Remark 10.2.6.26 (04SS)

Remark 10.2.6.26. Let $\iota : \operatorname{{\bf \Delta }}_{\mathrm{min}} \hookrightarrow \operatorname{{\bf \Delta }}$ denote the inclusion functor, and let

\[ C: \operatorname{{\bf \Delta }}\rightarrow \operatorname{{\bf \Delta }}_{\mathrm{min}} \quad \quad [n] \mapsto [0] \star [n] = [n+1] \]

denote the concatenation functor of Remark 10.2.6.2. There is a natural transformation $\eta : \operatorname{id}_{ \operatorname{{\bf \Delta }}} \rightarrow \iota \circ C$, which carries each object $[n] \in \operatorname{{\bf \Delta }}$ to the inclusion map

\[ [n] \hookrightarrow [n+1] \quad \quad i \mapsto i+1. \]

If $X_{\bullet }$ is a simplicial object of an $\infty $-category $\operatorname{\mathcal{C}}$, then composition with $\eta $ determines a natural transformation of simplicial objects $T_{\bullet }: \operatorname{Dec}(X)_{\bullet } \rightarrow X_{\bullet }$, given termwise by the face operator $\operatorname{Dec}(X)_{n} = X_{n+1} \xrightarrow { d^{n+1}_{0} } X_{n}$.

The natural transformation $\eta $ is the unit of an adjunction between $\iota $ and $C$; it admits a compatible counit $\epsilon : C \circ \iota \rightarrow \operatorname{id}_{ \operatorname{{\bf \Delta }}_{ \mathrm{min}} }$, which carries each object $[n]$ to the quotient map

\[ [n+1] \twoheadrightarrow [n] \quad \quad i \mapsto \mathrm{max}(0, i-1). \]

We therefore have a commutative diagram

10.17

\begin{equation} \begin{gathered}\label{equation:comparison-map-with-decalage} \xymatrix@C =50pt@R=50pt{ & C \circ \iota \circ C \ar [dr]^{ \operatorname{id}_ C \circ \epsilon } & \\ C \ar [ur]^{ \eta \circ \operatorname{id}_{C} } \ar [rr]^{ \operatorname{id}_{C} } & & C } \end{gathered} \end{equation}

in the functor category $\operatorname{Fun}( \operatorname{{\bf \Delta }}, \operatorname{{\bf \Delta }}_{\mathrm{min}} )$. If $\overline{X}$ is a splitting of the simplicial object $X_{\bullet }$, then precomposition with (10.17) determines a commutative diagram

\[ \xymatrix@C =50pt@R=50pt{ & \operatorname{Dec}( X )_{\bullet } \ar [dr]^{ T_{\bullet } } & \\ X_{\bullet } \ar [ur]^{ h_{\bullet } } \ar [rr]^{ \operatorname{id}} & & X_{\bullet } } \]

in the $\infty $-category of simplicial objects $\operatorname{Fun}( \operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}^{\operatorname{op}} ), \operatorname{\mathcal{C}})$. Here $h_{\bullet }$ is given termwise by the extra degeneracy map $h_{n}: X_{n} \rightarrow X_{n+1} = \operatorname{Dec}(X)_{n}$ appearing in Remark 10.2.6.4. In particular, if $X_{\bullet }$ is a split simplicial object of $\operatorname{\mathcal{C}}$, then it is a retract of the decalage $\operatorname{Dec}(X)_{\bullet }$.