In sheaf and topos theory, given any small category, its *trivial topology* is the coverage whose covering families are the identity morphisms, hence the Grothendieck topology for which only the sieves generated by identity morphisms are covering (i.e. those containing every morphism with the given codomain).

The sheaves for the trivial topology are precisely the presheaves on the underlying category. In this way every category of presheaves is realized as a category of sheaves and hence as a Grothendieck topos.

- Francis Borceux, Example 3.2.14.a in:
*Handbook of Categorical Algebra*, Vol. 3:*Categories of Sheaves*, Encyclopedia of Mathematics and its Applications**50**, Cambridge University Press (1994) [doi:10.1017/CBO9780511525872]

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