Group object - Knowlpedia

Let $\mathcal C$ be a category with finite products and terminal object $1$ . A group object in $\mathcal C$ is an object $G$ with morphisms

\begin{aligned} m&:G\times G\to G,\\ e&:1\to G,\\ i&:G\to G. \end{aligned}

For every object $T$ , write $!_T:T\to1$ for the unique morphism and define operations on generalized elements $f,g:T\to G$ using their product pairing $\langle f,g\rangle:T\to G\times G$ :

\begin{aligned} f*g&=m\circ\langle f,g\rangle,\\ e_T&=e\circ !_T,\\ f^{-1}&=i\circ f. \end{aligned}

The group-object axioms require, for all $f,g,h:T\to G$ ,

\begin{aligned} (f*g)*h&=f*(g*h),\\ e_T*f&=f=f*e_T,\\ f^{-1}*f&=e_T=f*f^{-1}. \end{aligned}

These are the associativity, identity, and inverse laws, tested on generalized elements from every object $T$ .

Generalized elements

For every object $T$ , composition with $m,e,i$ makes

\operatorname{Hom}_{\mathcal C}(T,G)

a group under the operations above. The construction is contravariantly natural in $T$ : precomposition with a morphism $S\to T$ is a group homomorphism. This is a reliable way to read the internal axioms, although ordinary elements alone may not detect all morphisms in every category.

Examples

Group objects in sets are ordinary groups.
Group objects in topological spaces are topological groups.
Group objects in smooth manifolds are Lie groups.
Group objects in schemes are group schemes.

References

Francis Borceux, Handbook of Categorical Algebra 2: Categories and Structures, Cambridge University Press, 1994. DOI record. Relevant: internal algebraic structures.