# Group Theory/Group objects and torsors

Definition (group object):

Let ${\displaystyle {\mathcal {C}}}$ be a category that has a terminal object, which we shall denote by ${\displaystyle I}$. A group object of ${\displaystyle {\mathcal {C}}}$ consists of

1. an object ${\displaystyle G}$ of ${\displaystyle {\mathcal {C}}}$ such that the categorical products ${\displaystyle G\times G}$, ${\displaystyle G\times (G\times G)}$ and ${\displaystyle (G\times G)\times G}$ exist
2. a morphism ${\displaystyle \mu :G\times G\to G}$, thought of as the multiplication morphism,
3. a morphism ${\displaystyle \eta :T\to G}$, thought of as the constant identity map (even though ${\displaystyle G}$ may not even be a set)
4. and a morphism ${\displaystyle \iota :G\to G}$, thought of as the inversion morphism

which satisfy the following equations:

1. ${\displaystyle \mu \circ (\iota ,\operatorname {Id} _{G})=\mu \circ (\operatorname {Id} _{G},\iota )=\operatorname {Id} _{G}}$ (identity law)
2. ${\displaystyle \mu \circ (\operatorname {Id} _{G},\iota )\circ \Delta =\mu \circ (\iota ,\operatorname {Id} _{G})\circ \Delta =\eta \circ t}$, where ${\displaystyle \Delta }$ shall denote the diagonal of ${\displaystyle G}$ and ${\displaystyle t:G\to T}$ is the unique morphism given by the definition of a terminal object (inverse law)
3. ${\displaystyle \eta \circ (\eta ,\operatorname {Id} _{G})=\eta \circ (\operatorname {Id} _{G},\eta )\circ \alpha }$, where ${\displaystyle \alpha :(G\times G)\times G\to G\times (G\times G)}$ is the canonical isomorphism (associative law)

Here, we wrote ${\displaystyle (f,g):A\times B\to C\times D}$ for the morphism that is induced by two maps ${\displaystyle f:A\to C}$ and ${\displaystyle g:B\to D}$ and the universal property of the product, by specifying that the map from ${\displaystyle A\times B}$ to ${\displaystyle C}$ shall be given by ${\displaystyle f\circ \pi _{A}}$, and that the map from ${\displaystyle A\times B}$ to ${\displaystyle D}$ shall be given by ${\displaystyle g\circ \pi _{B}}$.

Definition (action of a group object):

Let ${\displaystyle {\mathcal {C}}}$ be a category which possesses a terminal object, and let ${\displaystyle G}$ be a group object of ${\displaystyle {\mathcal {C}}}$. Let ${\displaystyle X}$ be another object of ${\displaystyle {\mathcal {C}}}$. An action of ${\displaystyle G}$