nLab group object

Redirected from "internal groups".
Contents

Context

Categorical algebra

Group Theory

Contents

Idea

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.

Definition

In a cartesian monoidal category

Definition

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

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

Remark

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.

Remark

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.

In a braided monoidal category

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 typically uses 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.

Hopf monoids may be defined in any symmetric monoidal category, or more generally any braided monoidal category, where the braiding is used in stating the fact that the comultiplication is a homomorphism of monoid objects.

In a monoidal category

A surprising fact reported by Tom Leinster is that in the category of sets, a group is the same as a monoid with the extra property that the associativity square

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 }

is a pullback. Presumably it is also true that a group object in a cartesian monoidal category is the same as a monoid object in that category where the associativity square is a pullback. This suggests that we can define a group object in any monoidal category to be a monoid object where the associativity square is a pullback. The category does not need to have all pullbacks for this definition to parse. However, the usefulness of this generalization remains to be studied.

In terms of presheaves of groups

Proposition

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,e S)\big(G(S), \cdot_S, \mathrm{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.

etc.

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:

Examples

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.

References

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

  • Alexander Grothendieck, p. 104 (7 of 21) of: FGA Techniques de construction et théorèmes d’existence en géométrie algébrique III: préschémas quotients, Séminaire Bourbaki: années 1960/61, exposés 205-222, Séminaire Bourbaki, no. 6 (1961), Exposé no. 212, (numdam:SB_1960-1961__6__99_0, pdf, English translation: web version)

following the general principle of internalization formulated in:

  • Alexander Grothendieck, p. 340 (3 of 23) in: FGA Technique de descente et théorèmes d’existence en géométrie algébriques. II: Le théorème d’existence en théorie formelle des modules, Séminaire Bourbaki : années 1958/59 - 1959/60, exposés 169-204, Séminaire Bourbaki, no. 5 (1960), Exposé no. 195 (numdam:SB_1958-1960__5__369_0, pdf, English translation: web version)

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:

Review:

In the broader context of internalization via sketches:

With focus on internalization in sheaf toposes:

Last revised on November 7, 2024 at 13:17:25. See the history of this page for a list of all contributions to it.