group object


Category theory

Group Theory


A group object in a category CC is a group internal to CC.


In terms of internal group objects

A group object or internal group in a category CC with binary products and a terminal object ** is an object GG in CC and arrows

1:*G 1:* \to G

(the unit map)

() 1:GG (-)^{-1}:G\to G

(the inverse map) and

m:G×GG m:G\times G \to G

(the multiplication map), such that the following diagrams commute:

G×G×G id×m G×G m×id m G×G m G \array{ G\times G\times G & \stackrel{id\times m}{\to} & G\times G\\ m\times id\downarrow && \downarrow m \\ G\times G & \stackrel{m}{\to} &G }

(expressing the fact multiplication is associative),

G (1,id) G×G (id,1) = m G×G m G \array{ G & \stackrel{(1,id)}{\to} & G\times G\\ (\id,1)\downarrow &\underset{=}{\searrow}& \downarrow m \\ G\times G & \stackrel{m}{\to} &G }

(telling us that the unit map picks out an element that is a left and right identity), and

G (id,() 1)Δ G×G (() 1,id)Δ 1 m G×G m G \array{ G & \stackrel{(id,(-)^{-1})\circ\Delta}{\to} & G\times G\\ ((-)^{-1},id)\circ\Delta\downarrow & \underset{1}{\searrow}& \downarrow m \\ G\times G & \stackrel{m}{\to} &G }

(telling us that the inverse map really does take an inverse), where we have let 1:GG1: G \to G denote the composite G*1GG \to * \stackrel{1}{\to} G and Δ\Delta is a diagonal morphism.

Even if CC doesn't have all binary products, as long as products with GG (and the terminal object **) exist, then one can still speak of a group object GG in CC.

In terms of presheaves of groups


Given a cartesian monoidal category CC, the category of internal groups in CC is equivalent to the full subcategory of the category of presheaves of groups Grp C opGrp^{C^{op}} on CC, spanned by those presheaves whose underlying set part in Set C opSet^{C^{op}} is representable.

This is a special case of the general theory of structures in presheaf toposes.

In other words, the forgetful functor from Grp C opGrp^{C^{op}} to Set C opSet^{C^{op}} (obtained by composing with the forgetful functor Grp \to Set) creates representable group objects from representable objects.

An object GG in CC with an internal group structure is a diagram

Grp (G,) C op Y(G) Set. \array{ && Grp \\ & {}^{(G,\cdot)}\nearrow & \downarrow \\ C^{op} &\stackrel{Y(G)}{\to}& Set } \,.

This equips each object SCS \in C with an ordinary group (G(S),)(G(S), \cdot) structure, so in particular a product operation

S:G(S)×G(S)G(S). \cdot_S : G(S) \times G(S) \to G(S) \,.

Moreover, since morphisms in GrpGrp are group homomorphisms, it follows that for every morphism f:STf : S \to T we get a commuting diagram

G(S)×G(S) S G(S) G(f)×G(f) G(f) G(T)×G(T) T G(T) \array{ G(S) \times G(S) &\stackrel{\cdot_S}{\to}& G(S) \\ \uparrow^{G(f)\times G(f)} && \uparrow^{G(f)} \\ G(T) \times G(T) &\stackrel{\cdot_T}{\to}& G(T) }

Taken together this means that there is a morphism

Y(G×G)Y(G) Y(G \times G) \to Y(G)

of representable presheaves. By the Yoneda lemma, this uniquely comes from a morphism :G×GG\cdot : G \times G \to G, which is the product of the group structure on the object GG that we are after.




The basic results of elementary group theory apply to group objects in any category with finite products. (Arguably, it is precisely the elementary results that apply in any such category.)

The theory of group objects is an example of a Lawvere theory.


Revised on December 22, 2016 07:46:36 by Urs Schreiber (