[Nottingham Internship] Globular sets and strict 𝜔-groupoids: welcome to infinity land!

6 minute read

Again, all of this has been explained to me by Paolo Capriotti: props to him!

We’re beginning to talk about the crux of the matter: here are weak 𝜔-groupoids!

  Monoids Categories 𝜔-Groupoids and Higher Categories
Stuff/Underlying carrier Sets Multigraphs (also called “quivers”) Globular Sets
Structure Binary operation Binary operation, Identity  
Laws Associativity, Unit Associativity  

Globular sets

As shown the above table, the underlying carrier of an 𝜔-groupoid a globular set.

Globular set:

it is a family of sets (Gn)n and functions sn,tn:GnGn1 (which stand for "source" and "target" respectively) such that

{ss=stts=tt

NB: to reduce the amount of notation, the subscripts in sn,tn are omitted.

What’s intuition behind and ?

Pretty simple, we want the sets to form kinds of cells (that’s what is meant by globular):

abθ

That is, for instance, in the above example, where

  • the points are members of G0
  • a,bG1
  • 𝜃G2
  • s𝜃a
  • t𝜃b

we have to make sure that:

{sa=sbta=tb

i.e.:

{ss𝜃=st𝜃ts𝜃=tt𝜃

As it happens, s𝜃=a and t𝜃=b are said to be parallel.

NB: Thus, and enable us to ensure that all sources and their corresponding targets are parallel.

Laws and Structure

The bottom line is that for usual mathematical objects such as monoids, the enriched layers over the carrier are twofold:

  1. one the one hand, you have everything that pertains to the structure:

    • for monoids: that encompasses the binary operation of the monoid
  2. on the other hand, there are the laws (which are stated as logical formulas):

    • for monoids: they’re comprised of

      • the associativity law:

        x,y,z.(xy)z=x(yz)
      • the unit law:

        e,x.xe=ex=x

For strict 𝜔-groupoids, the pattern remains the same: there are structures (composition, identity, inverse) and laws (identity laws, associativity, interchange law).

But when it comes to weak 𝜔-groupoids, there’s nothing but structure.

To be more precise, in weak 𝜔-groupoids, the laws and structures with regard to elements of Gn, such as:

  • [Structure] Composition:

    xfygz__xgfz
  • [Law] Associativity:

    (hg)f𝛼h(gf)

will now become elements of Gn+1 (and thus pertain to the structure).

That will lead to a combinatory explosion; if only for associativity: the number of ways of associating n applications of is the Catalan number Cn1n+1(2nn)4nn3/2π

(kh)(gf)αk(h(gf))((kh)g)fααk((hg)f)α(k(hg))fα

Strict 𝜔-groupoids

Let’s take a look at what the Gn (n) look like in Type Theory, if Gn0Gn is a strict 𝜔-groupoid.

First off, for all n,a,b:Gn1, let’s define:

Gn(a,b){fGnsf=a and tf=b}

Let n.

The elements of Gn will be called n-arrows.

Dimension skips:

If i>j>k,x,xGk,f,fGj(x,x) and 𝛼Gi(f,f), then 𝛼 can be seen as k-arrow of Gk(x,x), since

{ss=stts=tt
xffxα

Gn: the structure

In Gn, we’re given an identity, an inverse, and composition maps.

  • Identity:

    idn(_):a:Gn1Gn(a,a)
  • Inverse:

    invn(_,_):a,b:Gn1Gn(a,b)Gn(b,a)
  • Vertical Composition (going along the n-arrows):

    nn(_,_,_):a,b,c:Gn1Gn(a,b)×Gn(b,c)Gn(a,c)

However, we can’t just settle for this.

The only composition that was presented was the “obvious” one, but the thing is that there are different types of compositions in Gn:

1n,,nn

Basically, if 1in, the intuition is that the i-th composition “goes along the i-arrows”.

Let i such that 1in.

Remember that an n-arrow can be seen as an i-arrow, since ni.

NB: For the diagrams, it will be assumed that

  • the simple arrows are members of Gi
  • the double arrows are members of Gn

Here we go:

  • Horizontal Composition (it goes along the i-arrows):

    in:x,x,x:G0f,f:Gi(x,x),g,g:Gi(x,x)Gn(f,f)×Gn(g,g)Gn(fiig,fiig)xffxggxαα𝛼:Gi(x,x)𝛼:Gi(x,x)𝛼in𝛼:Gi(x,x)
  • Vertical Composition (it goes along the n-arrows):

    nn:x,x:G0f,f,f:Gi(x,x)Gn(f,f)×Gn(f,f)Gn(f,f)xfffxαα𝛼:Gn(f,f)𝛼:Gn(f,f)𝛼nn𝛼:Gn(f,f)

Gn: the laws

Now, things are becoming trickier, but fortunately G2 actually illustrates the general case, as long as you keep in mind the following general patterns, in which we’ll perform dimension skips.

NB: in the following, the diagrams formed by the squiggly arrows are to commute.

Identity (i)

Given that the point xGi, then

xxidx

Composition (i<j)

Given that

  • the points x,x,xGi
  • the simple arrows f,gGj

then

xfxgxxfjjgx

Identity Laws (i<j)

Given that

  • the points x,xGi
  • the simple arrow fGj

then

Left
xfxxidxfxxidxjjf=fx xfxxfxidxxfjjidx=fx

Associativity law (i<j)

Given that

  • the points x,x,x,xGi
  • the simple arrows f,g,hGj

then

xfxgxhxxfxgjjhxxfjjgxhxx(fjjg)jjh=fjj(gjjh)x

Interchange law (i<j<k)

Given that

  • the points x,x,xGi
  • the simple arrows f,f,f,g,g,gGj
  • the double arrows 𝛼,𝛼,𝛽,𝛽Gk

then

xfffxgggxhorizontalxfjjgfjjgfjjgxverticalverticalxffxggxhorizontalxfjjgfjjgxααββαjkβαjkβαkkαβkkβ(αjkβ)kk(αjkβ)=(αkkα)jk(βkkβ)

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