All Questions
Tagged with limits-colimits group-theory
36
questions
3
votes
1
answer
69
views
subgroups of $(\mathbf Q, +)$ as direct limits
This is a follow-up to this question.
A finitely generated subgroup of $(\mathbf Q, +)$ is isomorphic to the direct limit of the system
$$\mathbf Z\xrightarrow{1}\mathbf Z\xrightarrow{1}\mathbf Z\...
- 387
0
votes
0
answers
49
views
Why restricted product $\prod'$ is $\varinjlim_{S\subset I \text{ runs finite subset of} I} (\prod_{i\in S} X_{i}\times \prod_{v\in I-S}Y_i)$
This is a question related to this page.
https://ncatlab.org/nlab/show/restricted+product .
Let $I$ be a directed set.
Let $X_i(i\in I)$ be a group.
Let $\prod'_{i\in I}(X_i,Y_i)$ be a restricted ...
- 6,351
1
vote
0
answers
143
views
Does profinite completion preserve injectivity?
Let $G$ be an abelian group.
Let $\widehat{G}$ be a profinite completion of $G$.
Profinite completion means a inverse limit of $G$ by a system given by homomorphisms $G/N\to G/M$ where $N$ and $M$ are ...
- 6,351
1
vote
1
answer
80
views
Why is a short exact sequence the same as a bicartesian square?
Let $N \rightarrowtail G \twoheadrightarrow H$ be a sequence of group homomorphisms, with maps $\alpha: N \rightarrowtail G$ a monomorphism and $\beta: G \twoheadrightarrow H$ an epimorphism. I am ...
- 313
0
votes
1
answer
66
views
Direct limit of Cyclic Groups
With the usual order $I=\mathbb N$ is a directed set.
Suppose that for each $i\in \mathbb N$ we have a subset $\{s_i\}$ of real numbers, such that for each $i\leq j$, $s_i\mathbb Z$ is a subgroup of ...
5
votes
2
answers
285
views
Inverse limit of the group $\mathbb{Q}_p$ of $p$-adic numbers
I am studying $p$-adic numbers and inverse (aka projective) limits.
I am interested in characterising the group $\mathbb{Q}_p$ of $p$-adic numbers as an inverse limit. I already know the inverse ...
- 173
7
votes
1
answer
256
views
Colimit of symmetric groups
I don't yet know much about categorical limits and colimits, I have just started learning about them, and so I wanted to experiment a bit with this concept. And to that end, my first natural attempt ...
- 1,495
4
votes
0
answers
45
views
Direct limit of sequences induced by fusing together copies of $\mathbb{Z}$
Let $A=l^\infty(\mathbb Z,\mathbb Z)$ be the abelian group of bounded sequences $\mathbb Z\to\mathbb Z$. Define a homomorphism $f\colon A\to A$ by
$$f(a)(n)=a(2n)+a(2n+1),$$
for $a\in A$ and $n\in\...
- 4,861
7
votes
2
answers
157
views
A more succinct group object diagram (all axioms in one connected diagram), questions about its properties...
Here is the definition of group object from nLab. They give 3 associated maps $* \xrightarrow{1} G$, $m: G^2 \to G$, and $-^{-1}: G \to G$ and require 3 commutative diagrams to complete the axioms ...
1
vote
1
answer
98
views
Index of Directed Union of Groups
Let $I$ be a nonempty directed set, and $H_i \le G_i$ both be directed sequence of groups living in some ambient group. Is there a nice formula for $|\bigcup_{i \in I} G_i : \bigcup_{i \in I} H_i|$, ...
- 8,040
4
votes
1
answer
80
views
Showing the rank of a group $H$ is precisely the maximal rank $\alpha$ of a sequence of groups.
I'm working with an inverse limit of groups and I'm trying to prove some property. Let $F_n$ be the free group of rank $n$. We have a morphism from $f_n:F_n\to F_{n-1}$ that sends a word in $F_n$ to ...
- 1,048
6
votes
1
answer
374
views
The direct limit of roots of unity
Let $\mu_n$ denote the group of $n$-th roots of unity. In $\mathbb{C}$, this group has exactly $n$ elements. For a positive integer $n$ and a prime number $p$, using the canonical isomorphisms $\mu_{p^...
- 1,190
4
votes
1
answer
315
views
Direct limit and amalgamation (Serre's "Trees")
At the very beginning of Serre's Trees, it's taken that the groups $G_i$ (indexed over some set $I$, with no additional specifications) are equipped with homomorphisms $f_{ij}:G_i\to G_j$, collected ...
- 1,147
1
vote
0
answers
45
views
Reduced words in certain direct limit of groups
Let $\{ G_i \}_{i \in I}, \{ A_{ij} \}_{i,j \in I}$ be families of groups with injective homomorphisms $A_{ij} \rightarrow G_i$ and $A_{ij} \rightarrow G_j$. Consider the direct limit
$$
G:= \...
- 808
2
votes
0
answers
281
views
General Properties of Direct Limits of Groups
Anyone know a decent reference on the basic theory of direct limits of groups, from an elementary (meaning group theoretic, as opposed to categorical) perspective? Finitely generated abelian groups ...
- 233