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 whose morphisms are the commutative squares between these data.

For all objects $X$ in $\mathcal{C}$, the category $Pt_X \mathcal{C}$ has as objects all split epimorphisms over $X$, and morphisms the commutative triangles.

If $\mathcal{C}$ has pullbacks, any morphism $f: X \to Y$ in $\mathcal{C}$ induces a base change functor

$$f^{\ast}: Pt_Y \mathcal{C} \to Pt_X \mathcal{C},$$

which takes the pullback of a split epimorphism over $Y$ along $f$.

Denote by $\pi: Pt\mathcal{C} \to \mathcal{C}$ the functor associating to a split epimorphism its codomain. Since $\mathcal{C}$ has pullbacks, the functor $\pi$ is a fibration, called the fibration of points.

Theorem

Let $\mathcal{C}$ be a pointed, finitely complete category. The following are equivalent:

1. The Split Short Five Lemma holds. That is, given any commutative diagram in $\mathcal{C}$ where $f=\ker g$ and $f'=\ker g'$. If $g,g'$ are split epimorphisms and $\alpha, \gamma$ are isomorphisms, then $\beta$ is also an isomorphism;

2. For any object $Y$ in $\mathcal{C}$, the kernel functor

$$1^{\ast}: Pt_Y \mathcal{C} \to Pt_1 \mathcal{C}$$

is conservative (reflects isomorphisms);

3. For any morphism $f: X \to Y$ in $\mathcal{C}$, the change-of-base functor

$$f^{\ast}: Pt_Y \mathcal{C} \to Pt_X \mathcal{C}$$

reflects isomorphisms;

4. For any pullback of split epimorphisms, with $s: Y \to B$ the splitting of $p$, the pair $(q,s)$ is jointly extremally epic;

5. For any commutative diagram

in which $p,p'$ are split epimorphisms, the left-hand square and the outer rectangle are pullbacks, then so is the right-hand square.

Definition

A finitely complete category $\mathcal{C}$ is protomodular if it satisfies any of the equivalent conditions $(3)$–$(5)$ above. Moreover, when $\mathcal{C}$ is pointed, it is protomodular if it satisfies any of the equivalent conditions above.

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 $A$, 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 $i = 1,\ldots,n$.)

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 $\mathcal{C}/Z$ 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.

Example

Every cotopos.

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 any cotopos are strongly protomodular.

Related concepts

• homological category

• semi-abelian category

• regular category

References

• Francis Borceux, Dominique Bourn, Mal'cev, protomodular, homological and semi-abelian categories, Mathematics and Its Applications 566, Kluwer 2004 (doi:10.1007/978-1-4020-1962-3)

• Dominique Bourn, Marino Gran, Regular, Protomodular, and Abelian Categories, Chapter IV, pp.165-211 in: Maria Pedicchio, Walter Tholen (eds.), Categorical Foundations, Cambridge University Press 2004 (doi:10.1017/CBO9781107340985.007)

• 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