Proposition 3.5.9.24 (056Q)

Proposition 3.5.9.24. Let $f: X \rightarrow Y$ be an $n$-truncated Kan fibration between Kan complexes, let $k$ be an integer, and let $j: A \rightarrow B$ be a morphism of simplicial sets which is $(k-1)$-connective. Then the induced map

\[ \theta : \operatorname{Fun}(B, X ) \rightarrow \operatorname{Fun}(A, X) \times _{ \operatorname{Fun}(A,Y) } \operatorname{Fun}(B, Y) \]

is $(n-k)$-truncated.

Proof. Using Proposition 3.1.7.1, we can factor $j$ as a composition $A \xrightarrow {i} A' \xrightarrow {j} B$, where $i$ is anodyne and $j$ is a Kan fibration. In this case, $\theta $ factors as a composition

\[ \operatorname{Fun}(B, X) \xrightarrow {\theta '} \operatorname{Fun}(A', X) \times _{ \operatorname{Fun}(A',Y) } \operatorname{Fun}(B, Y) \xrightarrow {\rho } \operatorname{Fun}(A, X) \times _{ \operatorname{Fun}(A,Y) } \operatorname{Fun}(B, Y), \]

where $\rho $ is a trivial Kan fibration (Theorem 3.1.3.5). It will therefore suffice to show that $\theta '$ is $(n-k)$-truncated. Using Corollary 3.5.2.4 (or Exercise 3.1.7.11, in the case $k = 0$), we can factor $j'$ as a composition $A' \xrightarrow {\widetilde{j}} \widetilde{B} \xrightarrow {q} B$, where $\widetilde{j}$ is a monomorphism which is bijective on simplices of dimension $\leq k-1$ and $q$ is a trivial Kan fibration. In this case, we have a commutative diagram

\[ \xymatrix@R =50pt@C=50pt{ \operatorname{Fun}(B, X) \ar [r]^-{\theta '} \ar [d]& \operatorname{Fun}(A',X) \times _{ \operatorname{Fun}(A', Y) } \operatorname{Fun}(B, Y) \ar [d] \\ \operatorname{Fun}(\widetilde{B}, X) \ar [r]^-{\widetilde{\theta }} & \operatorname{Fun}(A', X) \times _{ \operatorname{Fun}(A',Y) } \operatorname{Fun}(\widetilde{B}, Y) } \]

where the vertical maps are homotopy equivalences. Consequently, to prove that $\theta $ is $(n-k)$-truncated, it will suffice to show that $\widetilde{\theta }$ is $(n-k)$-truncated. We may therefore replace $j$ by $\widetilde{j}$ and thereby reduce to proving Proposition 3.5.9.24 in the special case where $j$ is a monomorphism which is bijective on simplices of dimension $\leq k-1$.

If $j$ is a monomorphism, then $\theta $ is a Kan fibration (Theorem 3.1.3.1). Consequently, to show that $\theta $ is $(n-k)$-connective, it will suffice to show that every lifting problem

\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\partial \Delta }^{m} \ar [r] \ar [d] & \operatorname{Fun}(B, X ) \ar [d]^{\theta } \\ \Delta ^{m} \ar@ {-->}[ur] \ar [r] & \operatorname{Fun}(A, X) \times _{ \operatorname{Fun}(A,Y) } \operatorname{Fun}(B, Y) } \]

has a solution, provided that $m \geq n-k+2$ (Corollary 3.5.9.23) Unwinding the definitions, we can rewrite this as a lifting problem

\[ \xymatrix@R =50pt@C=50pt{ A \ar [r] \ar [d]^{j} & \operatorname{Fun}( \Delta ^{m}, X) \ar [d]^{\theta '} \\ B \ar@ {-->}[ur] \ar [r] & \operatorname{Fun}( \operatorname{\partial \Delta }^{m}, X) \times _{ \operatorname{Fun}( \operatorname{\partial \Delta }^{m}, Y) } \operatorname{Fun}( \Delta ^{m}, Y). } \]

Since $f$ is a Kan fibration, $\theta '$ is also a Kan fibration (Theorem 3.1.3.1), and our assumption that $f$ is $n$-truncated guarantees that $\theta '$ is $(n-m)$-truncated (Variant 3.5.9.19). In particular, $\theta '$ is $(k-2)$-truncated (Remark 3.5.9.6), so the existence of the desired solution follows from Corollary 3.5.9.23. $\square$