nLab
protomodular category

Contents

Context

Category theory

Group Theory

Contents

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

Any map f:XYf: X \to Y induces, by pullbacks, a base change functor denoted f *:Pt Y𝒞Pt X𝒞f^{\ast}: Pt_Y \mathcal{C} \to Pt_X \mathcal{C} between the fibres above YY and XX.

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

Examples

Example

The category Grp of all groups (including non-abelian groups) is pointed protomodular

(Borceux & Bourn 2004, Ex. 3.1.4)

Example

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 AA, the category of Heyting algebras, the varieties of Ω\Omega-groups. (It is shown in Bourn-Janelidze that a variety VV of universal algebras is protomodular if and only if it has 00-ary terms e 1,,e ne_1, \ldots ,e_n, binary terms t 1,,t nt_1,\ldots,t_n, and (n+1)(n+1)-ary term tt satisfying the identities t(x,t 1(x,y),,t n(x,y))=yt(x, t_1(x, y),\ldots,t_n(x, y)) = y and t i(x,x)=e it_i(x, x) = e_i for each i=1,,ni = 1,\ldots,n.)

Example

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

Example

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

Example

The dual of a topos.

Consequences of protomodularity

A pointed protomodular category is strongly unital, and

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

Strong protomodularity

A category 𝒞\mathcal{C} is strongly protomodular when it is protomodular and is such that any change of base functor f *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.

References

Last revised on November 9, 2021 at 06:27:30. See the history of this page for a list of all contributions to it.