J-irreducible object of a site ($J(C)=\{yC\}$)

An object of a site is J-irreducible when its only covering sieve on a category is the maximal sieve. In particular the empty sieve cannot cover it. Every arrow from a J-irreducible object belongs to any covering sieve on its codomain: the pulled-back sieve must contain the identity. This makes the induced topology on the full subcategory of J-irreducible objects trivial.