nLab trivial topology

Idea

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.

Related concepts

• canonical topology

• subcanonical topology

• Grothendieck topology

Literature

• 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]