Let \(U\) be a universe, let \(C\) be a \(U\)-logos, and let \(I\) be a partially ordered object in \(C\) with least and greatest elements. We say an object in \(C\) is \(I\)-categorically fibrant if it is internally local for the following morphisms.

• -\((\mathord {\textnormal {\textsf {d}}}^{2},\mathord {\textnormal {\textsf {d}}}^{0}):\Delta _{I}\lbrack 1\rbrack \mathbin {{}_{\mathord {\textnormal {\textsf {d}}}^{0}}\mathord {+_{\mathord {\textnormal {\textsf {d}}}^{1}}}}\Delta _{I}\lbrack 1\rbrack \rightarrow \Delta _{I}\lbrack 2\rbrack \)
• -\(\Delta _{I}\lbrack 3\rbrack \mathbin {{}_{(\mathord {\textnormal {\textsf {d}}}^{3}\circ \mathord {\textnormal {\textsf {d}}}^{1},\mathord {\textnormal {\textsf {d}}}^{0}\circ \mathord {\textnormal {\textsf {d}}}^{1})}\mathord {+_{\mathord {\textnormal {\textsf {s}}}^{0}+\mathord {\textnormal {\textsf {s}}}^{0}}}}(\Delta _{I}\lbrack 0\rbrack +\Delta _{I}\lbrack 0\rbrack )\rightarrow \Delta _{I}\lbrack 0\rbrack \)
• -\(((\mathord {\textnormal {\textsf {s}}}^{0},\mathord {\textnormal {\textsf {s}}}^{1}),(\mathord {\textnormal {\textsf {s}}}^{1},\mathord {\textnormal {\textsf {s}}}^{0})):\Delta _{I}\lbrack 2\rbrack \mathbin {{}_{\mathord {\textnormal {\textsf {d}}}^{1}}\mathord {+_{\mathord {\textnormal {\textsf {d}}}^{1}}}}\Delta _{I}\lbrack 2\rbrack \rightarrow \Delta _{I}\lbrack 1\rbrack \times \Delta _{I}\lbrack 1\rbrack \)