Kerodon

$\Newextarrow{\xRightarrow}{5,5}{0x21D2}$
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$

Remark 1.2.6.6. In the proof of Proposition 1.2.6.5, we have implicitly invoked the fact that every category $\operatorname{\mathcal{C}}$ satisfies the generalized associative law: every sequence of composable morphisms

\[ X_0 \xrightarrow {f_1} X_1 \xrightarrow {f_2} X_2 \rightarrow \cdots \xrightarrow {f_ n} X_ n \]

has a well-defined composition $f_ n \circ f_{n-1} \circ \cdots \circ f_1$, which can be computed in terms of the binary composition law by inserting parentheses arbitrarily. One might object that this logic is circular: the generalized associative law is essentially equivalent to Proposition 1.2.6.5 (applied to the directed graph $G$ described in Example 1.2.6.2). In ยง1.4.7, we will establish an $\infty $-categorical generalization of Proposition 1.2.6.5 (Theorem 1.4.7.1), whose proof will avoid this sort of circular reasoning (see Remark 1.4.7.4).