Let $U$ be a universe.

• -$(\mathord {\textnormal {\textsf {Q}}}(\mathord {\textnormal {\textsf {s}}}^{1}),\mathord {\textnormal {\textsf {Q}}}(\mathord {\textnormal {\textsf {s}}}^{0})):\mathord {\textnormal {\textsf {Q}}}(\Delta \lbrack 2\rbrack )\rightarrow \mathord {\textnormal {\textsf {Q}}}(\Delta \lbrack 1\rbrack )\times \mathord {\textnormal {\textsf {Q}}}(\Delta \lbrack 1\rbrack )$ determines a partial order in $\mathord {\textnormal {\textsf {Topos}}}(U)$. We refer to this partially ordered object in $\mathord {\textnormal {\textsf {Topos}}}(U)$ as $\mathbb {I}$.
• -$\mathord {\textnormal {\textsf {Q}}}(\mathord {\textnormal {\textsf {d}}}^{1}):\mathord {\textnormal {\textsf {Q}}}(\Delta \lbrack 0\rbrack )\rightarrow \mathord {\textnormal {\textsf {Q}}}(\Delta \lbrack 1\rbrack )$ determines the least element of $\mathbb {I}$.
• -$\mathord {\textnormal {\textsf {Q}}}(\mathord {\textnormal {\textsf {d}}}^{0}):\mathord {\textnormal {\textsf {Q}}}(\Delta \lbrack 0\rbrack )\rightarrow \mathord {\textnormal {\textsf {Q}}}(\Delta \lbrack 1\rbrack )$ determines the greatest element of $\mathbb {I}$.
• -$\mathord {\textnormal {\textsf {0}}}\simeq \mathord {\textnormal {\textsf {Q}}}(\Delta \lbrack 0\rbrack )\mathbin {{}_{\mathord {\textnormal {\textsf {Q}}}(\mathord {\textnormal {\textsf {d}}}^{1})}\mathord {\times _{\mathord {\textnormal {\textsf {Q}}}(\mathord {\textnormal {\textsf {d}}}^{0})}}}\mathord {\textnormal {\textsf {Q}}}(\Delta \lbrack 0\rbrack )$
• -Let $n\ge 3$ be an integer. Then the morphism $(\mathord {\textnormal {\textsf {Q}}}(\mathord {\textnormal {\textsf {s}}}^{n-1}),{(\mathord {\textnormal {\textsf {Q}}}(\mathord {\textnormal {\textsf {s}}}^{0}))}^{n-2}):\mathord {\textnormal {\textsf {Q}}}(\Delta \lbrack n\rbrack )\rightarrow \mathord {\textnormal {\textsf {Q}}}(\Delta \lbrack n-1\rbrack )\mathbin {{}_{{(\mathord {\textnormal {\textsf {Q}}}(\mathord {\textnormal {\textsf {s}}}^{0}))}^{n-2}}\mathord {\times _{\mathord {\textnormal {\textsf {Q}}}(\mathord {\textnormal {\textsf {s}}}^{1})}}}\mathord {\textnormal {\textsf {Q}}}(\Delta \lbrack 2\rbrack )$ is an equivalence. (This means that $\mathord {\textnormal {\textsf {Q}}}(\Delta \lbrack n\rbrack )$ is the object of increasing sequences in $\mathbb {I}$ of length $n$).