Sigma(C) Additive categories Antiadditive categories

The paradigmatic examples of respectively linear, additive, antilinear and antiadditive categories are CoM, Ab, the category PreH of preHeyting algebras, the category IMag of idempotent unitary magmas.

Finally the notion of strongly unital category is precised, where we have always \Sigma(C)=Z(C), i.e., where any central map has a symmetric.

We can now come back to our starting point. We recalled that any left exact category E could be represented by the fibration \pi of pointed objects all the fibres of which are pointed. This representation raises a quite natural question, namely : do the previous classifications extend, via this fibration, from pointed categories to any left exact category . This is effectively the case. We already knew that:

1) the fibration \pi is unital (= has its fibres unital) if and only if the fibration \pi is strongly unital if and only if the category E is Mal'cev (following tne definition of Carboni, Lambek, Pedicchio).

We can now add the following specification:

2) the fibration \pi is linear if and only if the fibration \pi is additive, which is the case if and only if the category E is Naturally Mal'cev (following the definiton of Johnstone).

3) the fibration \pi is antiadditive if and only if the fibration \pi is antilinear; provided moreover the category E is Barr exact, this is the case if and only if the category E is arithmetical (following the definition of Pedicchio).

Date received: August 2, 2002

Copyright © 2002 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cajf-46.