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\begin{document}

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\section*{globular operad}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{higher_algebra}{}\paragraph*{{Higher algebra}}\label{higher_algebra}

[[!include higher algebra - contents]]

\hypertarget{higher_category_theory}{}\paragraph*{{Higher category theory}}\label{higher_category_theory}

[[!include higher category theory - contents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{globular_collections}{Globular collections}\dotfill \pageref*{globular_collections} \linebreak
\noindent\hyperlink{globular_operads}{Globular operads}\dotfill \pageref*{globular_operads} \linebreak
\noindent\hyperlink{monad_of_a_globular_operad}{Monad of a globular operad}\dotfill \pageref*{monad_of_a_globular_operad} \linebreak
\noindent\hyperlink{CategoryOfOperators}{Categories of operators}\dotfill \pageref*{CategoryOfOperators} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{the_globular_operad_for_categories}{The Globular operad for $\omega$-categories}\dotfill \pageref*{the_globular_operad_for_categories} \linebreak
\noindent\hyperlink{weak_categories}{Weak $\omega$-categories}\dotfill \pageref*{weak_categories} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

The notion of \emph{globular operads} is a variant of that of [[operads]] on which certain [[algebraic definition of higher category|algebraic notions of higher category]] are based. The notion was introduced by [[Batanin]]; a globular operad is also called a \emph{Batanin operad}.

A globular operad gives rise to a [[monad]] on the [[category]] of [[globular sets]]; one example is the free [[strict ∞-category]] monad $T$ on globular sets. The monads which so arise may be characterized precisely as [[cartesian monads]] on globular sets over $T$ (itself a cartesian monad). This means that they are also examples of [[generalized multicategories]] relative to $T$.

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

\hypertarget{globular_collections}{}\subsubsection*{{Globular collections}}\label{globular_collections}

\begin{defn}
\label{}\hypertarget{}{}
A \textbf{globular collection} is a [[globular set]] $X$ equipped with a map

\begin{displaymath}
f: X \to T(1)
\end{displaymath}
to $T(1)$, the underlying globular set of the free strict $\omega$-category on the terminal globular set. Hence the category $Coll$ is the [[slice category]]

\begin{displaymath}
Coll = Set^{Glob^{op}}/T(1)
\end{displaymath}
\end{defn}
\begin{defn}
\label{MonoidalStructureOnCollections}\hypertarget{MonoidalStructureOnCollections}{}
The category of collections carries a [[monoidal category|monoidal product]] $\circ$ defined as follows. Given collections $f: X \to T(1)$, $g: Y \to T(1)$, the underlying globular set of $X \circ Y$ is given by [[pullback]]

\begin{displaymath}
\itexarray{
X \circ Y & \to & T(Y) \\
\downarrow & & \downarrow T(!) \\
X & \underset{f}{\to} & T(1)
}
\end{displaymath}
and the requisite map $X \circ Y \to T(1)$ is given by the composite

\begin{displaymath}
X \circ Y \to T(Y) \overset{T(g)}{\to} T T(1) \overset{\mu(1)}{\to} T(1)
\end{displaymath}
where $\mu: T T \to T$ denotes multiplication of the monad $T$. The monoidal unit is the collection $u(1): 1 \to T(1)$ where $u: Id \to T$ is the unit of $T$, and the associativity and unit constraints may be defined by means of universal properties, taking advantage of the fact that $T$ is cartesian.

\end{defn}
\hypertarget{globular_operads}{}\subsubsection*{{Globular operads}}\label{globular_operads}

\begin{defn}
\label{GlobularOperad}\hypertarget{GlobularOperad}{}
A \textbf{globular operad} is a [[monoid]] in the [[monoidal category]] $Coll$, (with the monoidal structure given by def. \ref{MonoidalStructureOnCollections}).

\end{defn}
\hypertarget{monad_of_a_globular_operad}{}\subsubsection*{{Monad of a globular operad}}\label{monad_of_a_globular_operad}

\begin{defn}
\label{}\hypertarget{}{}
Each globular operad $f: P \to T(1)$, (as in def. \ref{GlobularOperad}), gives rise to a globular [[monad]] $M_P$ on $Set^{Glob^{op}}$. Abstractly, $M_P(X)$ is just the pullback

\begin{displaymath}
\itexarray{
  M_P(X) & \to & T(X) \\
  \downarrow & & \downarrow T(!) \\
  P & \underset{f}{\to} & T(1)
}
\end{displaymath}
and the multiplication and unit for $M_P$ may be worked out from the multiplication and unit for the globular operad $P$.

