nLab trivial coverage

Definition

Definition

(trivial coverage)

For $\mathcal{C}$ a small category, the trivial coverage is the coverage with no covering families at all, meaning that the sheaf condition over the resulting site is empty, in that every presheaf is a sheaf for this coverage.

Hence the category of presheaves $[\mathcal{C}^{op},Set]$ over a site $\mathcal{C}_{triv}$ with trivial coverage is already the corresponding category of sheaves, hence the corresponding sheaf topos:

$$Sh\left( \mathcal{C}_{triv}\right) \;\simeq\; [\mathcal{C}^{op}, Set] \,.$$