2
$\begingroup$

Maybe there is no simple definition of $n$-category understandable for a physicist. Then I would like to know what are the trivial $0$-category, trivial $1$-category, trivial $2$-category, etc. How to obtain trivial $1$-category from trivial $0$-category? How to obtain trivial $2$-category from trivial $1$-category? etc

== add ==

After that I would like to know the definition of 0-category that allows us to obtain all the 0-categories, and the definition of 1-category that allows us to obtain all the 1-categories etc.

I am looking for a particular kind of n-categories. Based on the answer of André Henriques, it appears that the trivial 0-category that I am looking for is an one-dimensional vector space. A less trivial 0-category is a n-dimensional vector space, which is a composition of n trivial 0-categories. The trivial 1-category is a category of vector spaces, with only one simple object, which is a trivial object corresponding to an one-dimensional vector space, and the composition of several simple objects gives us a composite object which corresponds to a finite dimensional vector space.

I also guess that the trivial 0-category (ie an one-dimensional vector space) and the composite 0-categories (ie n-dimensional vector spaces) are all the 0-categories (of this type). The category of all vectors spaces is the "trivial" 1-category. All possible 1-categories (of this type) are fusion categories. Along this line, the "trivial" 2-category is the collection of all fusion categories. Then what are the most general 2-categories. We need a definition here.

The purpose of this question is not to find out an answer, but to find out a proper way to ask the question. I hope after some exchanges, I know what really is the question that I want to ask.

$\endgroup$
7
  • 6
    $\begingroup$ What do you mean by "trivial"? $\endgroup$ Feb 9, 2014 at 7:29
  • $\begingroup$ I have a guess: The trivial $0$-category is an one-dimensional vector space. A less trivial $0$-category is a $n$-dimensional vector space, which is a composition of $n$ trivial $0$-categories. The trivial $1$-category is a category of vector spaces, with only one simple object, which is a trivial object corresponding to an one-dimensional vector space, and the composition of several simple objects gives us a composite object which corresponds to a finite dimensional vector space. $\endgroup$ Feb 9, 2014 at 7:34
  • $\begingroup$ Two comments: 1) As a (sort of) physicist, I think the construction given in this nlab page provides an intuitive notion of how to construct an n-category. 2) It sounds like what you're interested in then is a parallel construction which begins with Vect rather than Set, i.e. the n-fold iterated enrichment over Vect. Is this the case? $\endgroup$ Feb 9, 2014 at 8:01
  • 3
    $\begingroup$ That is a strange way to use the word "trivial" (and in particular it doesn't agree with how mathematicians use it; to me the trivial $0$-category is either the zero-element set or the one-element set, for example). Maybe you're asking for a description of deloopings of $\text{Vect}$? You can start here: ncatlab.org/nlab/show/2-vector+space $\endgroup$ Feb 9, 2014 at 8:19
  • 2
    $\begingroup$ @Matthew: I don't think the OP is looking for iterated enrichments. The pattern that begins "a vector space, the category of vector spaces," together with the fact that the OP is a physicist, strongly suggests that the OP is thinking about target catgories for $n$-dimensional (Atiyah-Segal style) field theories, and one physically relevant (I think) choice for these are iterated deloopings of $\text{Vect}$. $\endgroup$ Feb 9, 2014 at 19:55

1 Answer 1

11
$\begingroup$

I'll interpret "definition understandable for a physicist" to mean "please give me a bunch of examples to keep in mind". Here you go:

Here are some examples of 0-categories:

  • a set with $n$ elements.

  • a vector space of dimension $n$ (strictly speaking, that's an example of a 0-category with extra structure, namely a linear structure)

Here are some examples of 1-categories:

  • the category of all sets.

  • the category of all groups.

  • the category of all Lie algebras.

  • the category of all vectors spaces (once again, that's an example of a category with extra structure, namely a linear structure)

  • the category Mod-$A$ of all modules over a fixed algebra $A$ (once again, this has a linear structure).

  • the category of d-dimensional cobordisms between (d-1)-dimensional manifolds.

  • the fundamental groupoid of a topological space.

Here are some examples of 2-categories:

  • the 2-category of algebras, bimodules, and maps between bimodules.

  • the 2-category of cobordisms, where you allow corners of codimension 2.

  • the fundamental 2-groupoid of a space (you look at points of the space, paths between points, and homotopies between paths).

  • the 2-category of categories, functors, and natural transformations.

  • the 2-category of $\mathcal C$-modules, where $\mathcal C$ is a fixed tensor category (again, there are notions of functors, and of natural transformations between those)

  • the 2-category of CFTs, their topological defects, and all possible (topological) field insertions.

Here are some examples of n-categories that work for any $n$:

  • the n-category of cobordisms, where you allow corners up to codimension $n$.

  • the fundamental n-groupoid of a topological space.

  • the category of $\mathcal C$-modules, where $\mathcal C=(\mathcal C,\otimes)$ is an $(n-1)$-category equipped with a monoidal product.

  • the collection of all $(n-1)$-categories.

  • the collections of all QFTs of dimension $n$, along with their (topological) defects of all possible dimensions (i.e. starting from domain walls, and going all the way down to point-like fields).

$\endgroup$
4
  • $\begingroup$ Thank you. This is very helpful! I am interested in the n-category related to the vector space. Here are my additional guesses: The trivial 0-category (ie an one-dimensional vector space) and composite 0-category (ie $n$-dimensional vector space) are all the 0-categories (of this type). The category of all vectors spaces is the "trivial" 1-category. All possible 1-categories (of this type) are unitary fusion categories. Along this line, the "trivial" 2-category is the collection of all unitary fusion categories. Then what are the most general 2-categories. We need definition $\endgroup$ Feb 9, 2014 at 8:21
  • $\begingroup$ I would like to have a definition that allows us to obtain all the 2-categories (and 3-categories, etc) of this type. Maybe "of this type" = the linear structure that you mentioned. It seems that "the collection of all (n−1)-categories" only give us the "trivial" n-category. How to define the the most general non-trivial n-categories (of this type)? $\endgroup$ Feb 9, 2014 at 8:42
  • 6
    $\begingroup$ I see... So really your question is how does one best continue the sequence: [0:finite dimensional Hilbert spaces], [1:unitary fusion categories], [2:????] (or maybe a somewhat easier question would be how to best continue the sequence: [0:finite dimensional vector spaces], [1:fusion categories], [2:????]). One thing to realize is that this sequence is really doubly indexed. Vector spaces sit in degree $(0,\infty)$, whereas fusion categories sit in degree $(1,1)$. Braided fusion categories sit in degree $(1,2)$. Have a look at ncatlab.org/nlab/show/periodic+table for more info. $\endgroup$ Feb 9, 2014 at 20:23
  • $\begingroup$ Finite dimensional vector spaces are the dualizable objects in all vector spaces. In arxiv.org/pdf/1312.7188.pdf it is shown that fusion categories are dualizable objects in a certain 3 category. I suspect therefore that what you are looking for is a sequence of monoidal n-categories in which the dualizable objects are interesting. $\endgroup$
    – Adam Gal
    Feb 10, 2014 at 0:52

Not the answer you're looking for? Browse other questions tagged or ask your own question.