nLab
globular operad

Context

Higher algebra

Higher category theory

higher category theory

Basic concepts

Basic theorems

Applications

Models

Morphisms

Functors

Universal constructions

Extra properties and structure

1-categorical presentations

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 ω-category monad TT on globular sets. The monads which so arise may be characterized precisely as cartesian monads on globular sets over TT (itself a cartesian monad). This means that they are also examples of generalized multicategories relative to TT.

Definition

Globular collections

Definition

A globular collection is a globular set XX equipped with a map

f:XT(1)f: X \to T(1)

to T(1)T(1), the underlying globular set of the free strict ω\omega-category on the terminal globular set. Hence the category CollColl is the slice category

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

The category of collections carries a monoidal product \circ defined as follows. Given collections f:XT(1)f: X \to T(1), g:YT(1)g: Y \to T(1), the underlying globular set of XYX \circ Y is given by pullback

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

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

XYT(Y)T(g)TT(1)μ(1)T(1)X \circ Y \to T(Y) \overset{T(g)}{\to} T T(1) \overset{\mu(1)}{\to} T(1)

where μ:TTT\mu: T T \to T denotes multiplication of the monad TT. The monoidal unit is the collection u(1):1T(1)u(1): 1 \to T(1) where u:IdTu: Id \to T is the unit of TT, and the associativity and unit constraints may be defined by means of universal properties, taking advantage of the fact that TT is cartesian.

Globular operads

Definition

A globular operad is a monoid in the monoidal category CollColl, (with the monoidal structure given by def. 2).

Monad of a globular operad

Definition

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

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

and the multiplication and unit for M PM_P may be worked out from the multiplication and unit for the globular operad PP.

Remark

A more concrete description of M P(X)M_P(X) may be worked out in terms of a concrete description of the free strict ω\omega-category T(X)T(X). To describe this, first notice that every element τ\tau of T(1)T(1), which is essentially a pasting diagram built up out of globes of 11, can be drawn as a globular set which we denote as [τ][\tau]. The globes of [τ][\tau] are instances of globular cells as they appear in the pasting diagram τ\tau, and their sources and targets are then also instances of cells in τ\tau. (Batanin describes T(1)T(1) in terms of trees, and the globular set τ\tau is given formally in the tree language.)

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

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

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

M P(X)(n)= xP(n)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 AA is (the syntactic category of) a homogeneous globular theory i A:Θ 0Θ Ai_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 ω\omega-categories

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

Proposition

The category StrωCatStr\omega 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 Θ 0Θ\Theta_0 \to \Theta that defines its globular theory: we have a pullback

StrωCat N Mod Θ PSh(Θ) U ωGraph Sh(Θ 0) PSh(Θ 0). \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 ω\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:KT(1)f: K \to T(1), where each element xK(n)x \in K(n) is thought of as a way of (weakly) pasting together the underlying shape f(x)f(x). The contractibility implies that for every two different ways of pasting together the same shape, i.e., two elements x,yK(n)x, y \in K(n) such that f(x)=f(y)f(x) = f(y) and such that xx and yy have the same source and have the same target, there is an (n+1)(n+1)-cell in K(n+1)K(n+1) mediating between them, with source xx and target yy, and which maps to the identity (n+1)(n+1)-cell on f(x)f(x).

A Batanin ω-category is a globular set with a KK-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)

Revised on January 26, 2014 08:36:16 by Tim Porter (2.26.18.36)