symmetric monoidal (∞,1)-category of spectra
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.
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.
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.
A globular theory (or rather its syntactic category) is a wide subcategory inclusion
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
for the full subcategory of the category of presheaves on the $\Theta_A$-models. This is the category of $\Theta_A$-models.
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…
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. .
The category of sheaves over the globular site is equivalent to the category of ∞-graphs
The (syntactic categories of) homogenous globular theories, def. are the categories of operators of globular operads:
A faithful monad $\underline{A}$ on ∞-graphs encodes algebras over a globular operad $A$ precisely if
the induced globular theory $\Theta_A \hookrightarrow Alg_{\underline{A}}$ is homogeneous, def. ;
every $\underline{A}$-algebras factors uniquely into an $A$-cover followed by an $\underline{A}$-free $\underline{A}$-algebra morphism.
The Theta category itself, equipped with the definition inclusion $i \colon \Theta_0 \to \Theta$, def. , is the globular theory of ∞-categories.
In particular:
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
Section 1 of
Last revised on November 13, 2012 at 12:38:29. See the history of this page for a list of all contributions to it.