6
votes
1answer
176 views

Describing the Wreath product categorically.

The Details: Let's have a recap of some definitions (taken from "Nine Chapters in the Semigroup Art" (pdf), by A. J. Cain). Definition 1: Let $P$ be a semigroup. The left action of $P$ on a set ...
2
votes
1answer
48 views

If direct limits of matrices are isomorphic, is the direct limit of the transpose matrices also isomorphic?

On the one hand, the following conjecture seems reasonable, but on the other hand it doesn't seem natural because some objects are being dualised while others are not. I would appreciate if anyone ...
4
votes
2answers
138 views

Different ways of constructing the free group over a set.

This could be too broad if we're not careful. I'm sorry if it ends up that way. Let's put together a list of different constructions of the free group $F_X$ over a given set $X$. It seems to be ...
4
votes
1answer
93 views

Category theory for graph theory research

I am doing research in algebraic graph theory, focusing on the relation between graphs and groups (especially the representing groups as graphs) for my Ph.D. In particular, one of the ideas is to ...
7
votes
1answer
69 views

Center of the categories $\mathbf{Grp}$ and $\mathbf{Ab}$.

This is Exercise II.5.8 from Mac Lane, Categories for the Working Mathematician. For the identity functor $I_C$ of any category, the natural transformations $\alpha:I_C\dot{\to}I_C$ form a ...
4
votes
0answers
58 views

Abelization of symmetric groups and its subgroups of bounded support

For $X$ a finite set one can show easily that the abelization of the symmetric group on $X$ is given by the group of order $2$ and that the commutator subgroup of $\operatorname{Sym}X$ is given by the ...
5
votes
3answers
158 views

Does it factor through?

Let $f:F\to G$ and $g:F\to H$ be group homomorphism between groups. If $\ker f \subset \ker g$ then does there exists $h:G\to H$ such that $hf = g$? I know the the above is true for vector spaces by ...
2
votes
1answer
74 views

Is possible to define the parity by a universal property?

Consider the parity homomorphism of the symmetric group $$ p:S_n\to Z/(2). $$ Is it possible to characterise this map by a pure universal property? This question occurred to me when I was reading ...
1
vote
1answer
96 views

What is the relationship between the second isomorphism theorem and the third one in group theory?

The second isomorphism theorem [wiki] in group theory is as follows: Let $G$ be a group. $H \triangleleft G, K \le G$. Then: $HK \le G$, $(H \cap K) \triangleleft K$, and $K/(H \cap K) ...
4
votes
0answers
46 views

~ The important use of Frobenius–Schur indicators and Frobenius-Schur exponents ~

I had asked a question on the uses of conjugacy class and centralizer. I had receive various helpful feedback. I appreciate them. Here I like to get some feedback on the Frobenius–Schur indicator. ...
3
votes
0answers
53 views

Existence of a natural transformation

Let $G$ be a group, $S$ a set and $\chi$ an action of $G$ on $S$. This defines a functor $F$ from the group as a single object category to the category of sets. Let $\phi$ be an automorphism of $G$, ...
3
votes
1answer
45 views

Pull-backs of diagrams of groups with free product.

Until recently I calculated only pull-back of diagrams of finite groups. Now I am trying to calculate the pull-back of diagram of groups when the groups are free products of other groups. It seems ...
2
votes
0answers
51 views

Push-out of product of push-out diagrams

Let $\pi(U\cup V)$ be the push-out of the diagram $\pi(U)\leftarrow \pi(U\cap V) \rightarrow \pi(V)$ that appears when we apply Vam-Kampen Theorem to the open sets $U,V$ in a topological space $X$. ...
2
votes
0answers
43 views

Push-out of product of diagrams

Im working on the category of groups. Let $D$ be the push-out of the diagram $B\leftarrow A \rightarrow C$. Let $D'$ be the push-out of the diagram $B' \leftarrow A' \rightarrow C'$. It is possible to ...
1
vote
1answer
38 views

Explanation of a natural transformation in group theoretic terms

In category theory, a functor looks like homomorphism between categories. Keeping that analogy in mind, can a natural transformation be described by (or restricted to) group theoretic terms? For ...
3
votes
1answer
46 views

Terminology concerning conjugation in groups of functions.

If there is a function $a$ such that $a\circ g\circ a^{-1}=h$ then the functions $g$ and $h$ are conjugate to each other. If one wished to identify $a$, would one say "$g$ and $h$ are conjugate "by ...
2
votes
1answer
63 views

is the abelianization functor (on groups) full?

