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globular theory

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

A globular theory is much like an algebraic theory / Lawvere theory only that where the former has objects labeled by natural numbers, a globular theory has objects labeled by pasting diagrams of globes. The models of “homogeneous” globular theories are precisely the algebras over globular operads.

Definition

Definition

Write

Definition

The globular site is the category Θ 0\Theta_0 from def. 1 equipped with the structure of a site by taking the covering families to be the jointly epimorphic families.

Definition

A globular theory (or rather its syntactic category) is a wide subcategory inclusion

i A:Θ 0Θ A i_A \colon \Theta_0 \to \Theta_A

of the globular site, def. 2, such that every representable functor Θ A(,T):Θ A op\Theta_A(-,T) \colon \Theta_A^{op} \to Set is a Θ A\Theta_A-model:

A Θ A\Theta_A-model is a presheaf XPSh(Θ A)X \in PSh(\Theta_A) which restricts to a sheaf on the globular site, i A *XSh(Θ 0)PSh(Θ 0)i_A^* X \in Sh(\Theta_0) \hookrightarrow PSh(\Theta_0).

Write

Mod APSh(Θ A) Mod_A \hookrightarrow PSh(\Theta_A)

for the full subcategory of the category of presheaves on the Θ A\Theta_A-models. This is the category of Θ A\Theta_A-models.

(Berger, def. 1.5)

Definition

Given an globular theory i A:Θ 0Θ Ai_A \colon \Theta_0 \to \Theta_A a morphism in Θ A\Theta_A is

  • an AA-cover if…;

  • an immersion if

Definition

A globular theory i A:Θ 0Θ Ai_A \colon \Theta_0 \to \Theta_A is homogeneous if it contains a subcategory Θ A covΘ A\Theta^{cov}_A \to \Theta_A of AA-covers, def. 4 such that every morphism in Θ A\Theta_A factors uniquely as an AA-cover, def. 4, followed by an immersion, def. 4.

(Berger, def. 1.15)

Properties

Relation to ω\omega-graphs

Proposition

The category of sheaves over the globular site is equivalent to the category of ω-graphs

ωGraphSh(Θ 0). \omega Graph \simeq Sh(\Theta_0) \,.

(Berger, lemma 1.6)

Relation to globular operads

The (syntactic categories of) homogenous globular theories, def. 5 are the categories of operators of globular operads:

Proposition

A faithful monad A̲\underline{A} on ω-graphs encodes algebras over a globular operad AA precisely if

  1. the induced globular theory Θ AAlg A̲\Theta_A \hookrightarrow Alg_{\underline{A}} is homogeneous, def. 5;

  2. every A̲\underline{A}-algebras factors uniquely into an AA-cover followed by an A̲\underline{A}-free A̲\underline{A}-algebra morphism.

(Berger, prop. 1.16)

Examples

The theory of ω\omega-categories

The Theta category itself, equipped with the definition inclusion i:Θ 0Θi \colon \Theta_0 \to \Theta, def. 1, is the globular theory of ω-categories.

In particular:

Proposition

The category StrωCatStr\omega Cat of strict ω-categories is equivalent to that of Θ\Theta-models, def. 3. Hence it is the full subcategory of that of ω-graphs which satisfy the Segal condition with respect to the canonical inclusion Θ 0Theta\Theta_0 \to Theta: 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)

References

Section 1 of

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

Revised on November 13, 2012 12:38:29 by Urs Schreiber (82.169.65.155)