group object

Unit: $e: {\operatorname{pt}}\to G$
Inverses: $({-})^{-1}: G\to G$
Pairing: $({-})\cdot({-}): G{ {}^{ \scriptscriptstyle\times^{2} } }\to G$
- Associativity: % https://q.uiver.app/?q=WzAsNCxbMCwwLCJHXFxjYXJ0cG93ZXJ7M30iXSxbMiwwLCJHXFxjYXJ0cG93ZXJ7Mn0iXSxbMCwyLCJHXFxjYXJ0cG93ZXJ7Mn0iXSxbMiwyLCJHIl0sWzIsMywibSJdLFsxLDMsIm0iLDJdLFswLDEsIihcXGlkX0csICBtKSJdLFswLDIsIihtLCBcXGlkX0cpIiwyXV0=
- Left and right identities: existence of sections % https://q.uiver.app/?q=WzAsMyxbMiwwLCJHIl0sWzQsMiwiR1xcY2FydHBvd2VyezJ9Il0sWzAsMiwiR1xcY2FydHBvd2VyezJ9Il0sWzAsMSwiKFxcaWRfRywgZSkiLDJdLFswLDIsIihlLCBcXGlkX0cpIl0sWzIsMCwibSIsMix7ImN1cnZlIjotNX1dLFsxLDAsIm0iLDAseyJjdXJ2ZSI6NX1dXQ==
- Inversion: % https://q.uiver.app/?q=WzAsNSxbMiwwLCJHIl0sWzQsMF0sWzIsMiwiR1xcY2FydHBvd2VyezJ9Il0sWzQsMiwiR1xcY2FydHBvd2VyezJ9Il0sWzAsMiwiR1xcY2FydHBvd2VyezJ9Il0sWzAsMiwiXFxEZWx0YSJdLFsyLDMsIihcXGlkX0csIChcXHdhaXQpXFxpbnYpIiwyXSxbMiw0LCIoKFxcd2FpdClcXGludiwgXFxpZF9HKSJdLFs0LDAsIm0iXSxbMywwLCJtIl1d
An equivalent characterization: $X\in \co \underset{ \mathsf{pre} } {\mathsf{Sh} }(\mathsf{C}, {\mathsf{Grp}})$ where $\tilde X\in \co \underset{ \mathsf{pre} } {\mathsf{Sh} }(\mathsf{C}, {\mathsf{Set}})$ is representable.

Equivalently, $X\in \mathsf{C}$ is a group object if $\mathop{\mathrm{Mor}}({-}, X): \mathsf{C} \to {\mathsf{Grp}}$ represents a functor to groups.