# Tagged Questions

**3**

votes

**1**answer

191 views

### How does associativity get twisted by elements of $H^3(G)$?

In Braided Monoidal Categories by Joyal and Street, §6 a monoidal category is $V =T(G,M,h)$ built using a recipe:
objects are are elements of $G$ ✓
$V_0(x,y) = M$ if $( x=y)$ or else ...

**10**

votes

**1**answer

345 views

### Categorical interpretation of disjoint cycle notation for tracing permutations

For a natural number $n\in\mathbb{N}$, let $\underline{n}$ denote the finite set $$\underline{n}:=\{1,2,\ldots,n\}.$$
A permutation $\sigma\in Aut(\underline{n})$ can be uniquely written (up to order) ...

**3**

votes

**0**answers

176 views

### Invariants of groups that are invariant under passage to finite index subgroups

This question is mostly idle curiosity.
Recall the following standard terminology: if $P$ is a property of groups, a group $G$ is said to be virtually $P$ if it has a subgroup of finite index which ...

**11**

votes

**1**answer

362 views

### Can one explain Tannaka-Krein duality for a finite-group to … a computer ? (How to make input for reconstruction to be finite datum?)

Consider a finite group. Tannaka-Krein duality allows to reconstruct the group from the
category of its representations and additional structures on it (tensor structure + fiber functor). Somehow ...

**6**

votes

**2**answers

742 views

### If $G \times G \cong H \times H$, then is $G \cong H$? [duplicate]

Let $G,H$ groups. Suppose that $G \times G \cong H \times H$. Then is necessarily $G \cong H$?
I know that if $G,H$ are finite groups then it is true (you show that the map $FinGrp \rightarrow ...

**3**

votes

**1**answer

351 views

### Does the internal axiom of choice imply Lagrange's theorem?

In a topos ${\mathcal{A}}$, given a group object $G$ and a subgroup $H$, the object $H\backslash G$ of right cosets is the coequalizer of two maps $G\times H\rightrightarrows G$, namely the group ...

**1**

vote

**1**answer

327 views

### “good” automorphisms of Galois classes of L functions

This question is a follow-up to Galois classes of L-functions. My goal here is to make things clearer.
Definition 1
Let $A$ be a subclass of the Selberg class containing $s\mapsto 1$, closed under ...

**5**

votes

**1**answer

308 views

### A naive question on eigensheaves for group actions on derived categories

In this Mathoverflow question, Examples of Eigensheaves outside of langlands, David Ben-Zvi says
" Given a G -space X you can recover quasicoherent sheaves on X from sheaves on X/G (ie equivariant ...

**11**

votes

**3**answers

2k views

### A “mother of all groups”? What kind of structures have “mother of all”s?

For ordered fields, we have a “mother of all ordered fields”, the surreal numbers $\mathbf{No}$, a proper-class “field” which includes (an isomorphic copy of) every other ordered field as a subfield. ...

**3**

votes

**3**answers

293 views

### What are the symmetries of a principal homogeneous bundle?

Let $\operatorname{Klein}$ denote the category of principal homogeneous bundles. An object in this category is a tuple $\mathbf Q = (Q, P; G, H; q, a, \tilde a)$, where:
$G$ is a Lie group, and $H$ ...

**7**

votes

**2**answers

312 views

### Projective arrows

We all know that a projective object in a category $\mathcal{C}$ is an object $P$ in $\mathcal{C}$ such that for every epimorphism $f: X\to Y$ in $\mathcal{C}$ and arrow $g\colon P\to Y$ there is a ...

**9**

votes

**2**answers

470 views

### Cogroups in the category of groups are free

The free group $F(S)$ on a set $S$ is a cogroup in the category of groups since $\hom(F(S),G) \cong G^S$ carries a natural group structure for every group $G$. I have read that these are the only ...

**1**

vote

**0**answers

235 views

### A generalization of a group isomorphism.

Let $H,K$ be two normal subgoups of a group $G$.
We know that there exists a group isomorphism:
$HK\diagup H\simeq H\diagup{H\cap K}$.
I want to generalize this statement in the language of category ...

**2**

votes

**1**answer

165 views

### A categorical framework for Freiman s-morphisms

Let $\mathfrak A_i$ be groups ($i = 1, 2$), written multiplicatively, and $s$ a non-negative integer (here, as usual, I am abusing notation and denoting the operations of $\mathfrak A_1$ and ...

**14**

votes

**2**answers

1k views

### Are semi-direct products categorical limits?

Products, are very elementary forms of categorical limits. My question is whether in the category of groups, semi-direct products are categorical limits.
As was pointed in:
...

