Lemma 4.2 (0MUQ) · ScalingStacks

0MUQ

Lemma 4.2. Let $S$ be a compact object of a triangulated category $\mathcal{T}$ and suppose we have a sequence of objects and morphisms

X_{0}\stackrel{{\scriptstyle\alpha_{0}}}{{\longrightarrow}}X_{1}\stackrel{{% \scriptstyle\alpha_{1}}}{{\longrightarrow}}X_{2}\stackrel{{\scriptstyle\alpha_% {2}}}{{\longrightarrow}}X_{3}\stackrel{{\scriptstyle\alpha_{3}}}{{% \longrightarrow}}\cdots

such that $\textnormal{Hom}(S,\alpha_{n}):\textnormal{Hom}(S,X_{n})\rightarrow\textnormal% {Hom}(S,X_{n+1})$ is an isomorphism for each $n\geqslant 0$ . Then $\textnormal{Hom}(S,X_{0})\cong\textnormal{Hom}(S,\textnormal{hocolim}(X_{n}))$ .

0MUR

Proof: It is well-known that the filtered colimit, $\textnormal{colim}\,\textnormal{Hom}(S,X_{n})$ , is isomorphic to $\textnormal{Hom}(S,X_{0})$ . The assertion now follows by Lemma 3.5. $\Box$