All Questions
Tagged with ra.rings-and-algebras monoidal-categories
26 questions
4
votes
1
answer
251
views
Connected Frobenius algebras non-semisimple as an object
A Frobenius algebra object $A$ in a tensor category $\mathcal C$ is said to be connected if $\text{Hom}_{\mathcal C}(\mathbb{1}, A)$ is a one dimensional vector space, where $\mathbb {1} $ denotes the ...
- 43
0
votes
0
answers
59
views
Bimodule endomorphisms of a bimodule over a noncommutative ring
Let $R$ be a noncommutative ring and $M$ an $R$-$R$-bimodule that is projective as left $R$-module. We know that the bimodule of left $R$-module endomorphisms ${}_REnd(M)$ is isomorphic to the tensor ...
2
votes
1
answer
178
views
In a monoidal category with duals is the coevaluation map determined by the evaluation?
For a monoidal category $(\mathcal{M},\otimes,1)$ and an object $X$ with a left dual $X^*$. Let $(e,c)$ be a pair of (co)evaluation maps for the pair $(X,X^*)$. Is it possible to have another map $c': ...
3
votes
2
answers
227
views
The evaluation and coevaluation maps for an object isomorphic to a dualisable object
Let $X$ be an object in a monoidal category with dual $X^*$ and evaluation and coevaluation maps $e$ and $c$. Now if we have an isomorphism $\sigma:X \to Y$, for some other object $Y$, then $Y$ must ...
3
votes
0
answers
202
views
Coevaluation for linear categories
For a field $k$ and an associative $k$-algebra $R$, the $k$-linear category $R\operatorname{-Mod}$ is self dual inside $\operatorname{DGCat}_k$, with the counit map sending $k$ to $R$ regarded as a ...
- 834
2
votes
1
answer
152
views
Coproduct for a Frobenius algebra
The definition of a Frobenius algebra given here describes it as a monoid and a comonoid in a monoidal category with a compatability condition. For the special case of the category of vector spaces a ...
4
votes
2
answers
585
views
A subalgebra of a Frobenius algebra that is not again a Frobenius algebra?
A Frobenius algebra is a vector space that is both an algebra and a coalgebra in a compatible way. (See here for a precise definition.) I guess that a subalgebra of a Frobenius algebra is not again a ...
0
votes
0
answers
254
views
The coevaluation map for a projective module and its dual
$\DeclareMathOperator\coev{coev}$Let $R$ be a noncommutative ring and let $P$ be a bimodule over $R$, that is finitely generated and projective as a left module. It is "well-known" that any ...
- 145
7
votes
1
answer
377
views
This is not a tensor: tensoring abelian groups over groups
$\newcommand{\Cats}{\mathsf{Cats}}\newcommand{\MonCats}{\mathsf{MonCats}}\newcommand{\BrMonCats}{\mathsf{BrMonCats}}\newcommand{\SymMonCats}{\mathsf{SymMonCats}}\newcommand{\CMon}{\mathsf{CMon}}\...
- 11.8k
4
votes
0
answers
190
views
Are PD-, $\lambda$-, $\psi$-, and $\delta$-rings monoids in a monoidal category?
$\newcommand{\pt}{\mathrm{pt}}\newcommand{\N}{\mathbb{N}}\newcommand{\Z}{\mathbb{Z}}$A number of algebraic structures can be defined as monoids in some appropriate monoidal category:
A monoid ...
- 11.8k
9
votes
1
answer
326
views
Are differential rings monoids in a monoidal category?
$\newcommand{\pt}{\mathrm{pt}}\newcommand{\N}{\mathbb{N}}\newcommand{\Z}{\mathbb{Z}}$A number of algebraic structures can be defined as monoids in some appropriate monoidal category:
A monoid ...
- 11.8k
18
votes
2
answers
1k
views
Why does Drinfeld Unitarization work?
In Drinfeld's paper "Quasi-Hopf Algebras" he illuminates a process by which you can replace the $R \in A \otimes A$ associated to a quasi-Hopf QUE-algebra $(A, \Delta, \varepsilon, \Phi)$ ...