**3**

votes

**0**answers

156 views

### Extensions of $Z/p$ by $Z/p$ and uniqueness of cokernels

I seem to run into something I cannot understand. Following Weibel (Homological Algebra), Ex. 3.4.1, p.76, it is claimed that if $p$ is prime, there are $p$ nonequivalent extensions of $Z/p$ by $Z/p$. ...

**15**

votes

**2**answers

568 views

### Is a retract of a free object free?

I wonder whether this is true in the categories of groups, monoids, commutative algebras, associative algebras, Lie algebras?

**9**

votes

**3**answers

489 views

### Group extensions and actions on categories

Let G and H be two groups. There is a one-to-one correspondence between:
(i) an (isomorphism class of) extension of G by H, i.e. an exact sequence of group morphisms $1\to H\to E\to G\to 1$;
(ii) an ...

**3**

votes

**2**answers

341 views

### a group from a family of bijections X->Y

Let $\Phi$ be a set of bijections $\phi_a:X\to Y$. To each pair of bijections $\phi_a$, $\phi_b$ one naturally relates a bijection $\psi_{ab}:=\phi_a^{-1}\circ\phi_b: X\to X$. In some cases the set of ...

**13**

votes

**6**answers

2k views

### Characterization of the transfer map in group theory

Let $i : H \to G$ be a subgroup of finite index. The transfer map is a special homomorphism $V(i) : G^\mathrm{ab} \to H^\mathrm{ab}$. The usual ad hoc definition uses a set of representatives of $H$ ...

**2**

votes

**0**answers

465 views

### Cancellation Theorem for groups

Cancellation theorem in group theory (for direct product) says that if $B$ is a finite group and $A \times B \simeq A_1 \times B_1$ and $B \simeq B_1$ then $A \simeq A_1.$
Of course, if $B$ is not ...

**2**

votes

**2**answers

226 views

### Automorphisms and Bicategories

Sorry if this isn't the right place for this, it hasn't gotten any answers on ME. I'm reading Lang's section on field theory and he stresses that, unlike typical "universal" constructions which are ...

**5**

votes

**2**answers

2k views

### Why SU(3) is not equal to SO(5)?

I am asking in the sense of isometry groups of a manifold. SU(3) is the group of isometries of CP2, and SO(5) is the group of isometries of the 4-sphere. Now, it happens that both manifolds are ...

**13**

votes

**6**answers

980 views

### Discrete-compact duality for nonabelian groups

A standard property of Pontrjagin duality is that a locally compact Hausdorff abelian group is discrete iff its dual is compact (and vice versa). In what senses, if any, is this still true for ...

**4**

votes

**0**answers

639 views

### $Ext$ functor, filtered complexes and spectral sequences

Let $\mathcal{A}$ an abelian category. Take $M$ an object of $\mathcal{A}$, and $K_*$ a bounded complex in $\mathcal{A}$ equipped with a bounded increasing filtration $F$. By using homological and ...

**5**

votes

**0**answers

250 views

### Are these notions of group object with weak inverses equivalent?

This is a follow-up to The definition of a group object is wrong?. The basic setup is as follows. Let $C$ be a category with finite products, $S : C \to D$ a product-preserving faithful functor, and ...

**8**

votes

**2**answers

2k views

### The definition of a group object is wrong?

An old MO answer by Noah Snyder makes a claim I don't completely understand, but mostly because I don't know any examples. The answer claims that in some examples of (things that one would want to ...

**1**

vote

**2**answers

299 views

### compact elements and continuous functors

Hi,
I am interested in abstracting the Scott topology from Domains to Categories. One can find a definition of a continuous functor which is just such an abstraction:
A functor F:C→D is continuous ...

**10**

votes

**1**answer

414 views

### Epimorphisms have dense range in TopHausGrp?

Consider the category of Topological Groups with continuous homomorphisms. Then a continuous homomorphism $f:G\rightarrow H$ with dense range is an epimorphism. Is the converse true? If not, what ...

**6**

votes

**1**answer

345 views

### Associativity with infinite nesting

I was trying to understand the Eilenberg-Mazur swindle (which I learned about here) especially as it could be used to show that if $A, B$ are compact (topological) $n$-manifolds whose connect sum is ...

**14**

votes

**1**answer

645 views

### For which categories does one have a Goursat Lemma?

Background
One of my favourite elementary results in group theory is Goursat's Lemma. This lemma characterises the subgroups of a direct product of groups in terms of fibred products.
Indeed, let ...

**5**

votes

**3**answers

729 views

### Are all group monomorphisms regular, constructively?

By "constructive" I mean something that would go through in CZF for example.
[added Oct 6]
A sketch of a standard proof (such as referenced in comment below), which is almost constructive: Let H be a ...

