Definition 3.4 (0MUK) · ScalingStacks

Definition 3.4. Let $\mathcal{T}$ be a triangulated category with set indexed coproducts. Let

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

be a sequence of objects and morphisms in $\mathcal{T}$ . The homotopy colimit $\textnormal{hocolim}(X_{i})$ is constructed by extending the map

\coprod_{i=0}^{\infty}X_{i}\stackrel{{\scriptstyle 1-\rm{shift}}}{{% \longrightarrow}}\coprod_{i=0}^{\infty}X_{i}

to a distinguished triangle:

\coprod_{i=0}^{\infty}X_{i}\stackrel{{\scriptstyle 1-\rm{shift}}}{{% \longrightarrow}}\coprod_{i=0}^{\infty}X_{i}\longrightarrow\textnormal{hocolim% }(X_{i})\longrightarrow\Sigma\coprod_{i=0}^{\infty}X_{i}{}{}.