Kerodon

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Definition 7.1.1.1. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category containing an object $Y$, let $K$ be a simplicial set, and let $u: K \rightarrow \operatorname{\mathcal{C}}$ be a diagram. We say that a natural transformation $\alpha : \underline{Y} \rightarrow u$ exhibits $Y$ as a limit of $u$ if the following condition is satisfied:

$(\ast )$

For each object $X \in \operatorname{\mathcal{C}}$, the composition

\[ \operatorname{Hom}_{\operatorname{\mathcal{C}}}( X, Y) \rightarrow \operatorname{Hom}_{ \operatorname{Fun}(K,\operatorname{\mathcal{C}})}( \underline{X}, \underline{Y} ) \xrightarrow { [ \alpha ] \circ } \operatorname{Hom}_{ \operatorname{Fun}(K, \operatorname{\mathcal{C}})}( \underline{X}, u ) \]

is an isomorphism in the homotopy category $\mathrm{h} \mathit{\operatorname{Kan}}$, where the second map is described in Notation 4.6.7.15.

We will say that a natural transformation $\beta : u \rightarrow \underline{y}$ exhibits $Y$ as a colimit of $u$ if the following dual condition is satisfied:

$(\ast ')$

For each object $Z \in \operatorname{\mathcal{C}}$, the composition

\[ \operatorname{Hom}_{\operatorname{\mathcal{C}}}( Y, Z) \rightarrow \operatorname{Hom}_{ \operatorname{Fun}(K,\operatorname{\mathcal{C}})}( \underline{Y}, \underline{Z} ) \xrightarrow { \circ [ \alpha ]} \operatorname{Hom}_{ \operatorname{Fun}(K, \operatorname{\mathcal{C}})}( u, \underline{Z} ) \]

is an isomorphism in the homotopy category $\mathrm{h} \mathit{\operatorname{Kan}}$.