Cauchy Completeness, Lax Epimorphisms and Effective Descent for Split Fibrations
Cauchy Completeness, Lax Epimorphisms and Effective Descent for Split Fibrations
For any suitable monoidal category $\mathcal{V}$, we find that $\mathcal{V}$-fully faithful lax epimorphisms in $\mathcal{V} \dash \mathcal{V}$ are precisely those $\mathcal{V}$-functors $F: \mathcal{A} \to \mathcal{B}$ whose induced $\mathcal{V}$-functors ${\mathfrak C} F: {\mathfrak C} \mathcal{A} \to {\mathfrak C} \mathcal{B}$ between the Cauchy completions are equivalences. For the case $\mathcal{V}= {\rm Set}$, …