symmetric monoidal (∞,1)-category of spectra
homotopy hypothesis-theorem
delooping hypothesis-theorem
stabilization hypothesis-theorem
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
for the category of strict ∞-categories;
for the the Theta category, the full subcategory on the strict -categories free on ∞-graphs(globular sets) (the pasting diagrams);
for the wide non-full subcategory on the morphism induced from morphisms of the underlying ∞-graphs (this means that these morphisms in send -globes to single -globes, not to pastings of them).
for the full subcategory on the pasting diagrams given by a single globe – the globe category.
The globular site is the category 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 Set is a -model:
A -model is a presheaf which restricts to a sheaf on the globular site, .
Write
for the full subcategory of the category of presheaves on the -models. This is the category of -models.
Given an globular theory a morphism in is
an -cover if…;
an immersion if…
A globular theory is homogeneous if it contains a subcategory of -covers, def. such that every morphism in factors uniquely as an -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 on ∞-graphs encodes algebras over a globular operad precisely if
every -algebras factors uniquely into an -cover followed by an -free -algebra morphism.
The Theta category itself, equipped with the definition inclusion , def. , is the globular theory of ∞-categories.
In particular:
The category of strict ∞-categories is equivalent to that of -models, def. . Hence it is the full subcategory of that of ∞-graphs which satisfy the Segal condition with respect to the canonical inclusion : 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.