[Nottingham Internship] Globular sets and strict -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
Monoids | 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
- Globular set:
-
it is a family of sets
and functions (which stand for "source" and "target" respectively) such that
NB: to reduce the amount of notation, the subscripts in
What’s intuition behind
Pretty simple, we want the sets to form kinds of cells (that’s what is meant by globular):
That is, for instance, in the above example, where
- the points
are members of
we have to make sure that:
i.e.:
As it happens,
NB: Thus,
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
of the monoid
- for monoids: that encompasses the binary operation
-
on the other hand, there are the laws (which are stated as logical formulas):
-
for monoids: they’re comprised of
-
the associativity law:
-
the unit law:
-
-
For strict
But when it comes to weak
To be more precise, in weak
-
[Structure] Composition:
-
[Law] Associativity:
will now become elements of
That will lead to a combinatory explosion; if only for associativity: the number of ways of associating
Strict -groupoids
Let’s take a look at what the
First off, for all
Let
The elements of
- Dimension skips:
-
If
and , then can be seen as -arrow of , since
: the structure
In
-
Identity:
-
Inverse:
-
Vertical Composition (going along the
-arrows):
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
Basically, if
Let
Remember that an
NB: For the diagrams, it will be assumed that
- the simple arrows are members of
- the double arrows are members of
Here we go:
-
Horizontal Composition (it goes along the
-arrows): -
Vertical Composition (it goes along the
-arrows):
: the laws
Now, things are becoming trickier, but fortunately
NB: in the following, the diagrams formed by the squiggly arrows
Identity ( )
Given that the point
Composition ( )
Given that
- the points
- the simple arrows
then
Identity Laws ( )
Given that
- the points
- the simple arrow
then
Left
Right
Associativity law ( )
Given that
- the points
- the simple arrows
then
Interchange law ( )
Given that
- the points
- the simple arrows
- the double arrows
then
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