Definition 8.5.6.14. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $e: X \rightarrow X$ be an endomorphism in $\operatorname{\mathcal{C}}$. We will say that $e$ is weakly split if it satisfies the following conditions:
- $(1)$
The diagram $T_{e}$ of Notation 8.5.6.11 can be extended to a limit diagram in $\operatorname{\mathcal{C}}$, which we depict as
\[ \xymatrix@R =50pt@C=30pt{ & & & Y \ar [dll] \ar [dl] \ar [d]^{ i} \ar [dr] \ar [drr] & & & \\ \cdots \ar [r] & X \ar [r]^-{e} & X \ar [r]^-{ e} & X \ar [r]^-{e} & X \ar [r]^-{e} & X \ar [r]^-{e} & \cdots } \]- $(2)$
The diagram $T_{e}$ of Notation 8.5.6.11 can be extended to a colimit diagram in $\operatorname{\mathcal{C}}$, which we depict as
\[ \xymatrix@R =50pt@C=30pt{ \cdots \ar [r] & X \ar [r]^-{e} \ar [drr] & X \ar [r]^-{ e} \ar [dr] & X \ar [r]^-{e} \ar [d]^{r} & X \ar [r]^-{e} \ar [dl] & X \ar [r]^-{e} \ar [dll] & \cdots \\ & & & Z. & & & } \]- $(3)$
The composition $Y \xrightarrow {i} X \xrightarrow {r} Z$ is an isomorphism in $\operatorname{\mathcal{C}}$.