All Questions
Tagged with free-groups category-theory
48
questions
0
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4
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147
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Free object is a free group in the category of groups
I have a question and would appreciate a clear answer.
Firstly, I will provide an introduction regarding my understanding, and then I will ask my question.
Let's begin with the definition of a ...
- 123
2
votes
1
answer
121
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Proof of the universal property of free abelian groups
Let $S$ be a set. The group with presentation $(S, R)$ , where $R = \{\ [s , t] \mid s,t\in S\ \}$ is called the free abelian group on $S$ -- denote it by $A (S)$ . Prove that $A (S)$ has the ...
- 351
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0
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33
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Are right-adjoints of a forgetful functor reflectors?
From what I understand, there is no formal definition of a forgetful and an inclusion functor, but more like "moral guidelines" with "good properties" of why we would call them ...
- 681
2
votes
2
answers
78
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Pushout of Z isomorphic to Z
$\mathbb{Z}\xleftarrow{\text{id}}\mathbb{Z}\xrightarrow{\text{x2}}\mathbb{Z}$
How do we prove this group pushout is isomorphic to $\mathbb{Z}$?-
I know we can take $\langle x|\rangle$ as a ...
1
vote
0
answers
85
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Construction of left adjoint
A very standard example of left adjoints would be the functor $F:Sets\to Grps$ that maps sets to their free groups is a left adjoint to $G:Grps \to Sets$ which forgets the group structure. My question ...
- 211
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1
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77
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Three questions about Wikipedia's definition of Van Kampen's theorem for fundamental groups
I'm studying Algebraic Topology off of Hatcher and (unfortunately as usual) I find his definition and explanation of Van Kampen's theorem to be carelessly written and hard to follow. I happen to know ...
- 894
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2
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89
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Question about functoriality of free group.
This is an example from Riehl's category theory in context specifically 1.3.2.ix. She writes the following:
Example: There is a functor $F:\mathbf{Set} \rightarrow \mathbf{Group}$ that sends a set $X$...
- 2,673
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2
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182
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Injective map needed in definition of free groups
I have the following definition:
Let $X$ be a set. The free group genereted by $X$ is a group $F$, if there exists an injection $i:X\to F$ s.t. for all Groups $G$ and (not neccesarly injectiv) ...
- 423
3
votes
0
answers
88
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Proof of Uniqueness of Free Group [closed]
I just wanted to verify if my proof is correct as I have to present this proof in my next class.
Consider the comm. diagram drawn above. We can verify that it commutes due to the categorical ...
1
vote
1
answer
117
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Details in existence of free groups proof (Clara Loeh,pg-22,23)
I have a few doubts in the proof of existence of free groups given by Clara Loeh in page-22,23 of her Geometric Group theory book.
Theorem 2.2.7: Let S be a set. Then there exists a group freely ...
1
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0
answers
151
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Rotman's Algebraic Topology Lemma 9.11
This is the Lemma 9.11 of Rotman's "An Introduction to Algebraic Topology".
The topic where I found this simple lemma of homological algebra is the Theorem of Acyclic Models. So we are ...
- 765
3
votes
2
answers
189
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Is the coproduct of any family of groups a free group
In Lang’s Algebra, he constructs the coproduct of a family of groups in a similar way to the construction of the free group of a set $S$, $F(S)$. He later constructs the free product of $n$ groups and ...
- 563
1
vote
1
answer
434
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Direct product of groups is not the coproduct
On page 66 of Lang’s Algebra he states the following:
Let $G$ = ∏ $G_i$ be a direct product of groups. Then each $G_j$ admits an injective homomorphism into the product, on the $j$-th component, ...
- 563
4
votes
1
answer
62
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Quotient of a free group on a set
Suppose I have a free group $F_S$ on a set $S,$ with a nonempty proper subset $T.$ If $N$ is the smallest normal subgroup of $F_S$ containing $T$, I would like to show that $F_S/N$ is also free. There'...
- 43
3
votes
0
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230
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Final object in the "free group" category $\mathscr{F}^A$
The following excerpts come from the book Algebra: Chapter 0's section II.5 - Free groups. My question is about an easy exercise.
First, to define the free group $F(A)$ on $A$, the author introduces ...
- 602