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 ∞-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.

Globular operads

Definition

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

Monad of a globular operad

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

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

(Berger, theorem 1.12)

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).