Questions tagged [categorification]

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13
votes
0answers
204 views

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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0answers
106 views

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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0answers
199 views

What is categorification? (version 2.0)

A decade ago, Gil Kalai asked the question What precisely Is "Categorification"? After seeing some answers and some online pages, I think one of the meanings of categorification is the ...
14
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1answer
584 views

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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0answers
132 views

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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0answers
137 views

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 $$ \Omega^{d}_{H}(BG)= \mathbb{Z}_m \oplus \dots, $$ ...
6
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0answers
150 views

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 ...
4
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1answer
211 views

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 ...
26
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6answers
2k views

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}(\...
9
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0answers
170 views

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 +\...
7
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0answers
197 views

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 ...
9
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1answer
251 views

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 ...
4
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2answers
283 views

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....
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0answers
143 views

Categorification of the Taylor polynomial of a function at a point

Can you give a categorification of the Taylor polynomial of a real function $ f : U \subset \mathbb{R} \to \mathbb{R} $ defined by : $$ f(x) = f(a) + f'(a) (x-a) + \frac{1}{2} (x-a)^{2} + \dots + \...
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0answers
99 views

Partial Flag Varieties and Quotients of Symmetric Polynomials

$\def\Q{\mathbf Q}\DeclareMathOperator{\Gr}{Gr}$First, consider a Grassmannian $\Gr(k, N)$ of $k$-dimensional subspaces in an $N$-dimensional space. It is known that its cohomology ring is $$H_k=\Q[...
15
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1answer
827 views

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 ...
2
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0answers
119 views

Unnormalized Kauffman homology of the unknot

Is the unnormalized Kauffman homology of the unknot known? The Poincare polynomial of HOMFLY homology of the unknot is known as $$\frac{1+at}{1-q}.$$ Is the Poincare polynomials of Kauffman homology ...
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0answers
516 views

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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2answers
615 views

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 ...
10
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1answer
299 views

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. ...
22
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1answer
1k views

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 ...
5
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2answers
380 views

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 ...
25
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1answer
2k views

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 ...
17
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1answer
795 views

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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0answers
225 views

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 ...
8
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3answers
938 views

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(\...
6
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1answer
416 views

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, ...
14
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2answers
1k views

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 ...
5
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0answers
424 views

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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1answer
1k views

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 ...
17
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2answers
2k views

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 ...
5
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3answers
1k views

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 ...
3
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1answer
421 views

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 ...
28
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4answers
5k views

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 ...
15
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1answer
529 views

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\...
29
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19answers
5k views

Examples of categorification

What is your favorite example of categorification?
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1answer
688 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) ...
3
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1answer
353 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 ...
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11answers
9k 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.
15
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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 ...
8
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4answers
603 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 ...
11
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2answers
538 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 ...
11
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4answers
658 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 ...
15
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2answers
859 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 ...
8
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1answer
488 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 &...
14
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3answers
902 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 ...
16
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3answers
2k views

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

Polynomials in 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 ...
1
vote
1answer
639 views

Are cyclotomic Khovanov-Lauda-Rouquier algebras symmetric?

Recall that for k a field, a finite dimensional k-algebra A is called symmetric if it is isomorphic to its dual as a bimodule of itself. Which is to say, there's a trace map t:A -> k such that t(ab)=...
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7answers
3k views

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 ...