nLab
protomodular category

Idea

An important aspect of group theory is the study of normal subgroups. A protomodular category, even a non-pointed one, 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 π’ž\mathcal{C} and denote by Ptπ’žPt\mathcal{C} the category whose objects are the split epimorphisms in π’ž\mathcal{C} with a given splitting and morphisms the commutative squares between these data. Denote by Ο€:Ptπ’žβ†’π’ž\pi: Pt\mathcal{C} \to \mathcal{C} the functor associating its codomain with any split epimorphism. Since the category π’ž\mathcal{C} has pullbacks, the functor Ο€\pi is a fibration which is called the fibration of points.

Any map f:Xβ†’Yf: X \to Y induces, by pullbacks, a change of base functor denoted f *:Pt Yπ’žβ†’Pt Xπ’žf^{\ast}: Pt_Y \mathcal{C} \to Pt_X \mathcal{C} between the fibres above YY and XX.

Then a left exact category π’ž\mathcal{C} is said to be protomodular when the fibration Ο€\pi has conservative change of base functors, i.e., reflecting the isomorphisms. A protomodular category is necessarily Mal'cev.

Examples

  • Certain categories of algebraic varieties, 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 Ξ©\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.)

  • 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 π’ž/Z\mathcal{C}/Z and the fibres Pt Zπ’žPt_Z \mathcal{C} of the fibration Ο€\pi of pointed objects for instance, or more generally the domain π’ž\mathcal{C} of any pullback preserving and conservative functor U:π’žβ†’π’ŸU : \mathcal{C} \to \mathcal{D}; when its codomain π’Ÿ\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 XX and equivalence relations on XX.

Strong protomodularity

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

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

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)

Revised on October 17, 2014 07:54:08 by David Corfield (129.12.18.116)