Questions tagged [categorification]
The categorification tag has no usage guidance.
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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.
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Examples of categorification
What is your favorite example of categorification?
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Intuitive explanation of Burnside's Lemma
Burnside's Lemma states that, given a set $X$ acted on by a group $G$,
$$|X/G|=\frac{1}{|G|}\sum_{g\in G}|X^g|$$
where $|X/G|$ is the number of orbits of the action, and $|X^g|$ is the number of ...
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Analogy between the exterior power and the power set
The symmetric algebra of an object exists in every cocomplete $\otimes$-category. For the category of sets $\mathrm{Sym}(X)$ is the set of multi-subsets of $X$.
The usual definition of the exterior ...
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Categorifications of the real numbers
For the purposes of this question, a categorification of the real numbers is a pair $(\mathcal{C},r)$ consisting of:
a symmetric monoidal category $\mathcal{C}$
a function $r\colon \mathrm{ob}(\...
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Categorification of the integers
I would like to know a natural categorification of the rig of integers $\mathbb{Z}$. This should be a $2$-rig. Among the various notions of 2-rigs, we obviously have to exclude those where $+$ is a ...
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Can adjoint linear transformations be naturally realized as adjoint functors?
Last week Yan Zhang asked me the following: is there a way to realize vector spaces as categories so that adjoint functors between pairs of vector spaces become adjoint linear operators in the usual ...
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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 ...
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Has this "backwards" perspective on toposes been studied?
Topos theory can be seen as a categorification of topology via the following analogies.
\begin{array}{|c|c|}
\hline
\text{locales}&\text{Grothendieck toposes}\\\hline
\text{open sets}&\text{...
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Categorification of Floer homology
Floer homology associates a vector space $HF^\ast(L_1,L_2)$ to any pair of Lagrangian submanifolds $L_1,L_2$ inside a symplectic manifold $X$. By a categorification of Floer homology, I mean a ...
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Which cluster algebras have been categorified?
In "Tilting Theory and Cluster Combinatorics" Buan, Marsh, Reineke, Reiten, and Todorov constructed cluster categories for mutation finite cluster algebras (without coefficients), and Amiot gives a ...
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Can we categorify the formula for the quadratic Gauss sum?
Background
Fix an odd prime $p$ and set $\zeta=e^{2\pi i/p}$. We define the quadratic Gauss sum as
$$g=\sum_{n=0}^{p-1} \zeta^{n^2}.$$
It's pretty easy to show that
$$g^2=
\begin{cases}
p & \...
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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 ...
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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 ...
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Is there a good notion of "induction" for representations of 2-categories?
One of the most important observations in the representation theory of algebras is that if one has a subalgebra $A\subset B$, then these is a functor $B\otimes_A -\colon A\operatorname{-pmod}\to B\...
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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 ...
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Where does the canonical basis differ from the KLR basis?
The question implicitly asked in Ben Webster's question is: Does the canonical basis of Uq(n+) agree with the basis coming from categorification via Khovanov-Lauda-Rouqier algebras?
Thanks to ...
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What is decategorification?
A decategorification is, roughly, some procedure $\Phi$ which inputs some sort of $n$-categorical data and outputs some sort of $(n-1)$-categorical data. Whatever this means, categorification is ...
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Adding inverses to a symmetric monoidal category (Reference?)
As we all know, the forgetful functor $\mathsf{Ab} \to \mathsf{CMon}$ from abelian groups to commutative monoids has a left adjoint, the Grothendieck group. I would like to categorify this ...
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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) ...
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Categorification of "Every domain embeds into a field"?
In the category of commutative rings, every domain embeds into a field. Is this true in the category of presentably symmetric monoidal stable $\infty$-categories? Here's what I mean by that.
Let $...
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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 ...
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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 ...
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Non-abelian freeness of SU_2
The distribution of the trace of a random element of $SU_2$ is the Sato-Tate distribution.
The analogue of the Gaussian distribution in free probability theory is the Wigner semicircle distribution.
...
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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 &...
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Is the instanton homology for webs and foams a categorified Chern-Simons?
In their paper, Kronheimer and Mrowka constructed an instanton homology $J^{\#}$ for webs and foams and conjectured that for planar webs, $\dim J^{\#}=\#\text{ of Tait colorings}$. According to my ...
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Horizontal categorification: Two questions
According to the nlab, horizontal categorification is a process in which a concept is realized to be equivalent to a certain type of category with a single object, and then this concept is generalized ...
