Let $C$ be a small category. Consider the class of all Grothendieck topologies on $C$, it is a preorder with the relation "finer". Does this preorder has all infima and suprema?


(For example, how do you prove that there is always the canonical topology?)



Infima and Suprema of Grothendieck topologies
Yes. In fact, Grothendieck topologies on any small category constitute a locale.
See Proposition 3.2.13 in Borceux's [Handbook of Categorical Algebra 3](http://gen.lib.rus.ec/get?nametype=orig&md5=C1C3ADB25F2F79762F44CE46C69DF2BC).


