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Definition Let $\widehat{\operatorname{\mathcal{C}}}$ be an $\infty $-category. We say that a full subcategory $\operatorname{\mathcal{C}}\subseteq \widehat{\operatorname{\mathcal{C}}}$ is weakly dense if the following condition is satisfied:

  • Let $f: Y \rightarrow Z$ be a morphism of $\widehat{\operatorname{\mathcal{C}}}$ such that, for every object $X \in \operatorname{\mathcal{C}}$, the induced map

    \[ \operatorname{Hom}_{\widehat{\operatorname{\mathcal{C}}}}( X, Y) \xrightarrow { [f] \circ } \operatorname{Hom}_{\widehat{\operatorname{\mathcal{C}}}}(X, Z) \]

    is a homotopy equivalence of Kan complexes. Then $f$ is an isomorphism.

We say that a collection of objects $\{ X_ i \} _{i \in I}$ of $\widehat{\operatorname{\mathcal{C}}}$ is weakly dense if it spans a weakly dense full subcategory of $\widehat{\operatorname{\mathcal{C}}}$.