Remark 5.70.
Let \(\varphi\colon T \to S\) be a morphism of logoi with kernel \(K = \ker(\varphi)\). Combining the quotient triple factorization with the image factorization gives
\[T \xrightarrow{(1)} T/ K^{\mono}
\xrightarrow{(2)} T/K
\xrightarrow{(3)} \lra{\varphi(T)}
\xhookrightarrow{(4)} S .\]
The four factors isolate the following properties:
\[\begin{array}{c|c|c}
\text{factor} & \text{property of the logos morphism} & \text{kernel or image condition} \\
\hline
(1) & \text{monogenic quotient} & \ker=K^{\mono} \\
(2) & \text{epigenic quotient} & \ker\subseteq\Conn_\infty \\
(3) & \text{conservative and algebraic} & \text{its image generates the target} \\
(4) & \text{fully faithful} & \text{inclusion of the image logos}.
\end{array}\]
The composite \((1)(2)\) is the quotient part of \(\varphi\), while \((3)(4)\) is its conservative part. The composite \((1)(2)(3)\) is algebraic in Lurie's terminology. Further terminology for composites, especially variants of “surjective”, depends on whether one works with classical or higher topoi, so we will use the explicit properties above.