Limit along Equivalences

Question

Let $F: I \to \mathscr{C}$ be a connected functor into an $\infty$-category which factors over $\mathscr{C}^{\mathrm{core}} \to \mathscr{C}$. Why is $\operatorname{lim}{F} \simeq F(i)$ for any $i \in I$?

Orally: Take a limit over a diagram of equivalences, then the limit is any of the equivalent objects. It seems evidently true, but I cannot prove it in the $\infty$-categorical formalism. What am I missing?

My particular motivation comes from verifying that $B^{\infty} : \mathscr{C} \to \mathbf{Sp}(\mathscr{C})$ is fully faithful which boils down to the above assertion.

Answer 1

I had helpful conversations with Nima Rasekh and Emma Brink.

In this generality, the statement is wrong. A diagram $F: BC_2 \to \mathbf{An}$ is an involution $\tau : X \to X$ for an anima $X$ and one can classically compute $\operatorname{holim}(\tau) = P_{\tau} \simeq *$.

A better assumption is $|I| \simeq *$. Let $i \in I$ and say $|I| \simeq \{i \}$. In that case, \begin{align*} \operatorname{Hom}_{\mathscr{C}} \left(c, \lim_{i \in I} F \right) &\simeq \operatorname{Hom}_I(\operatorname{const}{c}, F) \\ &\simeq \operatorname{Hom}_{|I|}(\operatorname{const}{c}, \widetilde{F}) \\ &\simeq \operatorname{Hom}_{\mathscr{C}} \left(c, \lim_{|I|} \widetilde{F} \right) \\ &\simeq \operatorname{Hom}_{\mathscr{C}}(c, \widetilde{F}(i)) \end{align*} where $\operatorname{const}{c}$ and $F$ factor over $|I|$ and we denote the factorization of $F$ by $\widetilde{F} : |I| \to \mathscr{C}$. This is because $F$ factors over $\mathscr{C}^{\mathrm{core}} \to \mathscr{C}$. The second step is then by the universal property of localizations. The last step uses $|I| \simeq \{i \}$.

Answer 2

For another counterexample that shows also that the sufficient criterium of $I$ being weakly contractible cannot be weakened without making the statement generally false: consider an $\infty$-category $I$. Then the constant functor on a terminal object $*\colon I\to\mathbf{An}$ satisfies $\mathrm{colim}_I\,*\simeq |I|$, the anima that is the localization of $I$ at all morphisms, and hence $\mathrm{colim}_I\,*\simeq *$ iff $I$ is a weakly contractible $\infty$-category. We can consider a similar functor $*\colon I\to\mathbf{An}^\mathrm{op}$ (but now $*$ is the initial object of $\mathbf{An}^\mathrm{op}$), and now find $\mathrm{lim}_I\,*\simeq|I^\mathrm{op}|\simeq |I|^\mathrm{op}\simeq |I|$, and hence this limit evaluates to (something equivalent to) $*$ iff $I$ is weakly contractible.