nLab
globular operad

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 ω-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

$$f:X\to T(1)$$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 $\mathrm{Coll}$ is the slice category

$$\mathrm{Coll}={\mathrm{Set}}^{{\mathrm{Glob}}^{\mathrm{op}}}/T(1)$$Coll = Set^{Glob^{op}}/T(1)

Definition

The category of collection 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

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

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

where $\mu :TT\to T$ denotes multiplication of the monad $T$. The monoidal unit is the collection $u(1):1\to T(1)$ where $u:\mathrm{Id}\to T$ 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.

Globular operads

Definition

A globular operad is a monoid in the monoidal category $\mathrm{Coll}$, def. 2.

Monad of a globular operad

Definition

Each globular operad $f:P\to T(1)$, def. 3, gives rise to a globular monad $M_P$ on ${\mathrm{Set}}^{{\mathrm{Glob}}^{\mathrm{op}}}$. Abstractly, $M_P(X)$ is just the pullback

$$\begin{array}{ccc}{M}_P(X)& \to & T(X)\\ \downarrow & & \downarrow T(!)\\ P& \underset{f}{\to }& T(1)\end{array}$$\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 $T(X)$:

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

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

$$M_P(X)(n)=\sum _{x\in P(n)}\mathrm{hom}([f(x)],X)$$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 $i_A:{\Theta }_0\to {\Theta }_A$ 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 $\mathrm{Str}\omega \mathrm{Cat}$ of strict ω-categories is equivalent to that of algebras over the terminal globular operad. Hence it is the full subcategory of that of ω-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

$$\begin{array}{ccccc}\mathrm{Str}\omega \mathrm{Cat}& \underset{\simeq }{\overset{N}{\to }}& {\mathrm{Mod}}_{\Theta }& \hookrightarrow & \mathrm{PSh}(\Theta )\\ {\downarrow }^U& & {\downarrow }^{}& & \downarrow \\ \omega \mathrm{Graph}& \stackrel{\simeq }{\to }& \mathrm{Sh}({\Theta }_0)& \hookrightarrow & \mathrm{PSh}({\Theta }_0)\end{array}.$$\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) } \,.

(Berger, theorem 1.12)

Weak $\omega$-categories

As a refinement of the above example:

In the Batanin (or Leinster) theory of $\mathrm{\infty }$-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 $f(x)$.

An Batanin ω-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)