Stabilization as a right adjoint

Proof. Let \(D\) be a stable \(\infty \)-category. We need to show that postcomposition with \(\Omega ^{\infty }\) induces an equivalence \[ \Fun ^{\lex }(D,\Sp (C)) \iso \Fun ^{\lex }(D,C). \]

We construct an inverse. Let \(F\colon D \to C\) be left exact. Since \(D\) is pointed, the functor \(F\) has a canonical lift to \(C_*\): the object \(F(d)\) is pointed by the map \(* \simeq F(0) \to F(d)\) induced by \(0 \to d\). For every \(n \geq 0\), define a functor \[ \widetilde {F}_n\colon D \to C_*, \qquad \widetilde {F}_n(d) := F(d[n]). \] The unit equivalences \(d[n] \to \Omega _Dd[n+1]\), together with left exactness of \(F\), provide natural isomorphisms \[ \widetilde {F}_n \iso \Omega \widetilde {F}_{n+1}. \] Explicitly, these are obtained by applying \(F\) to the unit equivalences and then using the natural isomorphisms \(F(\Omega _Dd[n+1]) \iso \Omega F(d[n+1])\) provided by left exactness. Thus the functors \(\widetilde {F}_n\) and these structure isomorphisms assemble into a functor \[ \widetilde {F}\colon D \to \Sp (C). \] This construction is natural in \(F\), and hence defines a functor \(\Fun ^{\lex }(D,C)\to \Fun ^{\lex }(D,\Sp (C))\). Since limits in \(\Sp (C)\) are computed levelwise, and since each shift \(d\mapsto d[n]\) is an equivalence of \(D\), the functor \(\widetilde {F}\) is left exact.

The composite \(\Omega ^{\infty }\widetilde {F}\) is \(F\), since evaluation at level zero gives \(\widetilde {F}(d)_0=F(d)\). Conversely, let \(G\colon D \to \Sp (C)\) be left exact. Since \(D\) and \(\Sp (C)\) are stable, \(G\) is exact by Corollary 4.2.27. Hence \[ G(d[n]) \simeq G(d)[n]. \] By Corollary 4.3.15, evaluation at level zero gives natural isomorphisms \[ (\Omega ^{\infty }G)(d[n]) = G(d[n])_0 \simeq G(d)_n. \] These isomorphisms are compatible with the structure maps, by naturality of the unit \(d[n]\to \Omega _Dd[n+1]\) and of the equivalence between suspension and shift in \(\Sp (C)\). Thus \(\widetilde {\Omega ^{\infty }G}\simeq G\), naturally in \(G\). □