Globular sets as models of (n,r)-categories

One of the concerns of mine is the presheaf theory of categories and higher categories, and the use of presheaf topoi to create models for higher categories. The basic presheaf topoi I would like to focus in on are the (n,r)-globular sets.

Definition. Let a 0-globular set be a set. Then an (n+1)-globular set is a quiver with an n-globular set as its hom classes.

So for example, a quiver is a 1-globular set and all of its hom classes are simply sets. A 2-globular set is simply a quiver such that each hom class is itself its own quiver. Equivalently, that means that a 2-globular set is like a 2-quiver with the condition that $s \circ t = s \circ s$ and $t \circ s = t \circ t$ so that all 2-morphisms are between parallel pairs of 1-morphisms. The relationship to topos theory is that these are presheaves over the globe category $G_n$.

Definition. let $G_n$ denote the nth globe category. This has source and target morphisms from (n+1)-morphisms to n-morphisms with the condition that $s \circ t = s \circ s$ and $t \circ s = t \circ t$ for each source and target morphism.

It follows that an n-globular set is actually an object of the presheaf topos $Sets^G_n$. It follows that we can use presheaf topos theory to reason about n-globular sets.

Proposition. the section preorder of an n-globular set is a height (e.g max chain length) n+1 partial order.

Proof. the n-globe category is an endotrivial partially ordered category, so it follows that the section preorder is a partial order. Its chains are determined by chains of n-morphisms followed by (n-1)-morphisms all the way to 0-morphisms. It follows that the chain can have length no more then $(n+1)$. $\square$

The subobject lattice $Sub(G)$ of a globular set $G$ in the topos $Sets^{G_n}$ is simply a distributive lattice of the max chain length n+1 partial order. The congruence lattice $Con(G)$ is formed in the standard way it is formed for any presheaf object in a topos. The first thing we see is that $Sets^{G_n}$ lacks identity morphisms so it is not a good model for n-categories. Instead it is a model for n-semicategories. To resolve this we define and implement unital n-globular sets.

Definition. we construct the unital n-globe category $U_n$ in the following manner:

• we have n-morphisms for all 0,...,n
• we have nth source and nth target morphisms for each m-morphism going to an (m-n) morphism. These are denoted $s^n$ and $t^n$
• we have nth identity morphisms $id^n$ that take an m-morphism to an (m+n) identity morphism
• every morphism in $U_n$ can canonically be represented as a pair of $id^n s^m$ or as a pair $id_n t^m$ representing the n-th identity morphism on an mth-source or an mth-target morphism.
• the composition of two morphisms $id^w s^x$ and $id^y t^z$ proceeds by canceling out as many identities as possible by removing pairs of $s$ and $id$. Then if identities are remaining they get $id^{y-x+w}$ by combining identity morphisms as the identity part in front of $t_z$. If $x > y$ then we cancel out $id^y$ and we get $id^w s^{x-z}$ because when we get $s^n$ composed with $t^n$ we combine the two with the source/target type the last to be called. So this can canonically define composition in the unital n-globe category.

It follows that a unital n-globular set is simply a copresheaf in the fundamental topos $Sets^{U_n}$. This presheaf definition may not be the exact one you always want so another approach is as follows:

Definition. a unital n-globular set can be defined from the data of a n-globular set as well as identity morphisms for each m-morphism with $m \lt n$. The identity morphisms for a morphism $m$ have the property that their source and target morphisms are both $m$.

We now have an underlying presheaf model for n-categories. The distinction between the two is readily apparent:

• A 2-semicategory can be defined as a compositionally enriched 2-globular set. An ordered semigroup, for example, is a 2-semicategory. As a not necessarily globular set, it doesn't require the definition of identities.
• A 2-category is a compositionally enriched unital 2-globular set. It has identity morphisms for each object, and two identity morphisms for each 1-morphism defined by its underlying globular structure.

We now have two different topoi, with the later one, $Sets^{U_n}$, providing a better presheaf topos theoretic model for n-categories, and yet our discussion still cannot end here. There is one more detail, which leads to the formation of $(n,r)$-categories. As we shall see each of these different $(n,r)$-categories actually has its own special type of underlying globular structure.

