# Tagged Questions

**6**

votes

**2**answers

200 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 ...

**2**

votes

**1**answer

272 views

### How to define a generating subset for algebra in a category?

As is well known, the definition of an monoid can be generalised to the notion of a monoid $A$ in a monoidal category $C$ (see the n-lab entry here). What I would like to know is if the notion of ...

**4**

votes

**2**answers

146 views

### Is there a purely module theoretic characterization of semiprimitive rings?

A ring (say unital for simplicity) is semiprimitive (or Jacobson semisimple) if its Jacobson radical is trivial, or equivalently it has faithful semisimple module. Semiprimitivity is a Morita ...

**15**

votes

**5**answers

2k views

### Origin of exact sequences

I have seen exact sequences appearing a lot in algebraic texts with different purposes. But I've never seen names of the people associated with it. Also I don't understand what's so good about showing ...

**4**

votes

**0**answers

119 views

### Which endomorphism algebras are not Morita-trivial?

Let $\mathcal C$ be a symmetric monoidal category — for example, $\mathcal C$ might be the category of $R$-modules for a commutative ring $R$. At a minimum, I request further that:
$\mathcal C$ is ...

**3**

votes

**2**answers

284 views

### Are algebraic structures uniquely identifed by their free objects?

It might be a naive question, as I am not a specialist in this field.
This is a follow-up to this question.
I want to study varieties of objects generalizing ordered monoids, in particular using an ...

**13**

votes

**2**answers

392 views

### Does base extension reflect the property of being isomorphic?

Let L/K be a (separable?) field extension, let A be a finite dimensional algebra over K, and let M and N be two A-modules. Let $A' = L \otimes_K A$ be the algebra given by extension of scalars, and ...

**1**

vote

**1**answer

158 views

### “order two sequence” in a paper of Waldhausen

In F. Waldhausen's paper "Algebraic K-theory of generalized free products, Part I", page 142,line 19, there is a term "order two sequence". Can anyone explain its meaning to me?
According to the ...

**6**

votes

**1**answer

326 views

### Morita theorem for simplicial rings

My question is the following: is there an analog of Morita theorem in the simplicial setting?
I mean, we can define two simplicial rings $A,B$ to be simplicially Morita equivalent is the categories ...

**4**

votes

**1**answer

239 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 ...

**-2**

votes

**1**answer

309 views

### For a ring $A$, is $A$ Morita equivalent to $M_\infty(A)$? [closed]

Let $A$ be a ring, let $M_n(A)$ be the ring of $n$-by-$n$ matrices with elements in $A$, $A$ is Morita equivalent to $M_n(A)$, I was wondering if this also applied to infinite matrices? That is, if ...

**3**

votes

**2**answers

471 views

### Shape of axioms in abstract algebra

When defining abstract algebraic structures (like monoids, groups, etc...), are there some constraints on the shape of the axioms, for the structure to have good properties that we implicitly use in ...

**4**

votes

**0**answers

104 views

### What is the composition in SesquiAlg?

To motivate my question, I will describe a famous 2-category. First, there is the 1-category $\text{Vect}$ of vector spaces (over some fixed field). This category has a tensor product $\otimes$, and ...

**5**

votes

**0**answers

185 views

### How to compute the abelianization of the representation theory of a Hopf algebra?

I will ask two versions of my question, which probably aren't precisely the same, and I am also interested in hearing about nuances between the two.
Version 1:
Let $(C,\otimes)$ be any monoidal ...

**3**

votes

**0**answers

147 views

### Iterated Tangent Category Construction

We can think of the "tangent category" of a category $\mathcal{C}$ at an object $A$ as being the abelian group objects in the overcategory $\mathcal{C}/A$ (with whatever conditions I need on ...

**3**

votes

**1**answer

221 views

### Resolutions chain homotopic to projective ones

Motivation. In my research I have a situation where a monoid $M$ is acting by nice cellular maps on a contractible cell complex and so the augmented chain complex is a resolution of the trivial module ...

**1**

vote

**0**answers

113 views

### Cotorsion theory and its relative homology

Let (F(R), Cot(R)) be a cotorsion theory, Such that F(R) is the class of flat R-modules and Cot(R) the cotorsion modules. Why this is true that, For $ N\in Cot(R) $,
$ \text{Ext}_{F(R)}^i(M, N)\cong ...

**9**

votes

**0**answers

457 views

### Is “being a full ring of quotients” a Morita invariant property?

Definition and context:
An (associative, unital, not necessarily commutative) ring $R$ is called classical if every regular element of $R$ is a unit. Equivalently, $R$ is its own classical ring of ...

