I'm working to understand the Grothendieck topology version of the Zariski topology of schemes. Explained simply, it replaces the notion of "open subschemes" with "open immersions". So instead of $U\subseteq X$, we have $U\hookrightarrow X$.
The intersection $U\cap V$ between two open subschemes is replaced with the canonical immersion of the fiber product $(U\times_X V)\hookrightarrow X$. Is there a correspondingly simple analogue of the union, or do I have to construct it explicitly?
On
Maybe the following can help motivate this definition. In the case of $\mathbb{R}$-valued functions on a topological space $X$, if $\mathcal{U}$ is a covering sieve on $X$, so that the union of its open sets is $X$, then defining a function $\mathcal{U}$-locally on X is equivalent to defining a continuous function on $X$ (SGA 4 1/2, I, 2.3). One also defines a notion of vector bundles given $\mathcal{U}$-locally, and in this case the corresponding statement is that if $\mathcal{U}$ is a covering sieve, then there is an equivalence between the categories of vector bundles on $X$ and vector bundles given $\mathcal{U}$-locally (SGA 4 1/2, I, 3.3). And finally in the case of (flat) schemes over $X$, the corresponding statement is that if $\mathcal{U}$ is the sieve generated by a family $\{U_i\}$ of flat schemes over $X$, and the union of the images of $U_i$ is $X$, then there is an equivalence between the categories of quasi-coherent modules on $X$ and quasi-coherent modules given $\mathcal{U}$-locally (SGA 4 1/2, I, 4.5).
A covering of $U$ is replaced by a family of morphisms $U_i\to U$ the union of whose images is $U$.