Questions tagged [epimorphisms]
For questions related to epimorphisms, which are categorical generalizations of surjective functions.
102
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About superfluous epimorphism in exact rows between R modules [Ker g is superfluous submodule]
I'm doing the following exercices about R-modules and superfluous epimorphism
Consider the following conmutative diagram in cathegory of R-modules, asume that both rows are exact:
Exact Rows like this
...
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1
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Show that $\overline\beta$ is epimorphism without diagram chasing
Let $\mathcal{A}$ be an abeilan category. Consider the following exact, commutative diagram in $\mathcal{A}$
$$\require{AMScd}\begin{CD}\ker a@>{\overline\alpha}>> \ker b@>{\overline \beta}...
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Eilenberg-Zilber's lemma existence
Let $X$ a simplicial set, $x\in X_m$ an $m$-simplex. We say that $x$ is degenerate if $\exists s:[m]\to[n]$ an epimorphism such that $n<m$ and $y\in X_n$ such that $X(s)(y)=x$.
Now I want to show ...
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Epimorphism in simplex category is split
Consider $\Delta$ the simplex category, with objects $[n]=\{0,\dots,n\}$ and morphisms $f:[n]\to [m]$ such that $i<j\implies f(i)\leq f(j)$ (my definition is with $i<j$). I have shown that ...
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Let $H$ be a normal of order $6$. If $f :G\to G_1$ be an epimorphism of groups s.t. $H\subset{\rm Ker}(f)$, then show $G_1$ is a hom. image of $G/H$.
Let $H$ be a normal subgroup of order $6$. If $f :G\longrightarrow G_1$ be an epimorphism of groups such that $H\subset \mathrm{Ker} (f)$, then show that $G_1$ is also a homomorphic image of $G/H$.
In ...
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Show that $(\Bbb Q, +)$ and $(\Bbb R, +)$ are not isomorphic groups.
Show that $(\Bbb Q, +)$ and $(\Bbb R, +)$ are not isomorphic groups.
I dont how to proceed here. My general strategies include: trying to show that if one group has an element of a particular order ...
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88
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Show that $(\Bbb Z, +)$ and $(\Bbb R, +)$ are not isomorphic groups .
Show that $(\Bbb Z, +)$ and $(\Bbb R, +)$ are not isomorphic groups.
What I did so far:
To show that these groups are not isomorphic, we need first assume that, there exists an isomorphism $f$ from $(...
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1
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94
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Monomorphisms and epimorphisms in abelian categories
Let $\mathsf{C}$ be a an abelian category. Let $f \colon X \rightarrow Y$ be a morphism in $\mathsf{C}$. I am looking at the following two statements:
The morphism $f$ is a monomorphism if and only ...
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70
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Find the set of all homomorphisms from $(\mathbb{Z},+)$ onto $(\mathbb{Z}_6,+)$.
Find the set of all homomorphisms from $(\mathbb{Z},+)$ onto $(\mathbb{Z}_6,+)$.
My solution goes like this:
We first try to find out all the homomorphisms from $(\mathbb{Z},+)$ to $(\mathbb{Z}_6,+)$....
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Find all homomorphisms from $(\mathbb{Z}_8,+)$ into $(\mathbb{Z}_6,+)$.
Find all homomorphisms from $(\mathbb{Z}_8,+)$ into $(\mathbb{Z}_6,+)$.
The solution given is as follows :
We have , $\mathbb{Z}_8=\langle [1]\rangle$ . Let $f:\mathbb{Z}_8\longrightarrow \mathbb{Z}...
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Prove a square is a pullback
Let $C$ an abelian category, and consider the following diagram:
$$
\require{AMScd}
\begin{CD}
P
@> \beta_1 >>
A_1
\\
@V \beta_2 VV
@VV \alpha_1 V
\\
A_2
...
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Pullback of an epimorphism in the category of Hausdorff spaces
Can anyone give me an example of a pullback of an epimorphism which is not an epimorphism, in the category of Hausdorff spaces?
