Higher category theory
higher category theory
Extra properties and structure
For the category – Joyal’s disk category or cell category – may be thought of as the full subcategory of the category of strict n-categories on those -categories that are free on pasting diagrams of -globes.
For instance contains an object that is depicted as
being the pasting diagram of two 2-globes along a common 1-globe and of the result with a 1-globe and another 2-globe along common 0-globes.
Such pasting diagrams may be alternatively be encoded in planar trees, the above one corresponds to the tree:
Accordingly, is also the category of planar rooted trees of level .
In low degree we have
is the point.
is the simplex category: the -simplex is thought of as a linear quiver and as such the pasting diagram of 1-morphisms
Dually, this is the planar rooted tree of the form
We discuss two equivalent definitions
Via the free strict ω-category
Let denote the free strict ω-category generated from the terminal globular set .
Notice that this terminal globular set consists of precisely one -globe for each : one point, one edge from the point to itself, one disk from the edge to itself, and so on.
So is freely generated under composition from these cells. As described above, this means that each element of the underlying globular set of may be depicted by a pasting diagram made out of globes, and such a pasting diagram itself may be considered as a globular set whose -cells are instances of the -globes appearing in the diagram.
We now describe this formally.
The n-cells of may be identified with planar trees of height , which by definition are functors
( is the category of simplices and is a simplex, i.e., ordered set , regarded as a category) such that . Such a is exhibited as a chain of morphisms in ,
and we will denote each of the maps in the chain by . Thus, for each , there is a fiber which is a linearly ordered set. (Need to fill in how composition of such trees is defined.)
To each planar tree we associate an underlying globular set , as follows. Given , define a new tree where we adjoin a new bottom and top , to every fiber of , for every :
Now define a -sector to be a triple where and are consecutive edges of . A -cell of the globular set is a -sector where . If , the source of a -cell is the -cell and the target is the -cell where are consecutive elements in . It is trivial to check that the globular axioms are satisfied.
Now let denote the free strict -category generated by the globular set .
is the full subcategory of on the strict ω-categories , as ranges over cells in the underlying globular set of .
Via iterated wreath product
(Berger, section 3)
For all there is a canonical embedding
given by .
Via duals of disks
In analogy to how the simplex category is equivalent to the opposite category of finite strict linear intervals, , so the -category is equivalent to the opposite of the category of Joyal’s combinatorial finite disks.
Embedding of grids (products of the simplex category)
For any small category there is a canonical functor
(Berger, def. 3.8)
Embedding into strict -categories
Write for the category of strict n-categories.
There is a dense full embedding
This was conjectured in (Batanin-Street) and shown in terms of free -categories on -graphs in (Makkai-Zawadowsky, theorem 5.10) and (Berger 02, prop. 2.2). In terms of the wreath product presentation, prop. 1 this is (Berger 05, theorem 3.7).
Under this embedding an object is identified with the -fold horizontal composition of the pasting composition of the -morphisms :
The pasting diagram
corresponds to the objects of given by
where in turn
Composing with the functor from remark 1 we obtain an embedding of -fold simplices into strict -categories
Under this embedding an object is sent to the -category which looks like (a globular version of) a grid of -cells.
for the inclusion of the gaunt strict -categoeries into all strict n-categories.
is the smallest full subcategory of containing the grids, the image of , example 3, and closed under retracts.
(B-SP, prop. 10.5)
The groupoidal version of is a test category (Ara).
In write for the unique object. Then write in
This is the strict n-category free on a single -globe.
The -categories were introduced in
A discussion with lots of pictures is in chapter 7 of
More discussion is in
- David Oury, On the duality between trees and disks, TAC vol. 24 (pdf)
The following paper proves that is a test category
Discussion of embedding of into strict -categories is in
- Michael Makkai, M. Zawadowsky, Duality for simple -categories and disks, Theory Appl. Categories 8 (2001), 114–243
- Clemens Berger, A cellular nerve for higher categories, Adv. Math. 169 (2002), 118–175.
The characterization in terms of -fold categorical wreath products is in
see also section 3 of
there leading over to the notion of Theta space.
The groupoidal version is discussed in
The relation of to configuration spaces of points in the Euclidean space is discussed in
Related discussion in the context of (infinity,n)-categories is also in