The Stacks project

Example 93.11.1 (Algebraic spaces). Let $\mathcal{F}$ be the category defined as follows

1. an object is a pair $(A, X)$ consisting of an object $A$ of $\mathcal{C}_\Lambda$ and an algebraic space $X$ flat over $A$, and

2. a morphism $(f, g) : (B, Y) \to (A, X)$ consists of a morphism $f : B \to A$ in $\mathcal{C}_\Lambda$ together with a morphism $g : X \to Y$ of algebraic spaces over $\Lambda$ such that

   $$\xymatrix{ X \ar[r]_ g \ar[d] & Y \ar[d] \\ \mathop{\mathrm{Spec}}(A) \ar[r]^ f & \mathop{\mathrm{Spec}}(B) }$$

   is a cartesian commutative diagram of algebraic spaces.

The functor $p : \mathcal{F} \to \mathcal{C}_\Lambda$ sends $(A, X)$ to $A$ and $(f, g)$ to $f$. It is clear that $p$ is cofibred in groupoids. Given an algebraic space $X$ over $k$, let $x_0 = (k, X)$ be the corresponding object of $\mathcal{F}(k)$. We set