nLab globular theory

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

• $Str \omega Cat \in Cat$ for the category of strict ∞-categories;

• $\Theta \hookrightarrow Str\omega Cat$ for the the Theta category, the full subcategory on the strict $\omega$-categories free on ∞-graphs(globular sets) (the pasting diagrams);

• $i \colon \Theta_0 \to \Theta$ for the wide non-full subcategory on the morphism induced from morphisms of the underlying ∞-graphs (this means that these morphisms in $\Theta_0$ send $n$-globes to single $n$-globes, not to pastings of them).

• $\mathbb{G} \hookrightarrow \Theta_0$ for the full subcategory on the pasting diagrams given by a single globe – the globe category.

Definition

The globular site is the category $\Theta_0$ from def. 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

$$i_A \colon \Theta_0 \to \Theta_A$$

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

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

Write

$$Mod_A \hookrightarrow PSh(\Theta_A)$$

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

(Berger, def. 1.5)

Definition

Given an globular theory $i_A \colon \Theta_0 \to \Theta_A$ a morphism in $\Theta_A$ is

• an $A$-cover if…;

• an immersion if…

Definition

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

(Berger, def. 1.15)

Properties

Relation to $\omega$-graphs

Proposition

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

$$\omega Graph \simeq Sh(\Theta_0) \,.$$

(Berger, lemma 1.6)

Relation to globular operads

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

Proposition

A faithful monad $\underline{A}$ on ∞-graphs encodes algebras over a globular operad $A$ precisely if

1. the induced globular theory $\Theta_A \hookrightarrow Alg_{\underline{A}}$ is homogeneous, def. ;

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

(Berger, prop. 1.16)

Examples

The theory of $\omega$-categories

The Theta category itself, equipped with the definition inclusion $i \colon \Theta_0 \to \Theta$, def. , is the globular theory of ∞-categories.

In particular:

Proposition

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

References

Section 1 of

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