By abelianization I mean, for any group $G$, its commutator subgroup is the subgroup $[G,G]$ generated by elements of the form $ghg^{-1}h^{-1}$ for $g,h\in G$. Then the abelianization of $G$ is ...
2
votes
1answer
58 views

Group categories with only one object with a defined product

Do you know how to deal with this kind of problem? Let $G$ be a group and $\mathcal {G} $ be the category with one object $G$ and $\mbox {Mor}( {G} ; {G} ) = {G} $. Find all groups $G$ such that ...
2
votes
0answers
32 views

Unipotent Group Scheme

Can someone give me a reference of why this is true? Let A an associative k-algebra in wich every element is nilpotent, (non unital) and free of finite rank k-module, then for every commutative ...
6
votes
1answer
95 views

Nonexistence of a functor from $Group$ to $Set$ taking each group to its set of automorphism

I am struggling with this question: show that there does not exist functor from $Group$ to $Set$ taking each group to its set of automorphisms. I have thought about it for a while now, not having any ...
1
vote
0answers
38 views

Indecomposable groups vs. indecomposable objects

An object $X$ in a category $\cal C$ with an initial object is called indecomposable if $X$ is not the initial object and $X$ is not isomorphic to a coproduct of two noninitial objects. A group $G$ ...
13
votes
5answers
857 views

Abelianization of free group is the free abelian group

How does one prove that if $X$ is a set, then the abelianization of the free group $FX$ on $X$ is the free abelian group on $X$?
11
votes
1answer
102 views

Field Extensions and their dimensions

There is a powerful theorem with respect to a field $F$ extensions and their dimensions. $F<E<K \ \Rightarrow [K:F] = [K:E][E:F] $ This is analogous to the famous Lagrange's theorem with ...
2
votes
1answer
152 views

Free product of groups as coproduct

Wikipedia says "The coproduct in the category of sets is simply the disjoint union with the maps ij being the inclusion maps. Unlike direct products, coproducts in other categories are not all ...
10
votes
0answers
126 views

Cogroup structures on the profinite completion of the integers

Let $\mathsf{ProFinGrp}$ be the category of profinite groups (with continuous homomorphisms). This is equivalent to the Pro-category of $\mathsf{FinGrp}$. Notice that $\widehat{\mathbb{Z}} = ...
10
votes
3answers
220 views

Can we rediscover the category of finite (abelian) groups from its morphisms?

It was a question on stackexchange approximately a month ago if in the category $(grp)^{fin}$ $|Hom(H,G_1)|= |Hom(H,G_2)|$ for all $H \Rightarrow G_1 \cong G_2$. Link to the previous question. So ...
4
votes
1answer
77 views

How general are $\mathrm{Aut}$ groups and $\mathrm{End}$ rings?

For any locally small category $X$ and object $A\in X$ the set $\mathrm{Aut}_X(A)$ is a group w.r.t. composition $\circ$. For any locally small abelian category $X$ and object $A\in X$ the set ...
5
votes
2answers
109 views

Functors and Groups

Let $\alpha$ be a functor from the category of groups in the category of groups which assigns to every group $G$ a characteristic subgroup $\alpha (G)$ of $G$ and to every homomorphism $\theta : H ...
2
votes
1answer
55 views

Group and preorder

Let $G$ be a group. Let $a, b$ be elements of $G$. We denote $\operatorname{Hom}(a, b) = \{ab^{-1}\}$. Then we get a category whose set of objects is $G$. We can regard this category as a preorder in ...
4
votes
2answers
87 views

What can we learn purely from the existence of a (non-constant) functor to the category of abelian groups?

I admit that the following is a very broad question. So if you feel that it is too vague please say so. It might also just be that I haven't read enough about category theory and my question is silly. ...
5
votes
1answer
90 views

A monomorphism of groups which is not universal?

Is there an injective homomorphism of groups $f_1\colon G\longrightarrow H_1$ together with another homomorphism $f_2\colon G\longrightarrow H_2$ such that the pushout $H_2\longrightarrow H_1\coprod_G ...
4
votes
1answer
79 views

Linear structure on the category of formal groups

Let $R$ be a commutative ring. If $R$ is a $\mathbb{Q}$-algebra, then the category of formal groups over $R$ (or the category of formal group laws) carries the structure of an $R$-linear category; ...
10
votes
0answers
149 views

Taking the automorphism group of a group is not functorial.

