coherent (resp. regular) object is compact (resp. supercompact), every atom is an indecomposable object, every irreducible object is supercompact, every supercompact object is indecomposable.*

Let us recall from [@El2] the following terminology.

A site $({\cal C}, J)$ is said to be locally connected if every $J$-covering sieve is connected i.e. for any $R\in J(c)$, $R$ is connected as a full subcategory of ${\cal C}\slash c$.

A site $({\cal C}, J)$ is said to be atomic if $\cal C$ satisfies the right Ore condition and $J$ is the atomic topology on $\cal C$.

Given a site $({\cal C}, J)$,

\(i) we say an object $c$ of $\cal C$ is $J$-irreducible if the only $J$-covering sieve on $c$ is the maximal sieve $M(c)$;

\(ii) we say $J$ is rigid if, for every $c\in {\cal C}$, the family of all morphisms from $J$-irreducible objects to $c$ generates a $J$-covering sieve. The site $({\cal C}, J)$ is said to be rigid if $J$ is rigid as a Grothendieck topology on $\cal C$.

We will say that a site $({\cal C}, J)$ is coherent (resp. regular) if $\cal C$ is cartesian and $J$ is a finite-type Grothendieck topology on $\cal C$ (resp. a Grothendieck topology 