Group object

In mathematics, group objectscertain generalizationsgroups whichbuilt on more complicated structures than sets. A typical example ofgroup object istopological group,group whose underlying set istopological space such thatgroup operationscontinuous.

Tablecontents

1 Definition 2 Examples 3 Group theory generalized

Definition

Formally, we start withcategory C which hasterminal object 1in which any two objects haveproduct. A group objectCan object GC togethermorphisms

• m : G × G → G (thoughtas"group multiplication")
• e : 1 → G (thoughtas"inclusion ofidentity element")
• inv: G → G (thoughtas"inversion operation")

such thatfollowing properties (modeled ongroup axioms)satisfied

• massociative, i.e. m(m × id_G) = m (id_G × m) as morphisms G × G × G → G; here we identify G × (G × G) incanonical manner(G × G) × G.
• e istwo-sided unitm, i.e. m (id_G × e) = p₁, where p₁ : G × 1 → G iscanonical projection, m (e × id_G) = p₂, where p₂ : 1 × G → G iscanonical projection
• inv istwo-sided inversem, i.e. if d : G → G × G isdiagonal map,e_G : G → G iscomposition ofunique morphism G → 1 (also calledcounit)e, then m (id_G × inv) d = e_Gm (inv × id_G) d = e_G.

Examples

• A group can be viewed asgroup object incategorysets. The map m isgroup operation,map e (whose domain issingleton) picks outidentity element ofgroup, andmap inv assignsevery group element its inverse. e_G : G → G ismap that sends every elementG toidentity element.
• A topological group isgroup object incategorytopological spacescontinuous functions.
• A Lie group isgroup object incategorysmooth manifoldsdifferentiable maps.
• An algebraic group isgroup object incategoryalgebraic varieties. In modern algebraic geometry, one considersmore general group schemes, group objects incategoryschemess.
• The group objects incategorygroups (or monoids)essentiallyAbelian groups. The reasonthisthat, if invassumedbehomomorphism, then G must be abelian. More precisely: if Aan abelian groupwe denote by mgroup multiplicationA, by einclusion ofidentity element,by invinversion operation on A, then (A,m,e,inv) isgroup object incategorygroups (or monoids). Conversely, if (A,m,e,inv) isgroup objectonethose categories, then m necessarily coincides withgiven operation on A, e isinclusion ofgiven identity element on A, inv isinversion operationA withgiven operationan abelian group.

Group theory generalized

Muchgroup theory can be formulated incontext ofmore general group objects. The notionsgroup homomorphism, subgroup, normal subgroup andisomorphism theoremstypical examples. However, resultsgroup theory that talk about individual elements, ororderspecific elements or subgroups, normally cannot be generalizedgroup objects instraight-forward manner.