\( \infty \)-Functors
Table of Contents
2.3.20
1. QUESTION Do we have \(h\Fun (\mathcal C, \mathcal D) = \Fun(h \mathcal C, h \mathcal D) \)? Or maybe we should talk about the nerve?
conservative functors
Let \(f \colon X \rightarrow Y \) be a morphism of simplicial sets such that \(f_0 \colon X_0 \rightarrow Y_0 \) is bijective. Then the functor \( f^* \colon \Fun(Y, \mathcal C) \rightarrow \Fun(X, \mathcal C) \) is conservative for every \(\infty \)-category \(\mathcal C \).
Let \(X \) be a simplicial set and \(\mathcal C \) an \(\infty \)-category. Then the functor \(\operatorname{ev} \colon \Fun(X, \mathcal C) \rightarrow \prod _{x \in K_0} \mathcal C \) is conservative. If \(F \colon X \times \Delta^1 \rightarrow \mathcal C \) is a natural transformation between functor, then \(F \) is a natural equivalence (equivalence in \(\Fun(X, \mathcal C) \)) if and only if \(F_x \colon \Delta^1 \rightarrow \mathcal C \) is an equivalence (in \(\mathcal C \)) for every \(x \in K_0 \).