Endomorphism spectra

Proof. We may assume that \(D\) is small by enlarging the ambient universe if necessary. Consider the fully faithful Yoneda embedding \[ D \hookrightarrow \widehat D := \Ind (D). \] The \(\infty \)-category \(\widehat D\) is presentable and stable, and the inclusion of \(D\) is exact; see Proposition 22.3.6 and [Lurie (2017), Proposition 1.1.3.6]. In particular, it preserves mapping spectra. Being presentable and stable, \(\widehat D\) is \(\Sp \)-local, so Theorem 18.5.3, Proposition 18.5.2 equip it with a unique \(\Sp \)-module structure in \(\PrL \); in particular \(\widehat D\) is canonically left-tensored over \(\Sp \). The resulting action functor \(-\otimes M\colon \Sp \to \widehat D\) is colimit-preserving and satisfies \(\S \otimes M \simeq M\), so by Theorem 4.4.14 it agrees with the tensoring of Proposition 4.4.9; its right adjoint is therefore \(\hom _{\widehat D}(M,-)\), and for spectra \(E\) there are natural isomorphisms \[ \Hom _{\Sp }(E,\hom _D(M,M)) \iso \Hom _{\widehat D}(E \otimes M,M). \] Consequently, \(A:=\hom _D(M,M)\) is an endomorphism object of \(M \in \widehat D\). Applying Corollary 19.4.7 gives \(A\) a preferred associative algebra structure and equips \(M\) with a preferred left \(A\)-module structure. This proves (1).

Using this module structure, we may form the two-sided bar construction in \(\widehat D\), just as in Section 19.2; see also [Lurie (2017), Construction 4.4.2.7]. Its geometric realization defines a colimit-preserving functor \[ F := - \otimes _A M\colon \RMod _A \to \widehat D. \] Since both categories are presentable, Theorem 22.2.5 gives a right adjoint \[ G\colon \widehat D \to \RMod _A. \] Let \(U\colon \RMod _A \to \Sp \) denote the forgetful functor. For \(K \in \Sp \) and \(X \in \widehat D\), the free-right-module adjunction and the tensor–Hom adjunction give natural isomorphisms \begin {align*} \Hom _{\Sp }(K,UG(X)) &\simeq \Hom _{\RMod _A}(K \otimes A,G(X)) \\ &\simeq \Hom _{\widehat D}((K \otimes A) \otimes _A M,X) \\ &\simeq \Hom _{\widehat D}(K \otimes M,X) \\ &\simeq \Hom _{\Sp }(K,\hom _{\widehat D}(M,X)). \end {align*}

The Yoneda lemma therefore provides a natural isomorphism \[ UG(X) \simeq \hom _{\widehat D}(M,X). \] Restricting \(G\) along \(D \hookrightarrow \widehat D\) gives the lift in (2). □