nLab
protomodular category

Idea

An important aspect of group theory is the study of normal subgroups. A protomodular category, even one which is not pointed, is defined in such a way that it possesses an intrinsic notion of normal subobject. The concept is due to Dominique Bourn and as such sometimes referred to as Bourn-protomodularity.

Definition

(Taken from Bourn04)

Consider any finitely complete category $\mathcal{C}$ and denote by $Pt\mathcal{C}$ the category whose objects are the split epimorphisms in $\mathcal{C}$ with a given splitting and morphisms the commutative squares between these data. Denote by $\pi: Pt\mathcal{C} \to \mathcal{C}$ the functor associating its codomain to any split epimorphism. Since the category $\mathcal{C}$ has pullbacks, the functor $\pi$ is a fibration which is called the fibration of points.

Any map $f: X \to Y$ induces, by pullbacks, a base change functor denoted $f^{\ast}: Pt_Y \mathcal{C} \to Pt_X \mathcal{C}$ between the fibres above $Y$ and $X$.

Then a left exact category $\mathcal{C}$ is said to be protomodular when the fibration $\pi$ has conservative base change functors, i.e., ones that reflect isomorphisms. A protomodular category is necessarily Mal'cev.

Examples

• Certain categories of varieties of algebras, such as the category of groups, the category of rings, the category of associative or Lie algebras over a given ring $A$, the category of Heyting algebras, the varieties of $\Omega$-groups. (It is shown in Bourn-Janelidze that a variety $V$ of universal algebras is protomodular if and only if it has $0$-ary terms $e_1, \ldots ,e_n$, binary terms $t_1,\ldots,t_n$, and $(n+1)$-ary term $t$ satisfying the identities $t(x, t_1(x, y),\ldots,t_n(x, y)) = y$ and $t_i(x, x) = e_i$ for each $i = 1,\ldots,n$.)

• Categories of algebraic varieties as above internal to a left exact category, for example, TopGrp.

• Constructions which inherit the property of being protomodular, such as the slice categories $\mathcal{C}/Z$ and the fibres $Pt_Z \mathcal{C}$ of the fibration $\pi$ of pointed objects for instance, or more generally the domain $\mathcal{C}$ of any pullback preserving and conservative functor $U : \mathcal{C} \to \mathcal{D}$; when its codomain $\mathcal{D}$ is protomodular.

• The dual of a topos.

Consequences of protomodularity

• a map is a monomorphism if and only if its kernel is trivial
• a reflective relation is an equivalence relation
• an internal category is always an internal groupoid
• a regular epimorphism is always the cokernel of its kernel
• an object is abelian when its diagonal is a normal subobject

A pointed protomodular category is strongly unital, and

• there is a bijection between normal subobjects of an object $X$ and equivalence relations on $X$.

Strong protomodularity

A category $\mathcal{C}$ is strongly protomodular when it is protomodular and is such that any change of base functor $f^{\ast}$ is a normal functor, that is, a left exact conservative functor which reflects the normal monomorphisms.

Grp, Ring and the dual of any topos are strongly protomodular.

Related concepts

• homological category
• semi-abelian category

References

• Francis Borceux, Dominique Bourn, Mal'cev, protomodular, homological and semi-abelian categories, Mathematics and Its Applications 566, Kluwer 2004

• Dominique Bourn, Protomodular aspect of the dual of a topos, Advances in Mathematics 187(1), pp. 240-255, 2004.

• Dominique Bourn, Action groupoid in protomodular categories, TAC

• Dominique Bourn, George Janelidze, Characterization of protomodular varieties of universal algebras, (TAC)

• Dominique Bourn, From Groups to Categorial Algebra : Introduction to Protomodular and Mal’tsev Categories, Compact Textbooks in Mathematics, Birkhäuser 2017