Once upon a time I proved that there is no functorial 'association' $$F:\ \mathbf{Grp}\ \longrightarrow\ \mathbf{Grp}:\ G\ \longmapsto\ \operatorname{Aut}(G).$$ A few days ago I casually mentioned ...
6
votes
2answers
133 views

Computing easy direct limit of groups

How do I start computing easy direct limit of groups: 1) $\mathbb{Z} \overset{1}\longrightarrow \mathbb{Z} \overset{2}\longrightarrow \mathbb{Z} \overset{3}\longrightarrow \mathbb{Z} ...
4
votes
1answer
136 views

Slicker construction of the free product of groups

The usual construction of the free product of groups $\{G_i\}_{i \in I}$ consists of taking the discriminated union $\coprod_{i \in I} G_i$ and taking the set of words satisfying a handful of ...
2
votes
1answer
43 views

Induced injections between free groups

Let $A$ and $B$ be non-empty sets with associated free groups $F(A),F(B)$. Given an injective function $f: A \to B$, is the induced homomorphism $\bar{f}: F(A)\to F(B)$ injective? Let $i_A: A \to ...
8
votes
0answers
178 views

Kernels in $\mathbf{Top}$

There is a following well-known theorem for abelian categories (at least the ones I know, Ab, $R$-mod and so on... not so familiar with categorical language to be honest) which states the following : ...
2
votes
1answer
97 views

Understanding Basic Categorical Duality with an Example from Group Theory

I am trying to understand the concept of duality in category theory, but I am having a problem, well illustrated by the following situation. Let $H$ be any nontrivial subgroup of the alternating ...
2
votes
1answer
78 views

pullback square of factor groups

Let H and K be normal subgroup of a group G. The following square is always a pullback square? $$\begin {matrix} G/H\cap K\rightarrow &G/K\\ \downarrow&\downarrow\\ G/H\rightarrow&G/HK\\ ...
5
votes
2answers
76 views

A generalization of abelian categories including Grp

The category of groups shares various properties with abelian categories. For example, the Five lemma and Nine lemma hold in Grp. Is there a weakened notion of abelian category which also includes Grp ...
3
votes
2answers
93 views

Proving that tensor distributes over biproduct in an additive monoidal category

I'm trying to prove that the tensor product distributes over biproducts in an additive monoidal category; namely that given objects $A,B,C$, we have $A \otimes (B \oplus C) \cong (A \otimes B) \oplus ...
3
votes
2answers
117 views

Counter examples on Categories

I'm reading Categories for the Working Mathematician by Saunders Mac Lane. At the section 5 from chapter 1, for a fixed category, he claims that every arrow with right inverse, is epic (right ...
5
votes
1answer
144 views

Grothendieck group of a symmetric monoidal category is a lambda ring?

I understand that taking the Grothendieck group of a braided monoidal (abelian) category gives us a commutative ring and that taking that of a symmetric monoidal (abelian) category gives us a ...
8
votes
1answer
184 views

Can a free group over a set be constructed this way (without equivalence classes of words)?

Denote category of monoids equipped with involution by $\textbf{invMon}$. Objects are pairs $\left(M,\iota\right)$ where $\iota$ is a map on the underlying set of $M$. Denoting $\iota$ by ...
9
votes
1answer
198 views

a group is not the union of two proper subgroups - how to internalize this into other categories?

A well-known fact from group theory is that a group cannot be the union of two proper subgroups. I wonder, does this statement internalize into other categories than the category of sets? That is, is ...
3
votes
1answer
144 views

Cocartesian squares in the category of abelian groups.

Recently, I've been doing a recap of (basic) category theory and found an old exercise I seem to be unable to solve. The question is as follows. Let $A, B$ be abelian groups, $A'<A$ and $B'<B$ ...
4
votes
1answer
124 views

Group actions and natural isomorphisms

Let $G$ be a group-as-category, and let $S$ be the image of $G$ by the $Hom(G,-)$ functor. The $Hom$ functor defines a bijection $\chi: G \to S$ between elements of $S$ and morphisms of $G$, and thus ...
5
votes
2answers
122 views

Free objects in $\mathrm{Set}(G).$

What are the free objects in the category of $G$-sets for a group $G$? After considerable deliberation (I'm not very bright), I'm pretty sure they are the $G$-sets $X$ on which $G$ acts freely, that ...
4
votes
2answers
122 views

About the category $\mathrm{Set}(G)$

I'm not good with categories. I've attempted several times to understand what a natural transformation is, and so far I've failed. But I'm trying to learn algebraic topology now, and it seems that I ...
17
votes
0answers
217 views

Existence of a certain functor $F:\mathrm{Grpd}\rightarrow\mathrm{Grp}$

Let $\mathrm{Grpd}$ denote the category of all groupoids. Let $\mathrm{Grp}$ denote the category of all groups. Are there functors $F\colon\mathrm{Grpd}\rightarrow \mathrm{Grp}, ...