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Categorification of a vector space such that a functor between these is a linear map?

A functor between two monoids seen as 1 object categories is essentially a monoid-homomorphism. What is the equivalent construction for vector spaces and linear maps?
rick's user avatar
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14 votes
1 answer
1k views

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 ...
Tim Campion's user avatar
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3 votes
1 answer
429 views

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 ...
მამუკა ჯიბლაძე's user avatar
2 votes
0 answers
116 views

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 ...
Alec Rhea's user avatar
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1 vote
1 answer
127 views

One object preadditive groupoids as a categorification of skew fields

Recall that a preadditive category is just a category $\mathcal{C}$ enriched in the category of abelian groups such that composition is linear with respect to the various group operations, so $$f\circ(...
Alec Rhea's user avatar
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1 vote
1 answer
242 views

Star-autonomous categories are categorifications of Boolean algebras?

I asked this question fourteen days ago on MathStackexchange (see here). I have not received any answers or comments until now. It seems to me that on MathStackexchange not many people are familiar ...
Max Demirdilek's user avatar
7 votes
2 answers
572 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 ...
მამუკა ჯიბლაძე's user avatar
8 votes
0 answers
207 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 ...
Jernej Grlj's user avatar
4 votes
1 answer
299 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 ...
Reza Rezazadegan's user avatar
13 votes
0 answers
239 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 $...
Tim Campion's user avatar
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9 votes
1 answer
310 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 ...
Martin Brandenburg's user avatar
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0 answers
264 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 ...
Praphulla Koushik's user avatar
19 votes
1 answer
861 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{...
Oscar Cunningham's user avatar
2 votes
0 answers
164 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 ...
Lazy Llama's user avatar
5 votes
0 answers
148 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 $$ \Omega^{d}_{H}(BG)= \mathbb{Z}_m \oplus \dots, $$ ...
wonderich's user avatar
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9 votes
0 answers
241 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 ...
Alec Rhea's user avatar
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5 votes
1 answer
361 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 ...
Jake B.'s user avatar
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27 votes
6 answers
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}(\...
Richard Hepworth's user avatar
9 votes
0 answers
225 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 +\...
Henry's user avatar
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7 votes
0 answers
245 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 ...
Henry's user avatar
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9 votes
1 answer
348 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 ...
Henry's user avatar
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4 votes
2 answers
325 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....
user avatar
1 vote
0 answers
165 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 + \...
YoYo's user avatar
  • 325
2 votes
0 answers
112 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[...
Bubaya's user avatar
  • 259
18 votes
1 answer
1k 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 ...
John Pardon's user avatar
  • 18.4k
2 votes
0 answers
128 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 ...
Satoshi  Nawata's user avatar
4 votes
0 answers
591 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 $Rep(G)$...
user41650's user avatar
  • 1,952
13 votes
3 answers
1k 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 ...
Martin Brandenburg's user avatar
10 votes
1 answer
341 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. ...
Will Sawin's user avatar
  • 139k
27 votes
2 answers
2k 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 ...
Martin Brandenburg's user avatar
6 votes
2 answers
622 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 ...
Martin Brandenburg's user avatar
28 votes
1 answer
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 ...
Martin Brandenburg's user avatar
17 votes
1 answer
1k 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 & \...
Aleksandar Bahat's user avatar
3 votes
0 answers
241 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 ...
Zhaoting Wei's user avatar
  • 8,737
8 votes
3 answers
1k 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(\...
Zhaoting Wei's user avatar
  • 8,737
6 votes
1 answer
477 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, ...
Paolo Aceto's user avatar
14 votes
2 answers
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 ...
Peter McNamara's user avatar
5 votes
0 answers
501 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 ...
7 votes
1 answer
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 ...
Sergey Melikhov's user avatar
17 votes
2 answers
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 ...
Steve's user avatar
  • 2,263
6 votes
3 answers
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 ...
Jan Weidner's user avatar
3 votes
1 answer
476 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 ...
Zhen Lin's user avatar
  • 15k
29 votes
4 answers
7k 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 ...
Zev Chonoles's user avatar
  • 6,752
15 votes
1 answer
548 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\...
Ben Webster's user avatar
  • 44.2k
31 votes
19 answers
6k views

Examples of categorification

What is your favorite example of categorification?
13 votes
1 answer
761 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) ...
Vivek Shende's user avatar
  • 8,653
3 votes
1 answer
414 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 ...
Ilya Nikokoshev's user avatar
75 votes
13 answers
12k 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.
Gil Kalai's user avatar
  • 24.4k
15 votes
6 answers
2k 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 ...
Ben Webster's user avatar
  • 44.2k
8 votes
4 answers
693 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 ...
Ben Webster's user avatar
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