nLab family fibration

Definition

For a category $C$, its family fibration is the Grothendieck fibration $Fam(C) \to Set$, where

• An object of $Fam(C)$ is a set-indexed family of objects of $C$, say $(X_i)_{i\in I}$ where each $X_i\in Ob(C)$ for some set $I$.

• A morphism of $Fam(C)$ from $(X_i)_{i\in I}$ to $(Y_j)_{j\in J}$ consists of a function $u:I\to J$ along with a family of morphisms $(f_i : X_i \to Y_{u(i)})_{i\in I}$.

The functor $Fam(C) \to Set$ sends $(X_i)_{i\in I}$ to the set $I$, and a morphism as above to the function $u$. This is a Grothendieck fibration, and its fiber over a set $I$ is the power category $C^I$.

Remarks

• Concepts in the category theory of fibrations (or equivalently indexed categories) are generally defined so that when applied to family fibrations, they specialize to the corresponding notions in ordinary category theory.

• When $C=Set$, the family fibration is equivalent to the codomain fibration of $Set$.