Proposition 10.2.6.21 (04TG)

Proposition 10.2.6.21. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category, let $n$ be an integer, let $\operatorname{Fun}'( \operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}_{+}^{\operatorname{op}}), \operatorname{\mathcal{C}})$ be the full subcategory of $\operatorname{Fun}( \operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}_{+}^{\operatorname{op}}), \operatorname{\mathcal{C}})$ spanned by the $n$-coskeletal augmented simplicial objects of $\operatorname{\mathcal{C}}$, and let $\operatorname{Fun}'( \operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}_{\mathrm{min}}^{\operatorname{op}} ), \operatorname{\mathcal{C}}) \subseteq \operatorname{Fun}( \operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}_{\mathrm{min}}^{\operatorname{op}} ), \operatorname{\mathcal{C}})$ be its inverse image. Then precomposition with the concatenation functor $C_{+}$ induces a trivial Kan fibration

\[ \theta : \operatorname{Fun}'( \operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}_{\mathrm{min}}^{\operatorname{op}}), \operatorname{\mathcal{C}}) \rightarrow \operatorname{Fun}'( \operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}^{\leq n+1}_{\mathrm{min}})^{\operatorname{op}}, \operatorname{\mathcal{C}}) \times _{ \operatorname{Fun}( \operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}^{\leq n}_{+})^{\operatorname{op}}, \operatorname{\mathcal{C}}) } \operatorname{Fun}'( \operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}_{+}^{\operatorname{op}}), \operatorname{\mathcal{C}}). \]

Proof. Let $\operatorname{Fun}'( \operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}^{\leq n}_{+})^{\operatorname{op}}, \operatorname{\mathcal{C}})$ denote the full subcategory of $\operatorname{Fun}( \operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}^{\leq n}_{+})^{\operatorname{op}}, \operatorname{\mathcal{C}})$ spanned by those functors which can be extended to $n$-coskeletal augmented simplicial objects of $\operatorname{\mathcal{C}}$, and define $\operatorname{Fun}'( \operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}^{\leq n+1}_{\mathrm{min}})^{\operatorname{op}}, \operatorname{\mathcal{C}})$ similarly. We then have a commutative diagram

\[ \xymatrix@R =50pt@C=50pt{ \operatorname{Fun}'( \operatorname{N}_{\bullet }(\operatorname{{\bf \Delta }}_{\mathrm{min}}^{\operatorname{op}} ), \operatorname{\mathcal{C}}) \ar [r] \ar [d] & \operatorname{Fun}'( \operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}_{+}^{\operatorname{op}}), \operatorname{\mathcal{C}}) \ar [d] \\ \operatorname{Fun}'( \operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}^{\leq n+1}_{\mathrm{min}})^{\operatorname{op}}, \operatorname{\mathcal{C}}) \ar [r] \ar [d] & \operatorname{Fun}'( \operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}^{\leq n}_{+})^{\operatorname{op}}, \operatorname{\mathcal{C}}) \ar [d] \\ \operatorname{Fun}( \operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}^{\leq n+1}_{\mathrm{min}})^{\operatorname{op}}, \operatorname{\mathcal{C}}) \ar [r] & \operatorname{Fun}( \operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}^{\leq n}_{+})^{\operatorname{op}}, \operatorname{\mathcal{C}}). } \]

Combining Lemma 10.2.6.19 with Corollary 7.3.6.15, we deduce that the upper vertical maps are trivial Kan fibrations; in particular, the upper half of the diagram is a categorical pullback square (Proposition 4.5.2.21). Variant 10.2.6.20 guarantees that the lower half of the square is a pullback diagram. Since bottom horizontal map is an isofibration of $\infty $-categories (Corollary 4.4.5.3), it is a categorical pullback square (Corollary 4.5.2.27). It follows that the outer rectangle is a categorical pullback square (Proposition 4.5.2.18), so that $\theta $ is an equivalence of $\infty $-categories (Proposition 4.5.2.26). Corollary 4.4.5.3 guarantees that $\theta $ is also an isofibration, so it is a trivial Kan fibration (Proposition 4.5.5.20). $\square$