3
votes
1answer
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
1answer
348 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
0answers
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
1answer
378 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
2answers
752 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
1answer
355 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
1answer
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
1answer
311 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
3answers
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
3answers
295 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
2answers
313 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
2answers
478 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
0answers
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
1answer
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
2answers
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
0answers
157 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
2answers
572 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
3answers
490 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
2answers
343 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 ...
14
votes
6answers
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
0answers
469 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
2answers
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
2answers
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 ...
14
votes
6answers
993 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
0answers
640 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
0answers
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
2answers
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
2answers
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
1answer
415 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
1answer
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
1answer
654 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
3answers
730 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
9answers
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
1answer
600 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
3answers
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
2answers
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
3answers
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
1answer
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
2answers
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
1answer
299 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
1answer
331 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
2answers
404 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
1answer
335 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
2answers
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
5answers
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
0answers
352 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
1answer
390 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
4answers
490 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
3answers
497 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
5answers
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 ...