Excisive approximation

Proof. For the reduction functor, let \(F\colon \An _*^{\fin }\to C\) be arbitrary. Since \(*\) is initial in \(\An _*^{\fin }\), the unique natural transformation \(\const _*\to \id \) induces \(\const _{F(*)}\to F\). Define \[ F^{\mathrm {red}}(X):=*\sqcup _{F(*)}F(X). \] This is reduced. If \(G\) is reduced, every natural transformation \(F\to G\) vanishes on \(F(*)\), and hence factors uniquely through \(F\to F^{\mathrm {red}}\). Thus \((-)^{\mathrm {red}}\) is left adjoint to the inclusion of reduced functors.

For \(P_1\), we may replace \(C\) by \(C_*\), which inherits finite and sequential colimits from \(C\), so assume \(C\) is pointed. Write \[ T(F):=\Omega F\Sigma \] for the endofunctor of \(\Fun _*(\An _*^{\fin },C)\), equipped with the natural transformation \(\eta \colon \id \to T\) from Construction 16.5.4. We first record the coherence needed below. For a finite set \(S\) and \(X\in \An _*^{\fin }\), let \(K_S(X)\) be the colimit of the star-shaped diagram with central object \(X\) and one map \(X\to 0\) for every element of \(S\). Thus \[ K_{\emptyset }(X)=X,\qquad K_{\{s\}}(X)=0,\qquad K_{\{0,1\}}(X)=\Sigma X. \] Injections of finite sets induce maps between these pointed cones. Since finite colimits commute with each other, interchanging the two colimit coordinates gives natural isomorphisms \(K_SK_{S'}\simeq K_{S'}K_S\), coherently in \(S\) and \(S'\). This simultaneous interchange is the coherence needed below; it does not require choosing an order in which to identify the two suspension coordinates. Moreover, \[ T(F)(X)\simeq \lim _{\emptyset \neq S\subseteq \{0,1\}}F(K_S(X)) \simeq F(0)\times _{F(\Sigma X)}F(0) \simeq \Omega F(\Sigma X). \] Under this description, the maps \(K_{\emptyset }(X)\to K_S(X)\) induce \(\eta _F\). The isomorphisms which interchange \(K_S\) and \(K_{S'}\) therefore identify the two natural transformations \[ T(\eta _F),\eta _{T(F)}\colon T(F)\longrightarrow T^2(F). \]

Now let \(F\colon \An _*^{\fin }\to C\) be reduced. Iterate \(\eta \) to form \[ F \xrightarrow {\eta _F} T(F) \xrightarrow {\eta _{T(F)}} T^2(F) \longrightarrow \cdots \] and define \[ P_1(F):=\colim _{n\geq 0}T^n(F). \] Thus \(T^n(F)\simeq \Omega ^nF\Sigma ^n\), with the transition maps specified coherently by the pointed-cone construction.

The functor \(T\) preserves sequential colimits by assumption. Consequently, \[ T(P_1(F))\simeq \colim _{n\geq 0}T^{n+1}(F), \] and under this isomorphism \(\eta _{P_1(F)}\) is the map from the defining telescope to its tail. The shift \(n\mapsto n+1\) is cofinal, so this map is an isomorphism. The excisiveness criterion now shows that \(P_1(F)\) is excisive.

It remains to verify the universal property. The coherence \(T(\eta _G)\simeq \eta _{T(G)}\) identifies the sequential diagram defining \(P_1(T(G))\) with the tail of the diagram defining \(P_1(G)\), compatibly with \[ P_1(\eta _G)\colon P_1(G)\longrightarrow P_1(T(G)). \] Thus \(P_1(\eta _G)\) corresponds to the cofinal tail map and is an isomorphism. Moreover, \(P_1\) preserves sequential colimits, since \(T\) does.

Let \(u_F\colon F\to P_1(F)\) be the map into the telescope. Since \(P_1\) preserves sequential colimits, the map \[ P_1(u_F)\colon P_1(F)\longrightarrow P_1(P_1(F)) \] is the sequential colimit of the maps \(P_1(F)\to P_1(T^n(F))\). Each of these is a composite of the isomorphisms \(P_1(\eta _{T^k(F)})\), so \(P_1(u_F)\) is an isomorphism. If \(G\) is already excisive, the excisiveness criterion shows that every map in the defining telescope for \(P_1(G)\) is an isomorphism, so \(u_G\colon G\to P_1(G)\) is an isomorphism. In particular, \(u_{P_1(F)}\) is an isomorphism. By Proposition 21.8.9, the functor \(P_1\) is left adjoint to the inclusion of reduced excisive functors. □