I've been thinking about it but I have no idea.
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Describing epimorphisms in Set-valued functor categories without using pointwise computation of colimits
Let $\mathscr{A}$ be a small category and consider the functor category $[\mathscr{A}, \mathbf{Set}]$.
Fact.
The epimorphisms in $[\mathscr{A}, \mathbf{Set}]$ are precisely those natural ...
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Let $M$ be a left $R$-module, every epimorphism $f:M\rightarrow R$ splits.
This problem comes from a exercise of Rings and Categories of Modules.
Let $M$ be a left $R$-module, proof that every epimorphism $f:M \rightarrow R$ splits.
(If $f:M \rightarrow N $ and $f':N \...
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How to prove a map is epic using generalized elements only?
I have a map $\require{AMScd}f\colon X \to Y$ in some category $\mathcal E$ which I would like to show is epic. However the only description I have of $X$, $Y$, and $f$ is through the Yoneda ...
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3
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An "easy" way to construct an epimorphism from S4 to S3
I'm trying to construct an epimorphism φ from S4 to S3 such that:
H = ker(φ) = {(1),(12)(34),(13)(24),(14)(23)}
where H is a normal subgroup of S4, contained in A4 and isomorphic to the Klein 4-...
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49
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Is there a standard notation for quotient maps akin to that for inclusions ($\hookrightarrow$)?
I'm having to write out lots of short exact sequences and commutative diagrams at the moment. There are standard notations for monomorphisms ($\rightarrowtail$) and epimorphisms ($\twoheadrightarrow$),...
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Prove that there does not exist an epimorphism from $S_3$ to $(\mathbb{Z}_6,+)$
Prove that there does not exist an epimorphism from $S_3$ to $(\mathbb{Z}_6,+)$.
My approach:
Let $\phi: S_3\to \mathbb{Z}_6$ be an epimorphism.
$\bar{1}\in \mathbb{Z}_6$ and $o(\bar{1})=6$. Since $\...
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For any ideal, why must a ring homomorphism be surjective for the ideal to be mapped to an ideal?
Problem:
Let $f$: $R \to S$ be a ring homomorphism
Show that if $f$ is surjective, then for any ideal $I \subset R$, the set $f(I)$ is an ideal of $S$.
My Solution:
Let $x,y\in I\subset R\implies f(x),...
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Is the corestriction of a commutative rings morphism to its epicenter an epimorphism?
Given a morphism of commutative rings $f:A\to B$, we define its epicenter, or dominion, by the set $$E=\{b\in B \;|\; b\otimes_A 1 = 1\otimes_A b\}.$$Note that if $u,v:B\to C$ are morphisms of ...
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What are the epimorphisms in the category of topological groups?
A morphism $f: X \to Y$ is an epimorphism if for all $g, h: Y \to Z$, if $g \circ f = h \circ f$ then $g = h$. The epimorphisms in the category of groups are the surjective group homomorphisms. The ...
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65
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Kernel pairs, coequalizers and epimorphisms
Suppose we have a category $C$ having kernel pairs and coequalizers. Suppose we have an epimorphism $f:Y\to X$ in $C$, and consider the Kernel pair $p_1,p_2:Y\times_X Y \rightrightarrows Y$; then $$X=...
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Prove that if $\mathcal{A}$ has enough projectives, then so does $Ch(\mathcal{A})$
This is exercise 2.2.2 in Weibel's AIHA. We already know that a chain complex $P_{\bullet}$ is projective in $Ch(\mathcal{A})$ iff it is a split exact complex of projectives. Here's my proof, but it ...
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Example of non-balanced category.
I have a claim that in the category of torsion-subgroup-free abelian groups any nonzero homomorphism from $\mathbb{Z}$ to $\mathbb{Z}$ is mono and epi. I am struggling to prove that it is indeed the ...
