nLab group object

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

Definition

In terms of internal group objects

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

$$1:* \to G$$

(the unit map)

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

(the inverse map) and

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

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

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

(expressing the fact multiplication is associative),

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

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

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

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

As a slight abuse of notation we have reused $1$ here to denote $1: G \to G$, defined as the composite $G \to * \stackrel{1}{\to} G$. Also, 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.

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 presheaves of groups

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 $Grp^{C^{op}}$ 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.

In other words, the forgetful functor from $Grp^{C^{op}}$ to $Set^{C^{op}}$ (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

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

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

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

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

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

Taken together this means that there is a morphism

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

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.

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 transl. pdf)

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 transl. pdf)

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)