nLab group object

Idea

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

• an object $G$ in $\mathcal{C}$

• and morphisms in $\mathcal{C}$ as follows:

  • neutral element:

    $\mathrm{e} \,\colon\, \ast \to G$

  • inverse elements:

    $(-)^{-1} \,\colon\, G\to G$

  • binary operation$m \colon G\times G \to G$,

such that the following diagrams commute:

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

(expressing the fact multiplication is associative),

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

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

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

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

Remark

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

The pairing $(f,g)$ denotes $(f\times g)\circ\Delta$ where $\Delta$ is a diagonal morphism.

Remark

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 $\mathcal{C}$, 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

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

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 $\mathcal{C}$, the category of internal groups in $\mathcal{C}$ (in the sense of Def. ) is equivalent to the full subcategory of the category of presheaves of groups $Grp^{C^{op}}$ on $C$, spanned by those presheaves whose underlying set part in $Set^{C^{op}}$ 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 $Func\big(\mathcal{C}^{op}, Grp\big)$ to the presheaf category $Func\big(\mathcal{C}^{op}, Set\big)$ (obtained by composing with the forgetful functor Grp $\to$ Set) creates representable group objects from representable objects.

We unwind how this works:

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

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

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

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

is the Yoneda embedding of $\mathcal{C}$ into its category of presheaves $PSh(C) \,\coloneqq\, Func(C^{op}, Set)$, which sends each object $G$ 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 $G(S)$ (for $S \in \mathcal{C}$) as shorthand for

$$G(S) \coloneqq y(G)(S) \coloneqq Hom_C(S,G) \,.$$

(This a also referred to as “$G$ seen at stage $S$”, or similar.)

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

$$\cdot_S \,\colon\, G(S) \times G(S) \longrightarrow G(S) \,.$$

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

$$\array{ G(S) \times G(S) &\stackrel{\cdot_S}{\to}& G(S) \\ \big\uparrow\mathrlap{^{G(f)\times G(f)}} && \big\uparrow\mathrlap{^{G(f)}} \\ G(T) \times G(T) &\underset{\cdot_T}{\longrightarrow}& G(T) \mathrlap{\,.} }$$

Taken together this means that there is a morphism

$$y(G \times G) \longrightarrow y(G)$$

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

• A group object in Sets is a group.

• A group object in TopologicalSpaces is a topological group.

• A group object in SimplicialSets is a simplicial 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$^{op}$ is a commutative Hopf algebra.

• A group object in a functor category is a group functor.

• A group object in schemes is a group scheme.

• A group object in an opposite category is a cogroup object.

• A group object in G-sets/G-spaces is a $G$-equivariant group, namely a semidirect product group.

• A group object in stacks is a group stack.

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

• group action, action object

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

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

• ring, ring object

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:

• Barbara Fantechi, Lothar Göttsche, Luc Illusie, Steven L. Kleiman, Nitin Nitsure, Angelo Vistoli, Section 2.2 of: Fundamental algebraic geometry. Grothendieck’s FGA explained, Mathematical Surveys and Monographs 123, Amer. Math. Soc. (2005) [MR2007f:14001, ISBN:978-0-8218-4245-4, lecture notes]

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

• Beno Eckmann, Peter Hilton, Structure maps in group theory, Fundamenta Mathematicae 50 (1961), 207-221 (doi:10.4064/fm-50-2-207-221)

• Beno Eckmann, Peter Hilton, Group-like structures in general categories I multiplications and comultiplications, Math. Ann. 145, 227–255 (1962) (doi:10.1007/BF01451367)

• Beno Eckmann, Peter Hilton, Group-like structures in general categories III primitive categories, Math. Ann. 150 165–187 (1963) (doi:10.1007/BF01470843)

With emphasis of the role of the Yoneda lemma:

• Saunders MacLane, chapter III, section 6 in: Categories for the Working Mathematician, Springer (1971)

Review:

• John Michael Boardman, Algebraic objects in categories, Chapter 7 of: Stable Operations in Generalized Cohomology [pdf, pdf] in: Ioan Mackenzie James (ed.) Handbook of Algebraic Topology Oxford (1995) [doi:10.1016/B978-0-444-81779-2.X5000-7]

• Magnus Forrester-Barker, Group Objects and Internal Categories [arXiv:math/0212065]

In the broader context of internalization via sketches:

• Michael Barr, Charles Wells, Section 4.1 of: Toposes, Triples, and Theories, Originally published by: Springer-Verlag, New York, 1985, republished in: Reprints in Theory and Applications of Categories, No. 12 (2005) pp. 1-287

With focus on internalization in sheaf toposes:

• Saunders Mac Lane, Ieke Moerdijk, Section II.7 of: Sheaves in Geometry and Logic, Springer 1992 (doi:10.1007/978-1-4612-0927-0)