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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.

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