[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:
-
one the one hand, you have everything that pertains to the structure:
- for monoids: that encompasses the binary operation $\circ$ of the monoid
-
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
\[\begin{cases} ss = st \\ ts = tt \end{cases}\]
$G_n$: 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
\[\begin{xy} \xymatrix{ x\ar@/^/[r]|{f}&x' & \ar@{~>}[r] && x\ar@(ul,ur)[]|{ {\rm id}_x}\ar@/^/[r]|{f} & x' \\ & \ar@{~>}[drr] &&&& \ar@{~>}[d] \\ && && && && \\ && && x\ar@/^1pc/[rr]|{ {\rm id}_x *_j^j f = f}&& x' } \end{xy}\]Right
\[\begin{xy} \xymatrix{ x\ar@/^/[r]|{f}&x' & \ar@{~>}[r] && x\ar@/^/[r]|{f} & x'\ar@(ul,ur)[]|{ {\rm id}_{x'}} \\ & \ar@{~>}[drr] &&&& \ar@{~>}[d] \\ && && && && \\ && && x\ar@/^1pc/[rr]|{f *_j^j {\rm id}_{x'} = f}&& x' } \end{xy}\]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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