We can start with groupoids which are $(1,0)$-categories. In the most basic case we might treat a groupoid as simply a special case of a 1-category. But in actually, a groupoid is a structured dependency quiver, which is to say a unital quiver with an extra inverse function $inv: Arrows(Q) \to Arrows(Q)$ that takes any edge and maps it to its inverse. These dependency quivers are actually used in the topos theory of undirected graphs, whilst ordinary quivers are used in the topos theory of directed graphs.

We now actually have a better terminology for dependency quivers. They can now be considered to simply be the unital (1,0)-globular sets. So the $(n,r)$ category theory terminology matches the underlying globular set terminology, and this same approach in fact generalizes onwards to infinity.

The next case is that of 2-categories. Here we distinguish between three cases: (2,0) categories which are simply the 2-groupoids, (2,1) categories which have all 2-morphisms invertible, and ordinary 2-categories. Each of these different types of (n,r) categories has their own corresponding class of globular set, in the same way that groupoids are distinguished by unital (1,0)-globular sets.

A (2,1)-globular set has an inverse for 2-morphisms, and a (2,0)-globular set has an inverse for both 2-morphisms and 1-morphisms. A (2,2)-globular set is just an ordinary unital 2-globular set. Each of these different types of globular sets correspond to special classes of presheaves, in the same way as we constructed presheaves for dependency quivers. This generalises to the theory of (n,r) globular sets.

• a (0,0)-globular set is just a set
• a (1,0) globular set is a dependency quiver
• a (1,1) globular set is a unital quiver
• a (2,0) globular set is a unital 2-globular set with an inverse function for 1-morphisms and 2-morphisms
• a (2,1) globular set is a unital 2-globular set with an inverse function for 2-morphisms
• a (2,2) globular set is a unital 2-globular set

So these different types of (n,r)-globular set determine the different types of (n,r)-category that can be formed over them. The type of an (n,r) category is determined by the underlying presheaf that it enriches.

Definition. the category $G_{(n,m)}$ is the unital globe category $U_n$ with an inverse adjoined for all morphisms with index greater then $m$.

Definition. an (n,r) globular set is a presheaf in the topos $Sets^{G_{(n,m)}}$.

Every $(n,r)$ type encodes its own type of globular set and it has its own topos. At the same time, the type of for semicategories, and n-semicategories is determined by having an n-globular set without identities, so much like how $(n,r)$ categories are distinguished from one another by their underlying globular set, the same is true for how semicategories are distinguished from their corresponding categories.

In fact, rather or not an object is a category, semicategory, groupoid, 2-semicategory, 2-category, 2-groupoid, (2,1)-category, etc can always be determined by the topos of its underlying globular set. So the highest level classification of categories comes from their globular topos. Categories and their generalisations can always be identified by two different aspects:

• Combinatorial structure: the underlying globular set of the category, rather or not it is an (n,r)-globular set, has identities, etc.
• Algebraic structure: the compositional aspect of the category, which allows you to extend a quiver or other globular set with the composition of morphisms.

The combinatorial structure of a category can always be understood with topos theory and selected topoi like $Sets^{G_n}$ and $Sets^{G_{(n,m)}}$. One of the things that I have strongly emphasized is the topos theoretic foundations of algebra, and that presheaves are the best models we have for algebraic stuctures, but the full realisation of this programme is a subject for further development.

We have noticed by this formalisation that the different types of $(n,r)$ categories are distinguished presheaf theoretically by the type of their underlying globular sets. A slightly different situation occurs when considering $\infty$-categories. $\infty$-categories are defined by other base topoi like $\infty$-globular sets and simplicial sets. An $\infty$-category is then just a different type of presheaf. Higher categories then fit nicely into our foundations in presheaf topos theory.

The development of higher category theory does not challenge our foundation in presheaf topos theory. If anything, it actually reinforces it because all the different types of higher category construction are determined by the different types of presheaves. We will model higher categories using presheaf topos theory.

References:
(n,r)-Categories
Globular sets
Geometric shapes for higher structures