slice category

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slice category ‎(plural slice categories)

  1. (category theory) Given a category C and an object X ∈ Ob(C), the slice category  C \downarrow X has, as its objects, morphisms from objects of C to X, and as its morphisms, morphisms connecting the tails of its own objects in a commutative way (i.e., closed under composition). The category  C \downarrow X is said to be "over X". (More formally, the objects of C over X are ordered pairs of the form (A, f) where A is an object of C and f is a morphism from A to X. Then the morphisms of C over X have such ordered pairs as their domains/codomains instead of objects of C directly.)
    If slice category C over X has two objects (A, f) and (B, g) and a morphism h : (A, f) → (B, g), then this morphism would correspond to a like-named morphism h : AB of C such that  g \circ h = f .