nLab group object



Categorical algebra

Group Theory



A group object in a cartesian category CC is a group internal to CC (see at internalization for more on the general idea).

Given a non-cartesian monoidal category one can still make sense of group objects in the dual guise of Hopf monoids, see there for more and see Rem. below.


In terms of internal group objects


(group object in cartesian monoidal category)
A group object or internal group internal to a category 𝒞\mathcal{C} with finite products (binary Cartesian products and a terminal object *\ast) is

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}{\longrightarrow} & G\times G \\ {}^{ \mathllap{ m\times id } } \big\downarrow && \big\downarrow m \\ G\times G & \stackrel{m}{\longrightarrow} & G }

(expressing the fact multiplication is associative),

G (e,id) G×G (id,e) id m G×G m G \array{ G & \stackrel{(\mathrm{e},id)}{\longrightarrow} & G\times G \\ {}^{\mathllap{(\id,\mathrm{e})}} \big\downarrow &\underset{\id}{\searrow}& \big\downarrow m \\ G\times G & \underset{m}{\longrightarrow} &G }

(telling us that the neutral element is a left and right unit element), and

G (() 1,id) G×G (id,() 1) id m G×G m G \array{ G & \overset{ ((-)^{-1},id) } {\longrightarrow} & G\times G \\ {}^{ \mathllap{ (id,(-)^{-1}) } } \big\downarrow & \underset{id}{\searrow} & \big\downarrow m \\ G\times G & \stackrel{m}{\longrightarrow} & G }

(telling us that the inverse map really does take an inverse).


The associativity law technically factors through the isomorphisms between (G×G)×G(G\times G)\times G and G×(G×G)G\times (G\times G).

The pairing (f,g)(f,g) denotes (f×g)Δ(f\times g)\circ\Delta where 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 𝒞\mathcal{C}, as above.


(group objects in general monoidal categories: Hopf monoids)

Notice that the use of diagonal maps (Rem. ) in Def. precludes direct generalization of this definition of group objects to non-cartesian monoidal categories, where such maps in general do not exist.

Hence, while the underlying monoid object may generally be defined in any monoidal category, the internal formulation of existence of inverse elements requires extra structure, such as that of a compatible comonoid object-structure to substitute for the missing diagonal maps.

Given this, inverses may be encoded by an antipode map and the resulting “monoidal group objects” are known as Hopf monoids. These subsume and generalize Hopf algebras, which are widely studied, for instance in their role as quantum groups.

In terms of presheaves of groups


Given a cartesian monoidal category 𝒞\mathcal{C}, the category of internal groups in 𝒞\mathcal{C} (in the sense of Def. ) 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.

It means that the forgetful functor from the functor category Func(𝒞 op,Grp)Func\big(\mathcal{C}^{op}, Grp\big) to the presheaf category Func(𝒞 op,Set)Func\big(\mathcal{C}^{op}, Set\big) (obtained by composing with the forgetful functor Grp \to Set) creates representable group objects from representable objects.

We unwind how this works:

An object GG in 𝒞\mathcal{C} equipped with internal group structure is identified equivalently with a diagram of functors of the form

(1) Grp (G,) 𝒞 op y(G) Set, \array{ && Grp \\ & \mathllap{{}^{(G,\cdot)}}\nearrow & \big\downarrow \\ \mathcal{C}^{op} &\underset{y(G)}{\longrightarrow}& Set } \,,

where 𝒞 op\mathcal{C}^{op} is the opposite category of CC, Grp is the category of groups with group homomorphisms between them, and Set is the category of sets with maps/functions between them. Finally,

y: C PSh(C) G Hom C(,G) \array{ y \colon & C &\xhookrightarrow{\phantom{--}}& PSh(C) \\ & G &\mapsto& Hom_C(-,G) }

is the Yoneda embedding of 𝒞\mathcal{C} into its category of presheaves PSh(C)Func(C op,Set)PSh(C) \,\coloneqq\, Func(C^{op}, Set), which sends each object GG to the representable presheaf that it represents.

Since the Yoneda embedding is fully faithful, it is natural to leave it notationally implicit and to write G(S)G(S) (for S𝒞S \in \mathcal{C}) as shorthand for

G(S)y(G)(S)Hom C(S,G). G(S) \coloneqq y(G)(S) \coloneqq Hom_C(S,G) \,.

(This a also referred to as “GG seen at stage SS”, or similar.)

Now, the lift (1) of such a presheaf of sets to a presheaf of groups equips for each object S𝒞S \in \mathcal{C} the set G(S)y(G)(S)Hom C(S,G)G(S) \coloneqq y(G)(S) \coloneqq Hom_C(S,G) with an ordinary group structure (G(S), S,athrme S)\big(G(S), \cdot_S, \athrm{e}_S\big), in particular with a product operation (a map of sets) of the form

S:G(S)×G(S)G(S). \cdot_S \,\colon\, G(S) \times G(S) \longrightarrow G(S) \,.

Moreover, since morphisms in Grp are group homomorphisms, it follows that for every morphism f:STf \colon S \to T in CC we get a commuting diagram of the form

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) \\ \big\uparrow\mathrlap{^{G(f)\times G(f)}} && \big\uparrow\mathrlap{^{G(f)}} \\ G(T) \times G(T) &\underset{\cdot_T}{\longrightarrow}& G(T) \mathrlap{\,.} }

Taken together this means that there is a morphism

y(G×G)y(G) y(G \times G) \longrightarrow y(G)

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


As data structure

In the language of dependent type theory (using the notation for dependent pair types here) the type of group data structures is:



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.


The general definition of internal groups seems to have first been formulated in:

following the general principle of internalization formulated in:

reviewed in:

On internalization, H-spaces, monoid objects, group objects in algebraic topology/homotopy theory and introducing the Eckmann-Hilton argument:

With emphasis of the role of the Yoneda lemma:


In the broader context of internalization via sketches:

With focus on internalization in sheaf toposes:

Last revised on February 16, 2024 at 11:24:10. See the history of this page for a list of all contributions to it.