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Case Study IV: omega-categories December 15, 2007

Posted by Alexandre Borovik in Uncategorized.
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In higher-dimensional algebra, also known as higher-dimensional category theory, you encounter a ladder which you are irresistibly drawn to ascend. Let us begin with a finite set. About two elements of this set you can only say that they are the same or that they are different. Thinking about sets a little harder, you are led to consider what connects them, namely, functions (or perhaps relations). Taken together, sets and functions form a category. Now, there are two levels of entity, the objects (sets) and the arrows (functions), satisfying some conditions, existence of identity arrows and associative composition of compatible arrows.

There are plenty of examples of categories. For example, categories of structured sets, such as groups and homomorphisms, but also spatial ones, such as 2-Cob, whose objects are sets of circles and whose arrows are (diffeomorphism) classes of surfaces between them. A pair of pants/trousers represents an arrow from a single circle (the waist) to a pair of circles (the trouser cuffs) in this latter category.

In a category, the arrows between two objects A and B, {\rm Hom}(A, B), form a set. The only choice for two arrows in {\rm Hom}(A, B) is whether they are the same or different. At the level of objects, however, there is a new option. A and B need not be the same, but there may be arrows between them which compose to the identity arrows on each object. This kind of sameness is often called isomorphism. So-called `structuralists’ philosophers of mathematics are aware of this degree of sameness.

Let us continue up the ladder. Consider categories long enough, and you will start to think of functors between categories, and natural transformations between functors. Functors are ways of mapping one category to another. If the first category is a small diagram of arrows, a copy of that diagram within the second category would be a functor. Natural transformations are ways of mediating between two images within the target category. Taking all (small) categories, functors, and natural transformations, we have an entity with three levels, which we draw with dots, arrows and double-arrows. This is an example of a 2-category, the next rung of the ladder. In this setting, whether objects are the `same’ is not treated most generally as isomorphism, but as equivalence. Between equivalent objects there is a pair of arrows which do not necessarily compose to give identity arrows, but do give arrows for which there are invertible 2-arrows to these identity arrows.

This is something structuralists have missed. At the level of categories, isomorphism is too strong a notion of sameness. Anything you can do with, say, the category of finite sets, you can do with the equivalent (full) sub-category composed of a representative object for each finite number and functions between them.

Then, one more step up the ladder, 2-categories form a 3-category, with four levels of entity. Another example of a 3-category is the fundamental 3-groupoid of a space. Take the surface of a sphere, such as the world. Objects are points on the globe. 1-arrows between a pair of objects, say the North and South Poles, are paths. 2-arrows between pairs of 1-arrows, say the Greenwich Meridian and the International Date Line, are ways of sweeping from one path to the other. Finally, a 3-arrow between a pair of 2-arrows, say one that proceeds at a uniform rate between the Greenwich Meridian and the International Date Line and the other that tarries a while over New Delhi, is represented by a way of interpolating between these sweepings.

Mathematicians are aiming to extend this process infinitely far to infinite-dimensional categories, so-called \omega-categories, by defining them at one fell swoop. The idea for doing so was inspired by Alexandre Grothendieck, who realised that there was a way of treating spaces up to homotopy in algebraic terms if infinitely many levels of path between path were allowed. Already at the 2-category level there are many choices of shape to paste together. There are thus many ways of defining an \omega-category. At present, twelve definitions have been proposed. It is felt, however, that the choice is in a sense immaterial, in that all ways will turn out to be the `same’ at the level of \omega-categories, although each may be best suited to different applications.

There is a deep question here of why a complete treatment of sameness requires the construction of infinitely layered entities.

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Comments»

1. Jonathan Vos Post - December 19, 2007

Okay, does that mean that theorems should have the ritual statement: “assuming that the choice of twelve definitions is in a sense immaterial, in that all ways will turn out to be the `same’ at the level of omega-categories” in the same way as papers sometimes comment: “assuming the Riemann Hypothesis?”

2. dcorfield - December 19, 2007

Ideally, it would have been shown that all definitions are equivalent. Then whenever faced with a problem, one could choose the most convenient setting for resolving it.

Considerable effort has gone into comparing the definitions and many results are known. I don’t know of any case where someone would like to know something in one setting, proves its equivalent in another, and states that the original problem would also be solved had the definitions been shown to be equivalent.

3. serg271 - December 19, 2007

What about negative categories ?
I remember there was an explanation at sci.physics.research
0-category is set with functions
-1 category is set with one or element or empty
-2 category is set with one element
-3 category is set with one element and one morphism ?
-4 ?? some kind of unique structure ?
-n ??? progressively more unique ?

4. dcorfield - December 20, 2007

Regarding negative dimensional categories, you’re right about dimensions -1 and -2. Current thinking is that that’s where it stops. You can read Baez, Dolan and Bartels talk about it here.

5. Ronnie Brown - February 3, 2008

The question of why `sameness’ requires infinitely many levels has some well known counterparts in group theory and homological algebra, in the form of `free resolutions’. The basic problem is to describe in finite terms an infinite object, say an infinite group G. One chooses a hopefully finite set X of generators of G. However there are also relations between these generators, so we may hope to find a finite set R of relations between the generators. But there may be `identities among the relations’, and so it goes on. (These ideas in invariant theory go back to Hilbert’s `chains of syzygies’.) A `full’ description of the group G may thus require a free resolution of infinite length, but possibly with finitely many generators in each dimension.

The tension between finite language and the infinite world is a deep but everyday problem.

I should also point out that the idea of paths of paths, and so on, goes back a long way in homotopy theory. Where the homotopy theorists missed out seems to me was that they insisted on using groups, which are an example of algebraic structures with operations always always defined. Once you move to groupoids, which are like groups but having a partial multiplication, whole new worlds open up of sets with partial operations defined under geometric conditions. This is my idea of higher dimensional algebra, and it should be a delight to the confirmed algebraist! These new forms of algebra allow many strange forms of structure, and new insights into traditional areas.


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