indecomposable if does not admit any non-trivial coproduct decompositions;

\(iii) $A$ is said to be irreducible if it is $J_{\cal E}$-irreducible, where $J_{\cal E}$ is the canonical topology on $\cal E$; in other words, if any sieve in $\cal E$ containing a small epimorphic family contains the identity on $A$;

\(iv) $A$ is said to be compact if every small covering family $\{A_{i} \to A \textrm{ | } i\in I \}$ contains a finite covering subfamily;

\(v) $A$ is said to be coherent if it is compact and, whenever we are given a morphism $f:B\to A$ with $B$ compact, the domain of the kernel-pair of $f$ is compact;

\(vi) $A$ is said to be supercompact if every small covering family $\{A_{i}\to A \textrm{ | } i\in I \}$ contains a cover;

\(vii) $A$ is said to be regular if it is supercompact and, whenever we are given a morphism $f:B\to A$ with $B$ supercompact, the domain of the kernel-pair of $f$ is supercompact.

Recall that an object in a locally connected topos is indecomposable if and only if it is connected (cfr. the discussion after the proof of Lemma C3.3.6 [@El2]).

\[Onetoposmany\_Rem\] *It readily follows from the definitions that every 