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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.