- 183
2
votes
1
answer
127
views
Non-counital coalgebras
For any unital algebra $A$, we have an associated dual coalgebra $A^{\circ}$. (Recall that it is defined to be the largest subalgebra of the $\mathbf{C}$-linear dual of $A$ such that the coproduct $\...
- 219
3
votes
1
answer
608
views
Examples of strict monoidal categories and monoidal categories with nontrivial associators
What are some "natural" motivating examples of the following:
i) A strict monoidal category,
ii) A monoidal with non-trivial associatots?
For i) the only examples I know are categories which ...
- 1,144
2
votes
1
answer
250
views
Example of a projective bimodule with isomorphic left and right duals
What is an example of a non-free finitely generated $R$-bimodule $M$ satisfying
i) $M$ is projective as both a left and right $R$-module
ii) the right dual $\mathrm{Hom}_R(M,R)$ and the left dual ...
6
votes
1
answer
339
views
Monoidal categories from the projective modules of a ring
Let $R$ be a not necessarily commutative ring, and denote by $_R\mathrm{lp}_R$ the category of $R$-bimodules, which are finitely generated projective as left modules, with morphism $R$-bimodule maps, ...
- 993
4
votes
1
answer
186
views
Is the category of rational Lie algebras monoidal?
I hate to ask such a naive question, but here goes. Suppose $A$ and $B$ are rational Lie algebras, i.e. rational vector spaces together with a bracket. Then, $A\otimes_{\mathbb{Q}} B$ is a rational ...
- 30.3k
11
votes
3
answers
327
views
Unbiased Hopf algebras
In category theory, a notion of monoidal category in which every sequence $X_1, \ldots , X_n$ ($n\ge 0$) of objects has a specified product is called an ``unbiased monoidal category'' (see Section 3.1 ...
- 43.2k
8
votes
2
answers
280
views
Graded rings with compatible S_n actions
Does the following mathematical gadget have a standard name? Let $R$ be an $\mathbb{N}$-graded ring together with an $S_n$ action on each $R_n$ which are compatible in the following sense. Let $i:...
- 28.1k
4
votes
1
answer
447
views
About a categorical definition of graded (coloured) algebra
The definition of graded algebra had a growing interest in algebra and mathematical physics (see $[GTC]$), I see that this topic has an elegant and simple categorical generalization, but I have not ...
- 4,165
3
votes
0
answers
179
views
Involution of unital based ring (Grothendieck ring of a fusion category)
Let $A$ be a unital based ring in the sense of [Ostrik, arXiv:math/0111139]. As part of the data we have a base $B = \{b_i\}_{i\in I}$, and an involution $i \mapsto \bar i$ of $I$ whose induced map $\...
- 665
6
votes
2
answers
440
views
String diagrams for bimodules over noncommutative algebras?
I'm trying to do some calculations with bimodules (over Azumaya algebras, as it happens), and I need a string diagram notation that mixes the tensor product over the base ring (a symmetric monoidal ...
- 35.5k
16
votes
1
answer
548
views
Does every commutative variety of algebras have a cogenerator?
By a commutative variety $\mathcal{V}$ I mean a classical variety of algebras for some $(\Sigma,E)$, such that each pair of operations in $\Sigma$ commutes.
Equivalently (i) every interpretation of ...
- 1,271
5
votes
1
answer
481
views
Two definitions of modules in monoidal category
The standard definition of a (left) simplicial module $V$ over some simplicial algebra $A$ is the map of simplicial vector spaces $A\otimes V\to V$ that gives the usual modules component-wise. Here $\...
- 1,276
7
votes
1
answer
485
views
Two questions about commutative theories
Let $\mathcal{T}$ be a commutative algebraic theory (for example sets, abelian groups, commutative monoids, but not groups etc.). References include the nlab and Borceux' Handbook of Categorical ...
- 63.1k
4
votes
1
answer
497
views
Is the functor of divided powers a weakly monoidal functor?
Let $R$ be a commutative ring with identity $e$. For every $R$-module $M$ the algebra of divided powers $D(M)$ is defined as follows. The generators of $D(M)$ are the symbolls $m^{(k)}$ for every $m\...
- 315