@pupshaw One way to see this map is that if X is a space, then we have a canonical Postnikov 1-truncation $\tau_{\leq 1} X$ which is functorial in X, so that in particular we have a morphism $Aut(X) \rightarrow Aut(\tau_{\leq 1} X)$ of group objects in spaces. If $X$ is connected, then $\tau_{\leq 1} X \simeq B \pi_{1} X$ and $\pi_{0}Aut(\tau_{\leq 1} X) \simeq Out(\pi_{1}X)$, recovering your observation.

@PiotrPstrągowski yes exactly, that seems like the right way to think about it. I suppose I'm asking the same question where instead of truncating $X$ to its fundamental groupoid, I want to truncate a stratified space $(X,S)$ to its exit-path 1-category, and try to understand whether I get anything comparably explicit.

If $E$ is an $\infty$-topos and $F: C\to E$ is a flat functor from a small $\infty$-category (i.e. a functor whose Yoneda extension preserves finite limits), is it true that the functor $F\otimes -: Fun(C^{op}, E)\to E$ (where the tensor product is the tensor product of functors, i.e. the weighted colimit functor) preserves finite limits?

I recently heard that a research paper was written (or say, inspired) with the help of this chatroom and it made me feel proud of MO

@epic_math can you be more specific? which paper? I'm curious

@S.carmeli this is the paper: arxiv.org/abs/1409.8192

See the acknowledgements

Cool thanks! it's very nice to have such outcomes of the chat :-)

The authors would have MO accounts

Can someone link them?

Here are more papers inspired by MO:

Haha, also all of my papers...

Actually probably not the ones with L.Z. Wong, he's basically a standalone category theory chat room all on his own.

@epic_math I don't know if Zhen Lin is on here but @AaronMazel-Gee definitely is.