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Categorifying Hyperoperations
Is there some categorical version of tetration or higher hyperoperations?
This is motivated by the fact that coproducts categorify addition of finite cardinals, and products/exponentials categorify ...
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What is the "classical limit" of Khovanov homology?
Let me first explain what I mean by the "classical limit".
For quantum group invariants of links and webs (such as colored Jones polynomials), the "classical limit" means the limit $k\rightarrow +\...
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Is there a categorification of topological K-theory?
For a compact Hausdorff topological space $X$, its K-theory $K^0(X)$ is defined to be the Grothendieck group of the isomorphism classes of finite dimensional vector spaces on $X$. For example $K^0(\...
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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 ...
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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 ...
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categorifying induction in homotopy type theory
In trying to understand homotopy type theory, I stumbled upon the following silly question, which is likely to be trivial for the experts.
Let $B=\sqcup_{n\in\Bbb N} BS_n$, which I'd like to think of ...
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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 ...
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What is the mathematical structure of 2d TQFT from the 2d foam category (instead of 2d cobordism category)?
It is well-known that the category of 2d TQFTs is equivalent to the category of commutative Frobenius algebras.
What about functors from the 2d foam category (instead of 2d cobordism category) to ...
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Categorification request
Possible Duplicate:
Can we categorify the equation (1 - t)(1 + t + t^2 + …) = 1?
Can you give a categorification of the geometric series identity:
$$1/(1-x)=1+x+x^2+...$$
Categorifications ...
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The 2-group of extensions
Let $A,B$ objects of an abelian category. Then we can define the abelian group $\mathrm{Ext}^1(A,B)$ as the set of isomorphism classes of extensions $0 \to B \to E \to A \to 0$, endowed with the Baer ...
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Computations and applications of Khovanov's functor valued invariant of tangles
Soon after his famous paper "A categorification of the Jones polynomial", Khovanov
introduced a "bordered" version. His theory assingns to every oriented even tangle
a complex of (H_n,H_m)-bimodules, ...
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Categorifying skein algebras?
We can obtain the Jones polynomial by the Temperly-Lieb algebra and the HOMFLYPT polynomial from the Hecke algebra. Were there attempts to categorify the algebras itself and obtain the Khovanov ...
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Categorification-like statement in the cobordism group?
Suppose we consider a $d$-cobordism group classifying manifolds with $H$-structure and with a classifying space $BG$ of a group $G$, written as
$$
\Omega^{d}_{H}(BG)= \mathbb{Z}_m \oplus \dots,
$$
...
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Categorifying idempotent relations
Generalizing partial orders: A relation $R$ is transitive if $R \circ R \subseteq R$ and interpolative if $R \subseteq R \circ R$. It is idempotent if $R \circ R = R$. Interpolativeness means that ...
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Categorification of spaces and models for set theory
One aspect of topos theory is that it provides an enlarged view of the classical concept of space. Indeed, one may thought that the notion of topos is a sort of categorification of the notion of space....
user95283
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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 ...
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Euler characteristic, character of group representation and Riemann Roch theorem
I am considering the following set up:Let $G$ be a finite group,let $Rep(G)$ denote the category of finite dimensional representations over $\mathbb{C}$. Let $V,W$ be representations of $G$ in $Rep(G)$...
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Indeterminates and forgetful functors
Forgive me if this is a well-known observation/result, but I'm quite new to graduate-level algebra and I was wondering if there are generalisations of the constructions I describe below.
It's ...
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What are meromorphic line bundles?
Initially I wanted to call this question "Categorification of meromorphic functions?" but discovered so many questions about categorification that I became scared and decided to replace it ...
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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 ...
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Reference Request: The Categorification of $\mathbb{Z}$ as cochain complexes of vector spaces
Just as the fact that a categorification of $\mathbb{N}$ is the category of finite dimensional vector spaces, a categorification of $\mathbb{Z}$ (in my mind) is the category of bounded cochain ...
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An alternative (?) approach to differential ringoids
Recall that a derivation on a ring $R$ is a function $\partial:R\to R$ satisfying the Leibniz rule $$\partial(ab)=\partial(a)b+a\partial(b),$$ and a differential ring is just a ring equipped with a ...
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The relationship between representations of groups and evaluation and coevaluation maps for $vect_{G}$ module categories
Let $G$ be a finite group and $vec_{G}$ be the monoidal category of finite dimensional $G$-graded vector spaces.
Given any $vec_{G}$ module category $\mathcal{M}$ we can define a dual module category ...
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