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\begin{document}

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\section*{k-simply connected n-category}

\hypertarget{simply_connected_categories}{}\section*{{$k$-simply connected $n$-categories}}\label{simply_connected_categories}

\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{special_cases}{Special cases}\dotfill \pageref*{special_cases} \linebreak
\noindent\hyperlink{discussion}{Discussion}\dotfill \pageref*{discussion} \linebreak


\hypertarget{definition}{}\subsection*{{Definition}}\label{definition}

An $n$-[[n-category|category]] is \textbf{$k$-simply connected} (or just \textbf{$k$-connected}) if any two parallel $j$-morphisms are [[equivalence|equivalent]] for $j \leq k$. You may interpret this definition as weakly or strictly as you like, by starting with weak or strict notions of $n$-category. Note that we include the case $j = -1$ to mean that the $n$-category is [[inhabited set|inhabited]] when $k \geq -1$.

\hypertarget{special_cases}{}\subsection*{{Special cases}}\label{special_cases}

Thus:

\begin{itemize}%
\item A $(-2)$-simply connected $n$-category is simply an $n$-category;
\item A $(-1)$-simply connected $n$-category is an [[inhabited set|inhabited]] $n$-category.
\item A $0$-simply connected $n$-category is a [[connected category|connected]] $n$-category: an inhabited $n$-category in which all objects are equivalent;
\item A $1$-simply connected $n$-category is a [[simply connected category|simply connected]] $n$-category: an inhabited $n$-category in which all objects and morphisms are equivalent;
\item etc.

\end{itemize}
The [[delooping hypothesis]] says that a $k$-[[k-tuply monoidal n-category|tuply monoidal]] $n$-category is the same thing as a [[pointed object|pointed]] $(k-1)$-simply connected $(n+k)$-category; more generally, a $(j+k)$-tuply monoidal, $i$-simply connected $n$-category is the same as a $j$-tuply monoidal, $(i+k)$-simply connected $(n+k)$-category.

One might want a stricter definition for $n$-categories, but this is certainly correct for $n$-[[n-groupoid|groupoid]]s. Indeed, we can say that an $\infty$-[[infinity-groupoid|groupoid]] $X$ is $k$-simply connected if and only if its [[fundamental n-groupoid]] $\Pi_n(X)$ is trivial for $n \leq k$. In particular, an $\infty$-simply connected $\infty$-groupoid is [[contractible space|contractible]].

\hypertarget{discussion}{}\subsection*{{Discussion}}\label{discussion}

This discussion on terminology occurred when the page was at [[k-tuply connected n-category]].

[[Mike Shulman|Mike]]: I'm pretty sure that in algebraic topology, 1-connected means simply connected, and 0-connected means connected. So your definitions make a space 0-connected when its fundamental $\infty$-groupoid is 1-tuply connected. I would prefer that we retain the topologists' numbering and call this a 0-connected $\infty$-groupoid (the `tuply' sounds weird to me for connectedness), with the off-by-one shift happening in the delooping: the $k$-fold delooping of a $k$-tuply monoidal $n$-category would be a $(k-1)$-connected $(n+k)$-category.

\emph{Toby}: If the topologists have a system for this, then we should probably use it, even if it is a poor system. (I don't suppose that any topologists say `$1$-simply connected' instead?) The previous usage on the Lab was inconsistent, although I didn't check whether that inconsistency was all my fault.

In general, a $1$-foo should always be the same as a foo, as with $1$-category, $1$-poset, $1$-groupoid, $1$-group, $1$-stuff, etc. Trying to match the `right' dimension isn't going to work consistently and runs into too many other existing terms. I'm only sorry that the topologists did it otherwise here.

(Not to mention that most homotopy-theoretic dimension counting is off by $1$ anyway. But that's another topic, indeed a topic that always making a $1$-foo a foo will safely avoid.)

I have no opinion about `tuply'; it just came naturally to me since I was thinking about $k$-tuple monoidality.

[[Mike Shulman|Mike]]: Yeah, I agree that in general it is better if a 1-foo is the same as an unadorned foo, but not everyone has adhered to that. \href{http://en.wikipedia.org/wiki/N-connected}{Wikipedia} agrees with me about the meaning of $k$-connected in topology.

\emph{Toby}: All right, I'll change the numbering and move the page. But what do you think about `$k$-simply connected'? This doesn't seem to have any conflicts, although search for `$m$-simply connected' to see what might be potential conflicts ---or might actually agree with me (I haven't been able to check).

[[!redirects k-simply connected n-category]] [[!redirects k-connected n-category]] [[!redirects k-tuply connected n-category]]



\end{document}