Categorical quotient
In algebraic geometry, given a category C, a categorical quotient of an object X with action of a group G is a morphism π : X → Y {\displaystyle \pi :X\to Y} that (i) is invariant; i.e., π ∘ σ = π ∘ p 2 {\displaystyle \pi \circ \sigma =\pi \circ p_{2}} where σ : G × X → X {\displaystyle \sigma :G\times X\to X} is the given group action and p2 is the projection. (ii) satisfies the universal property: any morphism X → Z {\displaystyle X\to Z} satisfying (i) uniquely factors through π {\displaystyle \pi } .