\end{defn}
\begin{remark}
\label{}\hypertarget{}{}
A more concrete description of $M_P(X)$ may be worked out in terms of a concrete description of the free strict $\omega$-category $T(X)$. To describe this, first notice that every element $\tau$ of $T(1)$, which is essentially a pasting diagram built up out of globes of $1$, can be drawn as a globular set which we denote as $[\tau]$. The globes of $[\tau]$ are instances of globular cells as they appear in the pasting diagram $\tau$, and their sources and targets are then also instances of cells in $\tau$. (Batanin describes $T(1)$ in terms of trees, and the globular set $\tau$ is given formally in the tree language.)

Similarly, we can think of an element of $T(X)$ as a pasting diagram built out of globes in $X$, and such a pasting diagram can be thought of as having an underlying shape given by an pasting diagram $\tau$ in $T(1)$, together with a labeling of the pasting cells in $\tau$ by elements on $X$. The labeling is in fact just a morphism $[\tau] \to X$ of globular sets. Therefore we have an explicit formula for the set of $n$-cells of $T(X)$:

\begin{displaymath}
T(X)(n) = \sum_{\tau \in T(1)(n)} \hom([\tau], X)
\end{displaymath}
and similarly, for a globular operad with underlying collection $f: P \to T(1)$,

\begin{displaymath}
M_P(X)(n) = \sum_{x \in P(n)} \hom([f(x)], X)
\end{displaymath}
\end{remark}
\hypertarget{CategoryOfOperators}{}\subsubsection*{{Categories of operators}}\label{CategoryOfOperators}

The [[category of operators]] of a globular operad $A$ is (the [[syntactic category]] of) a homogeneous [[globular theory]] $i_A \colon \Theta_0 \to \Theta_A$ and every globular operad is characterized by its [[globular theory]]. See there for more details

\hypertarget{examples}{}\subsection*{{Examples}}\label{examples}

\hypertarget{the_globular_operad_for_categories}{}\subsubsection*{{The Globular operad for $\omega$-categories}}\label{the_globular_operad_for_categories}

Write $\omega$ for the globular operad whose [[category of operators]], see \hyperlink{CategoryOfOperators}{above}, is the [[Theta category]] $\Theta$.

\begin{prop}
\label{}\hypertarget{}{}
The category $Str\omega Cat$ of [[strict ∞-categories]] is [[equivalence of categories|equivalent]] to that of algebras over the terminal globular operad. Hence it is the [[full subcategory]] of that of [[∞-graphs]] which satisfy the [[Segal condition]] with respect to the canonical inclusion $\Theta_0 \to \Theta$ that defines its [[globular theory]]: we have a [[pullback]]

\begin{displaymath}
\itexarray{
    Str\omega Cat 
     &\underoverset{\simeq}{N}{\to}& Mod_\Theta &\hookrightarrow&      
PSh(\Theta)
    \\
    \downarrow^{\mathrlap{U}} && \downarrow^{} && \downarrow
   \\
   \omega Graph &\stackrel{\simeq}{\to}& Sh(\Theta_0) &\hookrightarrow& PSh(\Theta_0)
  }
  \,.
\end{displaymath}
\end{prop}
(\hyperlink{Berger}{Berger, theorem 1.12})

\hypertarget{weak_categories}{}\subsubsection*{{Weak $\omega$-categories}}\label{weak_categories}

As a refinement of the above example:

In the Batanin (or Leinster) theory of $\infty$-categories, there is a universal contractible globular operad $f: K \to T(1)$, where each element $x \in K(n)$ is thought of as a \emph{way} of (weakly) pasting together the underlying shape $f(x)$. The contractibility implies that for every two different ways of pasting together the same shape, i.e., two elements $x, y \in K(n)$ such that $f(x) = f(y)$ and such that $x$ and $y$ have the same source and have the same target, there is an $(n+1)$-cell in $K(n+1)$ mediating between them, with source $x$ and target $y$, and which maps to the identity $(n+1)$-cell on $f(x)$.

A [[Batanin ∞-category]] is a globular set with a $K$-algebra structure.

\hypertarget{references}{}\subsection*{{References}}\label{references}

A review and characterization in terms of [[globular theories]] is in section 1 of

\begin{itemize}%
\item [[Clemens Berger]], \emph{A cellular nerve for higher categories}, Advances in Mathematics 169, 118-175 (2002) (\href{http://math1.unice.fr/~cberger/nerve.pdf}{pdf})

\end{itemize}
Other work on globular operads :

\begin{itemize}%
\item [[Camell Kachour]], Operads of higher transformations for globular sets and for higher magmas, Published in Categories and General Algebraic Structures with Applications (2015).

\end{itemize}
[[!redirects globular operads]]



\end{document}