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

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 $(G_n)_{n∈ℕ}$ and functions $s_n, t_n: G_n ⟶ G_{n-1}$ (which stand for "source" and "target" respectively) such that

\begin{cases} s \circ s = s \circ t && ⊛\\ t \circ s = t \circ t && ⊛⊛ \end{cases}

NB: to reduce the amount of notation, the subscripts in $s_n, t_n$ 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):

\begin{xy} \xymatrix{ \bullet\ar@/^2pc/[rr]|a="ab1"\ar@/_2pc/[rr]|b="ab2"&&\bullet } \ar@2{->}@/_/"ab1";"ab2"|\theta \end{xy}

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

• the points $\bullet$ are members of $G_0$
• $a, b∈ G_1$
• $𝜃 ∈ G_2$
• $s𝜃 ≝ a$
• $t𝜃 ≝ b$

we have to make sure that:

\begin{cases} sa = sb \\ ta = tb \end{cases}

i.e.:

\begin{cases} ss 𝜃 = st 𝜃 \\ ts 𝜃 = tt 𝜃 \end{cases}

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 $\circ$ 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. \; (x \circ y) \circ z = x \circ (y \circ z)
• the unit law:

∃e, ∀x. \; x \circ e = e \circ x = 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 $G_n$, such as:

• [Structure] Composition:

x \overset{f}{⟶} y \overset{g}{⟶} z \quad \overset{\_ \circ \_ }{⟹} \quad x \overset{g \circ f}{⟶} z
• [Law] Associativity:

(h \circ g) \circ f \overset{𝛼}{⟹} h \circ (g \circ f)

will now become elements of $G_{n+1}$ (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 $\circ$ is the Catalan number $C_n ≝ \frac{1}{n+1}{2n\choose n} \sim \frac{4^n}{n^{3/2}\sqrt{\pi}}$

\begin{xy} \xymatrix{ & & (k \circ h) \circ (g \circ f) \ar[dll]|\alpha \\ k \circ (h \circ (g \circ f)) && && ((k \circ h) \circ g) \circ f \ar[ddl]|\alpha \ar[ull]|\alpha \\ \\ & k \circ ((h \circ g) \circ f) \ar[uul]|\alpha && (k \circ (h \circ g) \circ) f \ar[ll]|\alpha } \end{xy}

## Strict $𝜔$-groupoids

Let’s take a look at what the $G_n$ ($n∈ℕ$) look like in Type Theory, if $G ≝ \bigsqcup\limits_{n≥0} G_n$ is a strict $𝜔$-groupoid.

First off, for all $n∈ℕ^\ast, \; a, b:G_{n-1}$, let’s define:

G_n(a, b) ≝ \lbrace f∈G_n \mid s f = a \text{ and } t f = b\rbrace

Let $n∈ℕ^\ast$.

The elements of $G_n$ will be called $n$-arrows.

Dimension skips:

If $i > j > k ∈ ℕ, \; \; x, x’ ∈ G_k, \; f, f’ ∈ G_j(x, x’)$ and $𝛼 ∈ G_i(f, f’)$, then $𝛼$ can be seen as $k$-arrow of $G_k(x, x’)$, since

\begin{cases} ss = st \\ ts = tt \end{cases}
\begin{xy} \xymatrix{ x\ar@/^2pc/[rr]|f="ab1"\ar@/_2pc/[rr]|{f'}="ab2"&& x' } \ar@2{->}@/_/"ab1";"ab2"|\alpha \end{xy}

### $G_n$: the structure

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

• Identity:

{\rm id}^n (\_ ) : \prod\limits_{a: G_{n-1}} G_n(a, a)
• Inverse:

{\rm inv}^n (\_ , \_ ) : \prod\limits_{a, b: G_{n-1}} G_n(a, b) ⟶ G_n(b, a)
• Vertical Composition (going along the $n$-arrows):

*^n_n (\_ , \_ , \_ ) : \prod\limits_{a, b, c: G_{n-1}} G_n(a, b) × G_n(b, c) ⟶ G_n(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 $G_n$:

*_1^n, \ldots, *_n^n

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

Let $i∈ℕ$ such that $1 ≤ i ≤ n$.

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

NB: For the diagrams, it will be assumed that

• the simple arrows are members of $G_i$
• the double arrows are members of $G_n$

Here we go:

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

*^n_i : \prod\limits_{x, x', x'': G_0}\prod\limits_{f, f': G_i(x, x'), g, g': G_i(x', x'')} G_n(f, f') × G_n(g, g') ⟶ G_n(f *_i^i g, \, f' *_i^i g') \\ \begin{xy} \xymatrix{ x\ar@/^2pc/[rr]|f="ab1"\ar@/_2pc/[rr]|{f'}="ab2"&& x'\ar@/^2pc/[rr]|g="bc1"\ar@/_2pc/[rr]|{g'}="bc2" && x'' } \ar@2{->}@/_/"ab1";"ab2"|\alpha \ar@2{->}@/_/"bc1";"bc2"|{\alpha'} \end{xy} \\ \cfrac{𝛼:G_i(x, x')\\ 𝛼': G_i(x', x'')}{𝛼 *_i^n 𝛼': G_i(x, x'')}
• Vertical Composition (it goes along the $n$-arrows):

*^n_n : \prod\limits_{x, x': G_0}\prod\limits_{f, f', f'': G_i(x, x')} G_n(f, f') × G_n(f', f'') ⟶ G_n(f, f'') \\ \\ \begin{xy} \xymatrix{ x\ar@/^3pc/[rr]|f="ab1"\ar[rr]|{f'}="ab2"\ar@/_3pc/[rr]|{f''}="ab3"&& x' } \ar@2{->}"ab1";"ab2"|\alpha \ar@2{->}"ab2";"ab3"|{\alpha'} \end{xy} \\ \\ \cfrac{𝛼:G_n(f, f')\\ 𝛼': G_n(f', f'')}{𝛼 *_n^n 𝛼': G_n(f, f'')}

### $G_n$: the laws

Now, things are becoming trickier, but fortunately $G_2$ 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 $\leadsto$ are to commute.

