homotopy hypothesis-theorem
delooping hypothesis-theorem
stabilization hypothesis-theorem
A strict -category is a strict omega-category all whose k-morphisms for are identities.
The category of strict -categories and n-functors between them can also be defined inductively by
starting by setting Set;
noticing that Set is canonically a (symmetric, in fact cartesian) closed monoidal category such that one can consider categories enriched over it;
noticing that for any complete and cocomplete closed monoidal category, also (the category of -enriched categories) has these same properties;
finally setting, recursively,
The category of strict -categories can then in turn be defined as a suitable colimit of the categories .
Write for the 1-category of strict n-categories.
Write
for the full subcategory on the gaunt -categories, those -categories whose only invertible k-morphisms are the identities.
This subcategory was considered in (Rezk). The term “gaunt” is due to (Barwick, Schommer-Pries). See prop. below for a characterization intrinsic to -categories.
For the -globe is gaunt, .
Write
for the full subcategory of the globe category on the -globes for .
Being a subobject of a gaunt -category, also the boundary of a globe is gaunt, i.e. the -skeleton of .
Write
for the “categorical suspension” functor which sends a strict -category to the object which has precisely two objects and , has , , and
We usually suppress the subscript and write , etc.
The -globe is the -fold suspension of the 0-globe (the point)
The boundary of the -globe is the -fold suspension of the empty category
Accordingly, the boundary inclusion is the -fold suspension of the initial morphism
The category is a locally presentable category and in fact a locally finitely presentable category.
For two categories, a profunctor is equivalently a functor equipped with an identification and .
This motivates the following definition.
A -profunctor / -correspondence of strict -categories is a morphism in . The category of -correspondences is the slice category .
The categories of -correspondences between gaunt -categories are cartesian closed category.
By standard facts, in a locally presentable category with finite limits, a slice is cartesian closed precisely if pullback along all morphisms with codomain preserves colimits (see at locally cartesian closed category the section Cartesian closure in terms of base change and dependent product).
Without the restriction that the codomain of in the above is a globe, the pullback in will in general fail to preserves colimits. For a simple example of this, consider the pushout diagram in Cat given by
Notice that this is indeed also a homotopy pushout/(∞,1)-pushout since, by remark , all objects involved are 0-truncated.
Regard this canonically as a pushout diagram in the slice category and consider then the pullback along the remaining face . This yields the diagram
which evidently no longer is a pushout.
(See also the discussion here).
Write
for the smallest full subcategory that
The following categories are naturally full subcategories of
the -fold simplex category ;
the th Theta-category.
This is discussed in more detail in (infinity,n)-category in Presentation by Theta-spaces and by n-fold Segal spaces.
The following pushouts in we call the fundamental pushouts
Gluing two -globes along their boundary gives the boundary of the -globle
Gluing two -globes along an -face gives a pasting composition of the two globles
The fiber product of globes along non-degenerate morphisms and is built from gluing of globes by
The interval groupoid is obtained by forcing in the morphisms and to be identities and it is equivalent, as an -category, to the 0-globe
and the analog is true for all suspensions of this relation
We say a functor on preserves the fundamental pushouts if it preserves the first three classes of pushouts, and if for the last one the morphism is an equivalence.
A strict 1-category is just a category.
Strict 2-categories are important, because the coherence theorem for bicategories states that every (“weak”) 2-category is equivalent to a strict one, and also because many 2-categories, such as Cat, are naturally strict. However, for , these two properties fail, so that strict -categories become less useful (though not useless). Instead, one needs to use (at least) semistrict categories.
With an eye towards the generalization to (∞,n)-categories, strict -categories are discussed in
and in section 2 of
Last revised on January 20, 2024 at 00:09:47. See the history of this page for a list of all contributions to it.