Contents

### Context

#### Higher algebra

higher algebra

universal algebra

## Theorems

#### Higher category theory

higher category theory

# Contents

## Idea

The notion of globular operads is a variant of that of operads on which certain algebraic notions of higher category are based. The notion was introduced by Batanin; a globular operad is also called a 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 $T$.

## Definition

### Globular collections

###### Definition

A globular collection is a globular set $X$ equipped with a map

$f: X \to T(1)$

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

$Coll = Set^{Glob^{op}}/T(1)$
###### Definition

The category of collections carries a 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

$\array{ X \circ Y & \to & T(Y) \\ \downarrow & & \downarrow T(!) \\ X & \underset{f}{\to} & T(1) }$

and the requisite map $X \circ Y \to T(1)$ is given by the composite

$X \circ Y \to T(Y) \overset{T(g)}{\to} T T(1) \overset{\mu(1)}{\to} T(1)$

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.

###### Definition

A globular operad is a monoid in the monoidal category $Coll$, (with the monoidal structure given by def. ).

###### Definition

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

$\array{ M_P(X) & \to & T(X) \\ \downarrow & & \downarrow T(!) \\ P & \underset{f}{\to} & T(1) }$

and the multiplication and unit for $M_P$ may be worked out from the multiplication and unit for the globular operad $P$.

###### Remark

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 $T(X)$:

$T(X)(n) = \sum_{\tau \in T(1)(n)} \hom([\tau], X)$

and similarly, for a globular operad with underlying collection $f: P \to T(1)$,

$M_P(X)(n) = \sum_{x \in P(n)} \hom([f(x)], X)$

### Categories of operators

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

## Examples

### The Globular operad for $\omega$-categories

Write $\omega$ for the globular operad whose category of operators, see above, is the Theta category $\Theta$.

###### Proposition

The category $Str\omega Cat$ of strict ∞-categories is 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

$\array{ Str\omega Cat &\underoverset{\simeq}{N}{\to}& Mod_\Theta &\hookrightarrow& PSh(\Theta) \\ \downarrow^{\mathrlap{U}} && \downarrow^{} && \downarrow \\ \omega Graph &\stackrel{\simeq}{\to}& Sh(\Theta_0) &\hookrightarrow& PSh(\Theta_0) } \,.$

### Weak $\omega$-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 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.

## References

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

• Clemens Berger, A cellular nerve for higher categories, Advances in Mathematics 169, 118-175 (2002) (pdf)

Other work on globular operads :

• Camell Kachour, Operads of higher transformations for globular sets and for higher magmas, Published in Categories and General Algebraic Structures with Applications (2015).

Last revised on October 22, 2015 at 20:13:13. See the history of this page for a list of all contributions to it.