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G. Stefanich
G. Stefanich
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Let $f: \mathcal{C} \rightarrow \mathcal{D}$ be a colimit-preserving functor between locally presentable $\infty$-categoriescategories. Assume that $f$ induces an equivalence between the $\infty$-groupoidsgroupoids underlying $\mathcal{C}$ and $\mathcal{D}$. Is $f$ necessarily an equivalence? What if $\mathcal{C}$ and $\mathcal{D}$ are assumed to be stable and $f$ exactlinear/dg/stable?

Let $f: \mathcal{C} \rightarrow \mathcal{D}$ be a colimit-preserving functor between locally presentable $\infty$-categories. Assume that $f$ induces an equivalence between the $\infty$-groupoids underlying $\mathcal{C}$ and $\mathcal{D}$. Is $f$ necessarily an equivalence? What if $\mathcal{C}$ and $\mathcal{D}$ are assumed to be stable and $f$ exact?

Let $f: \mathcal{C} \rightarrow \mathcal{D}$ be a colimit-preserving functor between locally presentable categories. Assume that $f$ induces an equivalence between the groupoids underlying $\mathcal{C}$ and $\mathcal{D}$. Is $f$ necessarily an equivalence? What if $\mathcal{C}$ and $\mathcal{D}$ are assumed to be linear/dg/stable?

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G. Stefanich
G. Stefanich
  • 898
  • 1
  • 8
  • 19

Are locally presentable categories determined by their objects?

Let $f: \mathcal{C} \rightarrow \mathcal{D}$ be a colimit-preserving functor between locally presentable $\infty$-categories. Assume that $f$ induces an equivalence between the $\infty$-groupoids underlying $\mathcal{C}$ and $\mathcal{D}$. Is $f$ necessarily an equivalence? What if $\mathcal{C}$ and $\mathcal{D}$ are assumed to be stable and $f$ exact?