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 \(n\)).