Denote the $(\infty ,1)$-category of $n$-connected $\infty$-groupoids by $\operatorname{Grpd}^{\infty}_{n}$. There is clearly a suitable inclusion $\iota:\operatorname{Grpd}^{\infty}_{n}\hookrightarrow\operatorname{Grpd}^{\infty}_{n-1}$. I'm pretty sure $\iota$ preserves limits and colimits (I'm not certain), but it seems analogous to the cohesive stuff going on between $n$-truncated $\infty$-groupoids and $(n-1)$-truncated $\infty$-groupoids, where we have the discrete and codiscrete functors, fundamental $(n-1)$-groupoid, et cetera. $n$-Truncated spaces and $n$-connected spaces are complimentary, so it would be really nice if $\iota$ admitted left and right adjoints. It seems like one of the adjoints should just functorially kill off a lower homotopy group like in the Whitehead tower, but I explicitly want to avoid working with pointed groupoids.*

Any reference recommendations are also very welcome. The $n$Lab is somewhat lacking in discussion on this inclusion $\iota$, making me worry that I am just completely confused and incorrect.

*Edit. I was just looking back over the $n$Lab article on the Goodwillie calculus and how the language of formal power series rings over a base field translates into the language of stable homotopy theory. I noticed that they associate the $(n+1)$'th power of the augmentation ideal $J$ to the full subcategory on pointed $n$-connected spaces, whereas the quotient ring $k[[x]]/J^{n+1}$ is associated to the unpointed full subcategory of $n$-truncated spaces. This is motivation enough for me to be interested in adjoints instead to the inclusion of $n$-connected pointed $\infty$-groupoids into $(n-1)$-connected ones; it looks like it is moral to talk about basepoints when your space is $n$-connected and forget about them when it is $n$-truncated, or at least when you are focusing on those two features of the space.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.