# Questions tagged [categorification]

The tag has no usage guidance.

52 questions
Filter by
Sorted by
Tagged with
2answers
396 views

### What is a most natural categorification of a vector space?

Few days ago I became excited when I learned from an answer to Examples of simple vertex operator algebras (VOAs) that The irreducible modules of the rank $d$ free boson are naturally parametrized by ...
0answers
149 views

### references on categorification of knot invariants

I am extremely sorry if this is not the right place for this kind of question. I have studied some knot theory, quantum invariants and would like to study more about categorification of knot ...
1answer
186 views

### Categorified probability and statistics?

To put it simply, what if the sample space underlying our probability space is a category instead of a mere set. Has a theory or probability and statistics been developed for such a situations in ...
0answers
223 views

0answers
185 views

19answers
5k views

### Examples of categorification

What is your favorite example of categorification?
1answer
706 views

### Are homological knot invariants of finite type?

It is well known that, after a change of variables, the quantum knot invariants (Jones, HOMFLY, Kauffman, etc.) can be written as power series whose coefficients are finite type (i.e., Vassiliev) ...
1answer
363 views

### Categorifying the group representations

I've heard about this construction on the lecture about higher representation theory: Given a Lie algebra $g$, one constructs $\mathcal A$, a category whose $K_0$ is the universal enveloping ...
13answers
11k views

### What precisely Is “Categorification”?

(And what's it good for.) Related MO questions (with some very nice answers): examples-of-categorification; can-we-categorify-the-equation $(1-t)(1+t+t^2+\dots)=1$?; categorification-request.
6answers
1k views

### What structure on a monoidal category would make its 2-category of module categories monoidal and braided?

So, many of us know the answer to "what kind of structure on an algebra would make its category of representations braided monoidal": your algebra should be a quasi-triangular Hopf algebra (maybe if ...
4answers
638 views

### What's the right object to categorify a braided tensor category?

The yoga of categorification has gained a lot of popularity in recent years, and some techniques for it have made a lot of progress. It's well-understood that, for example, a ring is probably ...
2answers
553 views

### What do decategorification and “compactification on a circle” have to do with each other?

Some physicists have told me that if you think about an extended n-dimensional TQFT $F$, then the decategorification is given by $F'(X)=F(X\times S^1)$, which I believe they call "compactification on ...
4answers
677 views

### Is there a categorification of the integers in terms of “graded sets”?

One way to categorify the non-negative integers is to consider the category FinSet whose objects are finite sets and whose morphisms are functions. The isomorphism classes of objects in FinSet can be ...
2answers
896 views

### Is there any meaning to a “nice bijective proof?”

From Zeilberger's PCM article on enumerative combinatorics: The reaction of the combinatorial enumeration community to the involution principle was mixed. On the one hand it had the universal ...
1answer
521 views

### Determinant of a pullback diagram

Suppose that X and Y are finite sets and that f : X → Y is an arbitrary map. Let PB denote the pullback of f with itself (in the category of sets) as displayed by the commutative diagram PB &...
2answers
951 views

### Categorifying the Reals via von Neumann Algebras?

So one way to categorify the natural numbers is to replace them with vector spaces. Then the dimension of the vector space reproduces the natural number. More generally you can categorify integers to ...
3answers
3k views

### Can we categorify the equation (1 - t)(1 + t + t^2 + …) = 1?

Polynomials in $\mathbb Z[t]$ are categorified by considering Euler characteristics of complexes of finite-dimensional graded vector spaces. Now, given a rational function that has a power series ...