**15**

votes

**3**answers

800 views

### Characterising categories of vector spaces

Consider the category $FdVect_k$ of finite dimensional $k$-vector spaces, for some given field. It is abelian, semisimple, in that each object is a finite sum of simple objects (of which there is only ...

**10**

votes

**4**answers

448 views

### Morita equivalence for *-algebras

This is a reference request. I'm looking for a definition of Morita equivalence of *-algebras, as described below. If anyone thinks that this is not the right way to define Morita equivalence of ...

**5**

votes

**0**answers

206 views

### Cocontinuous monadic functors

Some forgetful functors of algebraic categories preserve colimits, but most do not. In order to understand this phenomen in general, I have classified cocontinuous monadic functors whose target is an ...

**3**

votes

**3**answers

302 views

### When is this diagram of tensor powers an equalizer?

Let $A$ be a commutative ring with $1$, and $B\subseteq A$ be a subring.
Is there a simple condition on $B$ and $A$ guaranteeing that $B\to A\rightrightarrows A\otimes_B A$ is an equalizer?
In ...

**4**

votes

**2**answers

358 views

### When is a quantum affine space $\mathbb{A}^{n}$ Calabi-Yau?

I would like to ask a simple question. Let $A=\mathbb{C}\langle x_{1},\dots,x_{n} \rangle/I$, where $I$ is the two-sided ideal generated by $x_{i}x_{j}=a_{ij}x_{j}x_{i}$ for $1\le i,j\le n$. We say a ...

**1**

vote

**1**answer

242 views

### (Co)Universal Property of Quotients/Subs

I'm not completely sure if this bunch of questions is the appropriate Level of MO. However at the same time I think that it is at least slightly above the level of stackex. ...
The tensor algebra ...

**6**

votes

**1**answer

206 views

### Significance of the vanishing of $K_{-1}(A)$

In M. Schlichting's paper, he defines the negative $K$-theory for derived categories. In this he states that for $\mathcal{A}$ an idempotent complete (see below) triangulated category, ...

**7**

votes

**3**answers

632 views

### Module category equivalent to graded module category?

Main Question
Let $R$ be a graded ring, graded by the nonnegative integers. Denote by $\mathrm{gr}R-\mathrm{Mod}$ the category of $\mathbb{Z}$-graded left $R$-modules with morphisms that preserve ...

**7**

votes

**2**answers

528 views

### “Composition of Morita equivalences” or “Morita equivalence and the Nakayama functor”

This problem occured to me, when trying to find a Morita invariant for finite dimensional algebras.
Suppose $\Lambda$ and $\Gamma$ are two self-injective $k$-algebras ($k$ being a field) which are ...

**4**

votes

**2**answers

491 views

### Constructing a ring from an abelian group in a minimal way

I'm looking for a specific construction, taking an abelian group (with designated element) $(G,+,1)$ to a commutative ring $(R,+,\cdot,1)$, where $G\subset R$ as a pointed abelian group, and which ...

**3**

votes

**1**answer

379 views

### About the category of chain complexes and Grothendieck categories.

Given an abelian category $\mathcal{A}$ the category of chain complexes over $\mathcal{A}$ is again an abelian category. If $\mathcal{A}$ is a Grothendieck category then the category of chain ...

**3**

votes

**3**answers

766 views

### Fractional Quantum Hall Effect - Mathematics

Just to include something that starts to answer my own question Topological Quantum Computation Lecture notes covers a lot of the Mathematics of the Fractional Quantum Hall effect, or topological ...

**14**

votes

**2**answers

590 views

### Is the endomorphism algebra of a dualizable bimodule necessarily finite dimensional?

Let $k$ be field. Let $A$, $B$ be $k$-algebras, and let ${}_AM_B$ be a dualizable bimodule.
Pre-Question (too naive): Is the algebra of $A$-$B$-bilinear endomorphisms of $M$ necessarily finite ...

**3**

votes

**1**answer

531 views

### Is there a way to define a prime ideal object via diagrams in the category of rings?

I like to think in terms of commutative diagrams rather than referring to elements. So to me a group is really a group object, i.e. an object with some maps satisfying certain commutative diagrams. ...

**4**

votes

**0**answers

220 views

### When are all modules direct factors of a direct product of a fixed one?

Background: For a ring $R$, we denote by ${\rm\mathop Mod}(R)$ the category of all (say right) $R$-modules. If $R$ is pure semisimple, then it is known that ${\rm\mathop Mod}(R)={\rm\mathop Add}(M)$, ...

**3**

votes

**2**answers

396 views

### Higher categories and semirings

Maybe my thinking here is completely wrong headed, but this seems like something that ought to have been answered before. Here is my question:
What is the (n - )categorical analogue of a ...

