Algebras and modules
Model category presentations
Geometry on formal duals of algebras
Higher category theory
higher category theory
Extra properties and structure
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 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
of the globular site, def. 2, such that every representable functor Set is a -model:
A -model is a presheaf which restricts to a sheaf on the globular site, .
for the full subcategory of the category of presheaves on the -models. This is the category of -models.
(Berger, def. 1.5)
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. 4 such that every morphism in factors uniquely as an -cover, def. 4, followed by an immersion, def. 4.
(Berger, def. 1.15)
Relation to -graphs
(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 on ω-graphs encodes algebras over a globular operad precisely if
the induced globular theory is homogeneous, def. 5;
every -algebras factors uniquely into an -cover followed by an -free -algebra morphism.
(Berger, prop. 1.16)
The theory of -categories
The Theta category itself, equipped with the definition inclusion , def. 1, is the globular theory of ω-categories.
The category of strict ω-categories is equivalent to that of -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 : we have a pullback
(Berger, theorem 1.12)
Section 1 of
- Clemens Berger, A cellular nerve for higher categories, Advances in Mathematics 169, 118-175 (2002) (pdf)