So on nLab the definition of a trivial (Grothendieck) topology is the following: "The Grothendieck topology on any category for which only the identity morphisms are covering is the trivial topology. Its sheaves are all the presheaves."


I am having trouble understanding what is meant exactly by only the "identity morphisms are covering." After reading a bit, I was under the impression that those are of the form $\hom(-,X)$, but then I don't understand what the link is with what I traditionally understand as "identity morphisms", which to me are of the form $1\_X$ for objects $X\in\operatorname{Obj}(C)$. Am I missing something obvious or is the definition of identity morphisms different in this context?



Trivial Grothendieck topology and identity morphisms
Technically, a Grothendieck topology is specified by its covering sieves, not its covers, so it would have been more accurate to say the covering sieves of the trivial topology are those generated by identity morphisms. The cover of $X$ by the sieve $hom(-,X)$ is indeed the sieve generated by the identity morphism of $X$. Taken literally, most categories do not admit a Grothendieck topology in which the *only* covers are identity maps, since whenever an identity morphism is a cover so is every isomorphism with the same target.