**24**

votes

**7**answers

2k views

### Why don't ideals and quotients work well for categories?

Ideals are intimately related to quotients and congruence relations. They clearly play a very important role in ring theory and order theory. So do normal subgroups in group theory. (Enriched) ...

**16**

votes

**7**answers

1k views

### “Sums-compact” objects = f.g. objects in categories of modules?

Hello,
Let us call an object of an additive category sumpact (contraction of "sums" and "compact") if taking $Hom$ from it (considered as functor from the category to $Ab$) commutes with coproducts. ...

**7**

votes

**3**answers

1k views

### What does Rng^{op} look like?

There are several well-known dualization results in category theory, i.e. that such-and-such a well-known category D is isomorphic to the opposite C^{op}. Does anyone know of such a result concerning ...

**15**

votes

**12**answers

2k views

### Constructions unique up to non-unique isomorphism

1) Fields have algebraic closures unique up to a non-unique isomorphism.
2) Nice spaces (without base point) have universal covering spaces unique up to a non-unique isomorphism.
3) Modules have ...

**1**

vote

**1**answer

254 views

### Realizing a restriction as direct/inverse image of sheaves

Consider the inclusion $j$ of ${\mathbb{R}}$ as the real axis of ${\mathbb{C}}$. On ${\mathbb{C}}$ I have a real polynomial algebra ${\mathbb{R}}[x,\bar{x}]$, where $\bar{x}$ denotes conjugation. ...

**10**

votes

**1**answer

524 views

### Why not _co_free modules?

Let $R$ be a ring, and $R\text{-Mod}$ its category of all left modules. There is a "forgetful" functor $\operatorname{Forget}: R\text{-Mod} \to \text{AbGp}$, which is additive, continuous, and ...

**6**

votes

**2**answers

449 views

### What are examples of cogenerators in R-mod?

Fill in the blank, please :)
Let $\mathcal C$ be a complete and cocomplete abelian category. A generator in $\mathcal C$ is an object $X \in \mathcal C$ such that every object $Y \in \mathcal C$ ...

**3**

votes

**2**answers

339 views

### Universal functors according to Cohn.

In section III.1 of P.M. Cohn's Universal Algebra a notion of universal functor ${\cal L} \rightarrow {\cal K}$ is defined for a representation of one category in another given by a (covariant) ...

**10**

votes

**4**answers

578 views

### When are modules and representations not the same thing?

I've been trying for a while to get a real concrete handle on the relationship between representations and modules. To frame the question, I'll put here the standard situation I have in mind:
A ring ...

**7**

votes

**2**answers

902 views

### A question on curved algebras, papers by Positselski and E. Segal

I am trying to understand something about curved dg algebras as studied by Positselski, E. Segal. These come up in mirror symmetry and when one wants to study Kozsul duality for algebras that are more ...

**2**

votes

**3**answers

617 views

### Lattice of subcategories: subobject classifier in Cat

Two short questions:
Is there any work classifying the lattice of subcategories of an arbitrary (sufficiently small) category $\mathcal{C}$, similar to the way that the set of subsets of set ...

**28**

votes

**3**answers

3k views

### Why is there no Cayley's Theorem for rings?

Cayley's theorem makes groups nice: a closed set of bijections is a group and a group is a closed set of bijections- beautiful, natural and understandable canonically as symmetry. It is not so much a ...

**7**

votes

**2**answers

401 views

### Is the tensorproduct of a triangulated category with a ring again triangulated?

$\underline{Background}$ :
Suppose $\tau$ is a preadditive category and $R$ a ring. Then one may form a new preadditive category $\tau \otimes R$ in the following way:
$\tau \otimes R$ has the same ...

**6**

votes

**1**answer

207 views

### For which rings does a projectivization of modules exist?

Let $R$ be a ring. Consider the inclusion functor from the category of finitely generated, projective $R$-modules to the category of all finitely generated $R$-modules. For which rings does it have a ...

**0**

votes

**1**answer

446 views

### F is ultrafilter over a Boolean algebra implies that for every b, either b or not-b is in F?

I'm trying to teach myself category theory from Steve Awodey's Category Theory. Chapter 2 asserts:
It is not hard to see that a filter F is an ultrafilter just if for every element b ∈ B, either b ...

**3**

votes

**1**answer

346 views

### Is there a good version of Artin-Wedderburn for semisimple algebra objects?

Artin-Wederburn says that if you have a semisimple algebra then it is a product of matrix algebras over division rings.
Suppose that C is a fusion category over the complex numbers (if you want to ...