Is there a notion of "base" $\mathcal{B}$ for a site, and "$\mathcal{B}$-sheaf", such that a presheaf is a sheaf iff it is a "$\mathcal{B}$-sheaf"?

Question

For a base $\mathcal{B}$ for the topology on a topological space $X$, there is the notion of a presheaf on $X$ being a "$\mathcal{B}$-sheaf", and a presheaf on $X$ is a sheaf iff it is a $\mathcal{B}$-sheaf.

I was wondering, is there a similar notion for sites? I.e., is there such a thing as a "base" $\mathcal{B}$ for a site or a Grothendieck topology, and a notion of a presheaf being a $\mathcal{B}$-sheaf which depends only on the values of the presheaf on $\mathcal{B}$, such that a presheaf is a sheaf iff it is a $\mathcal{B}$-sheaf.

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Large sites provide some motivating examples; e.g., is it possible to say (perhaps with some conditions) that a sheaf on the site of smooth manifolds is determined by e.g. its values on just the Euclidean/Cartesian spaces?

Answer 1

As user Zhen Lin explains in the comments, the notion I was looking for is a dense sub-site.

Answer 2

There is the notion of a coverage. In general the gluing conditions for a presheaf being a (separated pre-)sheaf, or more generally for a fibration to be a (pre)stack, are distinct conditions for each family of morphisms $U_i\to X$. Being a sheaf (or a stack) for a site then means being a sheaf (or a stack) for certain families of morphisms $U_i\to X$. For a site given by a Grothendieck topology, these are exactly the families that generate the covering sieves (the sieve generated by $\{U_i\to X\}$ consists of all $V\to X$ that factor through some $U_i\to X$). The notion of coverage is then a collection of families $U_i\to X$ with the property that the sieves they generate satisfy the axioms of a Grothendieck topology.

Explicitly, a coverage is a collection of families $\{U_i\to X\}$, called covering families, such that for any morphism $X\to Y$ and any covering family $\{V_j\to Y\}$, there exists a covering family $\{U_i\to X\}$ such that each composite $U_i\to X\to Y$ of $X\to Y$ with a morphism $U_i\to X$ in the covering family $\{U_i\to X\}$ factors through some morphism $V_j\to Y$ in the covering family $\{V_j\to Y\}$.