Skip to content

Slice Category

The slice category for a category \(C\) under an element \(c \in C\) is the category of elements of the hom-set functor \(C(c, -)\), denoted \(c/C\). Likewise, the slice category of \(C\) over \(c\), \(C/c\), is the category of elements of \(C(-, c)\). The former is often called an undercategory, whereas the latter is called an overcategory.

Let \(C\) be a category and \(c \in C\). The slice category \(c/C\) is a category such that:

  • Objects are morphisms in \(C\) with domain \(c\).
  • A morphism from \(f: c \to x\) to \(g: c \to y\) is a morphism \(h: x \to y\) such that \(g=hf\) such that the [[Triangle Diagram]] commutes. \(h\) is said to be a morphism under \(c\).

Examples of Slice Categories

  1. Let \(\text{Pos}\) be the Poset Category on \(\mathbb{R}\), and let \(x \in \mathbb{R}\). Then \(x/\text{Pos}\) is the smaller poset category with \(x\) as its minimum element, and \(\text{Pos}/x\) is the smaller poset category with \(x\) as its maximum element.
  2. Let \(X\) be a topological space. Then the objects in \(\text{Top}/X\) are pairs \((Y, \phi)\) where \(Y\) is a topological space and \(\phi: Y \to X\) is a continuous map, and morphisms are continuous maps \(f: Y \to Z\) where \(Y, Z\) are topological spaces such that the structure over \(X\) is preserved.