globular theory


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higher category theory

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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.





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.


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).


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)


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…


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)


Relation to ω\omega-graphs


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:


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)


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:


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)


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 (