homotopy hypothesis-theorem
delooping hypothesis-theorem
stabilization hypothesis-theorem
An -category is a higher category such that, essentially:
all k-morphisms for are trivial.
all k-morphisms for are reversible.
Put another way: given a sequence of (higher) categories in which each is of the form for some -cells and from , let us say that is a depth- Hom-category of . (We can also cleanly extend this notion to depth- Hom-categories, by taking the position that there are none). An -category, then, is one in which every depth- Hom-category is an -groupoid, and, furthermore, every depth- Hom-category is a point. (The appearance of here rather than allows us to make sense of this definition even when is as low as , and suggests that perhaps, had history gone differently, the conventions would be to number these differently.)
So -categories are a generalisation of both -categories and -groupoids, covering some of the ground in between (and a bit beyond). As increases, there are many more possibilities, until there are infinitely many kinds of -categories.
This definition does not attempt to cover all possible scenarios regarding the triviality or invertibility of morphisms at different dimensions. In particular:
In general, if a morphism at dimension always uniquely exists, it need not follow that a morphism at dimension always uniquely exists. A counterexample are monoid-enriched categories, where we view monoids as single-object categories, so that we can see monoid-enriched categories as 2-categories whose sets of 1-morphisms are singletons, but whose sets of 2-morphisms are not.
In general, if all morphisms at dimension are invertible, it need not follow that all morphisms at dimension are invertible. A counterexample is ordered groupoids, and in particular the single object case of ordered groups such as . They have invertible 1-morphisms, but non-invertible 2-morphisms.
A more fine-grained classification of higher categories could thus be imagined, where we would not specify two numbers and , but two subsets of .
Given a notion of -category (as weak or strict as you like), then an -category can be defined to be an -category such that
As explained below, we may assume that and .
For finite , we can also define this inductively in terms of (∞,r)-categories as follows:
For , an (n,0)-category is an ∞-groupoid that is n-truncated: an n-groupoid.
For , an (n,r)-category is an (∞,r)-category such that for all objects the -categorical hom-object is an -category.
(Even for , this definition makes sense, taking to be , as long as we know that an -category is the same thing as a -category. But this may be overkill.)
You can also start with a notion of -poset, then define an -category to be an -poset such that any -morphism is an equivalence for . Or, for , you can start with a notion of -category, then define an -category to be an -category such that any -morphism in an equivalence for .
To interpret the first definition above correctly for low values of , we must assume that all objects (-morphisms) in a given -category are parallel, which leads us to speak of the two -morphisms that serve as their common source and target and to accept any object as an equivalence between these. In particular, any -morphism is an equivalence for , so if , then the condition is satisfied for any smaller value of . Thus, we assume that .
To say that parallel -morphisms must be equivalent is meaningful; it requires that there be an object. One can continue to -morphisms and so on, but there is nothing to vary about these; so we assume that . In other words, a -category will automatically be an -category for any smaller value of .
If any two parallel -morphisms are equivalent, then any -morphism between equivalent -morphisms is an equivalence (being parallel to an identity for and automatically for ). Accordingly, any -category for is also an -category. Thus, we assume that (except when , where it would conflict with the convention and so we simply take .)
From the point of view of homotopy theory, the notion of -categories may be understood as a combination of the notion of homotopy n-type and that of directed space.
Recall that an (∞,0)-category is an ∞-groupoid. In light of the homotopy hypothesis – that identifies -groupoids with (nice) topological spaces and n-groupoids with homotopy n-types – and in view of the notion of directed space, the following terminology is suggestive:
An -category is an -directed homotopy -type.
Here we read
and
Then, indeed, we have for instance that
a (1,0)-category is an undirected 1-type: a 1-groupoid,
a (2,0)-category is an undirected 2-type: a 2-groupoid,
etc.
a (1,1)-category is directed 1-type : a category,
an (n,n)-category is an -directed -type: an n-category,
etc.
an (∞,0)-category is an undirected space: an ∞-groupoid,
an (∞,1)-category is a directed space: a quasi-category,
an (∞,n)-category is an -directed space
etc.
An (n,n)-category is simply an -category. An -category is an -poset. Note that an -category and an -poset are the same thing. An -category is an -groupoid. Even though they have no special name, -categories are widely studied.
For low values of , many of these notions coincide. For instance, a -groupoid is the same as a -category, namely a set. And -groupoid, -category, and -poset all mean the same thing (namely, a truth value) while -groupoid, -category, and -poset likewise all mean the same thing (namely, the point).
Of particular importance is the case where . See
An analogous systematics exists for -categories that in additions have the property of being a topos or higher topos.
a (0,1)-topos is a Heyting algebra
a -topos is a topos
an (∞,1)-topos is what Higher Topos Theory calls an -topos
There is a periodic table of -categories:
→ ↓ |
… | ||||||
---|---|---|---|---|---|---|---|
point | truth value | set | groupoid | 2-groupoid | ... | ∞-groupoid | |
" | " | poset | category | (2,1)-category | ... | (∞,1)-category | |
" | " | " | 2-poset | 2-category | ... | (∞,2)-category | |
" | " | " | " | 3-poset | ... | (∞,3)-category? | |
⋮ | ⋮ | ⋮ | ⋮ | ⋮ | ⋮ | ⋱ | ⋮ |
point | truth value | poset | 2-poset | 3-poset | ... | (∞,∞)-category/∞-poset? |
There are various model category models for collections of -categories.
The standard model structure on simplicial sets models (∞,0)-categories.
The Joyal-model structure on simplicial sets models (∞,1)-categories.
The Charles Rezk-model structure for Theta spaces models general -categories.
(n,r)-category
Last revised on April 30, 2024 at 01:57:29. See the history of this page for a list of all contributions to it.