#### Identity ($i$)

Given that the point $x∈ G_i$, then

\begin{xy} \xymatrix{ x & \ar@{~>}[r] && x\ar@(ul,ur)[]|{ {\rm id}_x} } \end{xy}

#### Composition ($i < j$)

Given that

• the points $x, x’, x’’ ∈ G_i$
• the simple arrows $f, g ∈ G_j$

then

\begin{xy} \xymatrix{ x\ar@/^/[r]|{f}& x'\ar@/^/[r]|{g} & x'' & \ar@{~>}[r] && x\ar@/^1pc/[r]|{f *_j^j g} & x'' } \end{xy}

#### Identity Laws ($i<j$)

Given that

• the points $x, x’ ∈ G_i$
• the simple arrow $f∈ G_j$

then

##### Left
\begin{xy} \xymatrix{ x\ar@/^/[r]|{f}&x' & \ar@{~>}[r] && x\ar@(ul,ur)[]|{ {\rm id}_x}\ar@/^/[r]|{f} & x' \\ & \ar@{~>}[drr] &&&& \ar@{~>}[d] \\ && && && && \\ && && x\ar@/^1pc/[rr]|{ {\rm id}_x *_j^j f = f}&& x' } \end{xy}
\begin{xy} \xymatrix{ x\ar@/^/[r]|{f}&x' & \ar@{~>}[r] && x\ar@/^/[r]|{f} & x'\ar@(ul,ur)[]|{ {\rm id}_{x'}} \\ & \ar@{~>}[drr] &&&& \ar@{~>}[d] \\ && && && && \\ && && x\ar@/^1pc/[rr]|{f *_j^j {\rm id}_{x'} = f}&& x' } \end{xy}

#### Associativity law ($i < j$)

Given that

• the points $x, x’, x’’, x’’’ ∈ G_i$
• the simple arrows $f, g, h ∈ G_j$

then

\begin{xy} \xymatrix{ x\ar@/^/[r]|{f}& x'\ar@/^/[r]|{g} & x''\ar@/^/[r]|{h} & x''' & \ar@{~>}[r] && x\ar@/^/[r]|{f} & x'\ar@/^1pc/[r]|{g *_j^j h} & x''' \\ && \ar@{~>}[d] && && & \ar@{~>}[d] \\ && && && && \\ x\ar@/^1pc/[r]|{f *_j^j g} & x''\ar@/^/[r]|{h} & x''' & \ar@{~>}[r] && x\ar@/^2pc/[rrr]|{(f *_j^j g) *_j^j h = f *_j^j (g *_j^j h)} &&& x''' } \end{xy}

#### Interchange law ($i < j < k$)

Given that

• the points $x, x’, x’’ ∈ G_i$
• the simple arrows $f, f’, f’’, g, g’, g’’ ∈ G_j$
• the double arrows $𝛼, 𝛼’, 𝛽, 𝛽’ ∈ G_k$

then

\begin{xy} \xymatrix{ x\ar@/^3pc/[rr]|f="ab1"\ar[rr]|{f'}="ab2"\ar@/_3pc/[rr]|{f''}="ab3"&& x'\ar@/^3pc/[rr]|g="bc1"\ar[rr]|{g'}="bc2"\ar@/_3pc/[rr]|{g''}="bc3" && x'' \ar@{~>}@/_1pc/[rr]|{\text{horizontal}} && x\ar@/^3pc/[rr]|{f *_j^j g}="ac1"\ar[rr]|{f' *_j^j g'}="ac2"\ar@/_3pc/[rr]|{f'' *_j^j g''}="ac3"&& x' \\ && \ar@{~>}[dd]|{\text{vertical}} && && && \ar@{~>}[dd]|{\text{vertical}} \\ && && && && \\ && && && && \\ x\ar@/^2pc/[rr]|f="a'b'1"\ar@/_2pc/[rr]|{f''}="a'b'2"&& x'\ar@/^2pc/[rr]|g="b'c'1"\ar@/_2pc/[rr]|{g''}="b'c'2" && x'' \ar@{~>}@/^1pc/[rr]|{\text{horizontal}}&& x\ar@/^2pc/[rrrr]|{f *_j^j g}="a'c'1"\ar@/_2pc/[rrrr]|{f'' *_j^j g''}="a'c'2"&&&& x'' } \ar@2{->}"ab1";"ab2"|\alpha \ar@2{->}"ab2";"ab3"|{\alpha'} \ar@2{->}"bc1";"bc2"|\beta \ar@2{->}"bc2";"bc3"|{\beta'} \ar@2{->}"ac1";"ac2"|{\alpha *_j^k \beta} \ar@2{->}"ac2";"ac3"|{\alpha' *_j^k \beta'} \ar@2{->}"a'b'1";"a'b'2"|{\alpha *_k^k \alpha'} \ar@2{->}"b'c'1";"b'c'2"|{\beta *_k^k \beta'} \ar@2{->}"a'c'1";"a'c'2"|{(\alpha *_j^k \beta) *_k^k (\alpha' *_j^k \beta') = (\alpha *_k^k \alpha') *_j^k (\beta *_k^k \beta')} \end{xy}

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