The category number
A common joke is that everyone has a minimal number
such that, if someone mentions an -category,
one stops thinking. My personal number is maybe :
afterward my pet example “-dimensional cobordisms embedded into an
-dimensional-space” (see below) gets quite hard to imagine
(left aside the problem to draw these cobordisms).
Notation issuesI must
admit that I tend to be quite sloppy with the notation of
an -category. To be more precise,
I should write -category instead
of -category, i.e. I never assume that higher
morphisms are invertible.
But I am even sloppier sometimes, i.e. if I speak of an -category
I tend to ignore the difference between a weak and a strict -category.
Note that in case this can be (sometimes) justified by a corresponding
I am also very sloppy when it comes to more fundamental set theoretical and logical questions, e.g. the
subtleties one needs to define the “categories of categories” and related concepts.
Much more details, e.g. a discussion of weak and strict -categories,
can be, for example, found in Leinster's book here.
Higher categoriesThe idea behind higher category
theory is easy to explain. It is a generalization of category theory, which can be
seen as generalization of set theory.
The slogan for me is that a category is a collection of “set like structures”
with the category of sets as a blueprint example. A $2$-category is
a collection of “category like structures”
with the $2$-category of categories as a blueprint example. A
a collection of “$2$-category like structures”...
Imagine a set as a collection of points (aka -cells),
then a -category, also know as category, is a
one-dimensional structure connecting the objects with some one-dimensional cells called morphisms. The usual
notation for this is
Note that the morphisms come with an extra structure, i.e. they can be composed as
The composition should satisfy some standard axioms. The idea of higher
categories is to “add new structure, i.e. higher arrows between arrows”.
For example, if one starts with a usual category, then one can add two-dimensional
cells between the one-dimensional cells and one gets a picture like
These new arrows can be composed in two ways called vertical and horizontal.
This gives a -category a topological structure.
Note that the horizontal composition is a -cell
(both composition arrows are not pictured). The composition should satisfy some standard
axioms again. If one continues in this fashion, i.e. adding -cells
with ways of composition, and if one adds some axioms, then
one gets a -category.
Some basic examples of (strict and weak) -categories are:
A -category is defined in the
spirit from above. It consists of possible non-trivial -cells
for and only -cells for
are allowed to be non-invertible. For example a
-category is a groupoid and, more general, a
-category is a -groupoid.
Of course, it is possible to
consider the limits .
- The $2$-category of categories CAT, i.e. objects are categories,
-cells are functors and
-cells are natural transformations.
- The Bimodule $2$-category BiMOD i.e. objects are rings
, -cells are bimodules over two
bimodule homomorphisms. The one-dimensional composition is given by “tensoring
over the middle ring” (e.g. )
and the two-dimensional compositions are standard composition and “tensoring of morphisms”.
- Two-dimensional cobordism $2$-categories, e.g. one can
take points as objects, one-dimensional cobordisms as -cells and
cobordisms as -cells.
I am still a fool.
The arXiv version of this paper
The arXiv version of this paper
"There are two ways to do mathematics.
The first is to be smarter than everybody else.
The second way is to be stupider than everybody else - but persistent." -
based on a quotation from Raoul Bott.
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