fiber-of

Let $F : \mathcal{C}^{\mathrm{op}} \to \mathbf{Set}$ be a functor (presheaf) and let $t$ be an object of $\mathcal{C}$.

Definition. The fiber of $F$ at $t$ is the set $F(t)$ — the value of the functor $F$ evaluated at the object $t$.

The two fields fiber-of: and base: are inseparable: $F(t)$ requires specifying both $F$ and $t$.

Proposition. For a morphism $g : t' \to t$ in $\mathcal{C}$, functoriality gives a restriction map $F(g) : F(t) \to F(t')$. The fiber $F(t)$ together with these restriction maps determines the local behavior of $F$ at $t$.

Distinction from related notions:

• component-of: X — structural projection $\pi : X \to A$ (a retraction with named section)
• fiber-of: F / base: t — categorical fiber: evaluation of a functor at an object
• element-of: X — algebraic membership (global element $1 \to X$)

Open questions

• Whether fiber-of should also require the restriction maps (naturality squares) to be named.