nLab
group object

A group object in a category $C$ is a group internal to $C$.

Definition

A group object or internal group in a category $C$ with binary products and a terminal object $*$ is an object $G$ in $C$ and arrows

$$1:*\to G$$1:* \to G

(the unit map)

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

(the inverse map) and

$$m:G\times G\to G$$m:G\times G \to G

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

$$\begin{array}{ccc}G\times G\times G& \stackrel{\mathrm{id}\times m}{\to }& G\times G\\ m\times \mathrm{id}\downarrow & & \downarrow m\\ G\times G& \stackrel{m}{\to }& G\end{array}$$\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),

$$\begin{array}{ccc}G& \stackrel{(1,\mathrm{id})}{\to }& G\times G\\ (id,1)\downarrow & \underset{=}{\searrow }& \downarrow m\\ G\times G& \stackrel{m}{\to }& G\end{array}$$\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

$$\begin{array}{ccc}G& \stackrel{(\mathrm{id},(-{)}^{-1})\circ \Delta }{\to }& G\times G\\ ((-{)}^{-1},\mathrm{id})\circ \Delta \downarrow & \underset{1}{\searrow }& \downarrow m\\ G\times G& \stackrel{m}{\to }& G\end{array}$$\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 $1:G\to G$ denote the composite $G\to *\stackrel{1}{\to }G$ and $\Delta$ is a diagonal morphism.

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

In terms of generalized group elements

Proposition

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

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

An object $G$ in $C$ with an internal group structure is a diagram

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

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

$${\cdot }_S:G(S)\times G(S)\to G(S).$$\cdot_S : G(S) \times G(S) \to G(S) \,.

Moreover, since morphisms in $\mathrm{Grp}$ are group homomorphisms, it follows that for every morphism $f:S\to T$ we get a commuting diagram

$$\begin{array}{ccc}G(S)\times G(S)& \stackrel{{\cdot }_S}{\to }& G(S)\\ {\uparrow }^{G(f)\times G(f)}& & {\uparrow }^{G(f)}\\ G(T)\times G(T)& \stackrel{{\cdot }_T}{\to }& G(T)\end{array}$$\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

$$Y(G\times G)\to Y(G)$$Y(G \times G) \to Y(G)

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

etc.

Examples

• A group object in Set is a group.
• A group object in Top is a topological group.
• A group object in Ho(Top) is an H-group.
• A group object in Diff is a Lie group.
• A group object in SDiff is a super Lie group.
• A group object in Grp is an abelian group (using the Eckmann-Hilton argument).
• A group object in Ab is an abelian group again.
• A group object in Cat is a strict 2-group.
• A group object in Grpd is a strict $2$-group again.
• A group object in CRing${}^{\mathrm{op}}$ is a commutative Hopf algebra.
• A group object in some functor category is a group functor.
• A group object in the category of schemes is a group scheme

Theory

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.

Related concepts

• monoid, monoid object, monoid object in an (∞,1)-category

• group, group object, group object in an (∞,1)-category

• groupoid, groupoid object, groupoid object in an (∞,1)-category

• infinity-groupoid, infinity-groupoid object, groupoid object in an (∞,1)-category

• ring, ring object