Kerodon

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

Remark 7.6.1.5. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $\{ Y_ i \} _{i \in I}$ be a collection of objects of $\operatorname{\mathcal{C}}$, which we will identify with a diagram

\[ F: I \rightarrow \operatorname{\mathcal{C}}\quad \quad F(i) = Y_ i \]

indexed by the constant simplicial set associated to $I$ (Remark 1.1.4.3). Suppose we are given another object $Y \in \operatorname{\mathcal{C}}$ together with a collection of morphisms $\{ q_ i: Y \rightarrow Y_ i \} _{i \in I}$. The following conditions are equivalent:

$(1)$

The collection of morphisms $\{ q_ i \} _{i \in I}$ exhibits $Y$ as a product of the collection $\{ Y_ i\} _{i \in I}$, in the sense of Definition 7.6.1.3.

$(2)$

Let $\underline{Y}: I \rightarrow \operatorname{\mathcal{C}}$ denote the constant diagram taking the value $Y$, so that the collection $\{ q_ i \} _{i \in I}$ can be identified with a natural transformation $q: \underline{Y} \rightarrow F$. Then $q$ exhibits $Y$ as a limit of the diagram $F$, in the sense of Definition 7.1.1.1.

$(3)$

Let $\overline{F}: I^{\triangleleft } \rightarrow \operatorname{\mathcal{C}}$ be the diagram carrying each edge $\{ i\} ^{\triangleleft } \subseteq I^{\triangleleft }$ to the morphism $q_ i$. Then $\overline{F}$ is a limit diagram in $\operatorname{\mathcal{C}}$, in the sense of Definition 7.1.2.4.

The equivalence $(1) \Leftrightarrow (2)$ is immediate from the definitions (see Remark 4.6.1.9) and the equivalence $(2) \Leftrightarrow (3)$ follows from Remark 7.1.2.6.