**10**

votes

**9**answers

1k views

### Proving the impossibility of an embedding of categories

A number of topological invariants take the form of functors $\mathscr{T}\to\mathscr{G}$, where $\mathscr{T}$ is the category of all topological spaces and continuous functions, and $\mathscr{G}$ is ...

**2**

votes

**1**answer

595 views

### Dualizing the definition of a free group

In most basic abstract algebra courses, the free group is directly constructed, a process that I find rather unwieldy. An alternate method of characterizing the free group is by means of its universal ...

**1**

vote

**3**answers

1k views

### Infinite Field Theory and Category Theory

I should start by saying that I have not studied field theory in depth, so if this question is totally off base, I apologize. Something I noticed as I studied group theory is many concepts that were ...

**11**

votes

**2**answers

2k views

### What's so special about the forgetful functor from G-rep to Vect?

The following is some version of Tannaka-Krein theory, and is reasonably well-known:
Let $G$ be a group (in Set is all I care about for now), and $G\text{-Rep}$ the category of all $G$-modules ...

**16**

votes

**3**answers

1k views

### Does subgroup structure of a finite group characterize isomorphism type?

Question
Suppose there is a bijection between the underlying sets of two finite groups $G, H$, such that every subgroup of $G$ corresponds to a subgroup of $H$, and that every subgroup of $H$ ...

**3**

votes

**1**answer

217 views

### A functor that comes from a morphism in a bigger category

My loose question is like this: what would you say about an equivalence of categories where both are concrete categories, and the equivalence functor is induced from a set-theoretic bijection at the ...

**34**

votes

**2**answers

1k views

### What algebraic group does Tannaka-Krein reconstruct when fed the category of modules of a non-algebraic Lie algebra?

Let $\mathfrak g$ be a finite-dimensional Lie algebra over $\mathbb C$, and let $\mathfrak g \text{-rep}$ be its category of finite-dimensional modules. Then $\mathfrak g\text{-rep}$ comes equipped ...

**9**

votes

**1**answer

297 views

### Is an invertible biset necessarily a bitorsor?

Question
Let $G$ be a group, and let $X$ be a $G$-biset that is (weakly) invertible with respect to the contracted product. Is $X$ necessarily a bitorsor?
Background
By $G$-biset, I mean a set ...

**9**

votes

**1**answer

330 views

### Jordan Hölder decomposition for group objects

Is there some generalization of the Jordan-Hölder decomposition for group objects in a category $\mathcal{C}$?
If $\mathcal{C}$ is the category Sch$(S)$ of schemes over a base scheme $S$ then (I ...

**4**

votes

**2**answers

401 views

### groups as categories and their natural transformations

If one views a group as a one object category with the elements of the group as morphisms then a natural transformation between functors of such categories is an inner automorphism, i.e. if we have ...

**6**

votes

**1**answer

334 views

### References on functorially-defined subgroups

I'm interested in results about functorially-defined subgroups (in a loose sense), especially in the non-abelian case, and would like to know about references I may have missed.
The question, it ...

**1**

vote

**2**answers

279 views

### Relative Frobenius Structure on the Category of G-modules

Let $G$ be a group $H\leq G$ a subgroup of finite index. Further, let ${\mathcal E}^G_H$ denote the class of those short exact sequences of $G$-modules (over some fixed base ring) which split when ...

**14**

votes

**5**answers

1k views

### two conjugate subgroups and one is a proper subset of the other? plus, a covering space interpretation.

Recently I've been reading J.P. May's A Concise Course in Algebraic Topology. In the section on the classification of covering groupoids, he mentions that sometimes a group G may have two conjugate ...

**1**

vote

**0**answers

347 views

### Automorphisms of category of groups [duplicate]

Possible Duplicate:
What are the auto-equivalences of the category of groups?
Does the category of groups have any nontrivial automorphisms? (an automorphism of a category being a functor ...

**1**

vote

**1**answer

388 views

### Module categories over $Rep(G)$.

Related to this question I also had some troubles to understand the classification of module categories over $Rep(G)$. Specifically, on page 12 of Ostrik's paper what is the category ...

**4**

votes

**4**answers

489 views

### projections of finitely presented groups

let's call an object $x$ of a cocomplete category (categorical) finitely generated if $\hom(x,-)$ commutes with filtered colimits of monomorphisms, and finitely presented if $\hom(x,-)$ even commutes ...

**11**

votes

**3**answers

492 views

### What are the auto-equivalences of the category of groups?

My question is motivated by Are the inner automorphisms the only ones that extend to every overgroup?
What are the auto-equivalences of the category of groups? What kind of structure do they form?
...

**11**

votes

**5**answers

1k views

### Abstract nonsense versions of “combinatorial” group theory questions

In particular, I'm just curious whether there's a version of the Sylow theorems (which are very combinatorially-flavored) which allows horizontal and/or vertical categorification? Or at least can be ...