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218
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A proof that epimorphisms in the category of Hausdorff spaces have dense image
I'm trying to understand the final step in the following argument showing the epimorphisms in $\textbf{Haus}$ have dense image. Let $C$ be a topological space. For $x, y \in C$, let $x \sim y$ if and ...
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Regular epimorphisms in the category of topological rings
I believe regular epimorphisms in the category of topological groups are precisely the surjective open maps. Is this also true for the category of topological rings? If not, is there some other ...
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showing regular epimorphisms are not stable under composition in general
The following counterexample is given in response to this question Is composition of regular epimorphisms always regular?:
"Let $\mathbf{2} =\textbf{{0→1}}$ be the category with two objects and ...
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quotient of an object 2 [duplicate]
Here I've asked what is a quotient of an object and the answer was that it is an equivalence class of epis.
But here on the first page they claim that regular quotient is an coequalizer of 2 morphisms....
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quotient of an object
In category theory
What from these 2 things is called quotient:
epi
or rather
split epi
Whats the difference of a usage of these 2.
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Epic morphisms in the category of vector spaces. Is AC needed?
In $\mathsf{FinVect}_k$, the category of finite-dimensional $k$-vector spaces, all epis are surjective, by the argument given in this answer. I know how to generalize this argument to $\mathsf{Vect}_k$...
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Epic does not always imply surjective?
I'm asking about one type of proof of epic $\implies$ surjective like Clive Newstead's answer in Epic implies Surjective or Thomas Andrew's answer in Morphism epimorphism if and only if surjective. ...
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correspondence between epi and monos
I want to know if there is a problem with this argument:
Let $C$ be a category and $F$ be a faithful functor $C^{op}\rightarrow C$, with left adjoint $F^{op}: C\rightarrow C^{op}$
If $f: A\rightarrow ...
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base change functor preserves internally projective objects
Let $E$ be a topos . an internally projective object is an object $P$ for which $(-)^P:E\rightarrow E$ preserves epi morphisms.
Equavalently: $E/P\rightarrow E$ preserves epis.
Prove that for ...
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Internally projective object gives a commuting square
Let $\epsilon$ be a topos and $P$ be an object of $\mathcal{E}$ such that $(-)^P: \mathcal{E} \rightarrow \mathcal{E}$ preserve epis then the right adjoint to pull back $\Pi _P : \mathcal{E} /P \...
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Do pull backs preserve epi morphisms in a topos?
Do pull backs preserve epi morphisms in a topos?
I know epi morphisms are not always preserved by pull back, but what if the category is a topos?
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Skewed duality between mono- and epi- morphisms
The duality between monomorphisms and epimorphisms is well understood in terms of the opposite category (reversing the arrows). However, it is much less obvious on intuitive level. To cite Notes on ...
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Relation between epi and mono in a category
Let $f: A \rightarrow B$ be an epic morphism in category $C$. Then is it true that the morphism $h: \text{Hom}(B,-)\rightarrow \text{Hom}(A,-)$ is mono?
If yes why?
What if $B$ is the initial object? ...
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The axiom of choice for a category
I am currently studying the counterpart of axiom of choice in ETCS which is the axiom that every surjective function has a right inverse.
In category of sets the surjective functions are epimorphsims ...
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3
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203
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Unique maximal ideal and ring epimorphism kernel with prime numbers equivalence
Let $A$ be a ring. Let $f: \mathbf{Z} \to A$ be a surjective ring homomorphism. Prove that $A$ has a unique maximal ideal iff there exists $n\in \mathbf{N}$ and $p\in\mathbf{N}$ a prime number such ...
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Determine if the function is epimorphism
Let $U_{n}$ be the multiplicative group of the $n$th roots of unity; this group is cyclic of order $n$ and is generated by $w = \cos(\frac{2\pi}{n})+i\sin(\frac{2\pi}{n})$. If we define $f: (\mathbb Z,...
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Epimorphism between unital algebras is unital
How could you prove the following statement?
Let A and B are unital algebras. If $f:A\to B$ is an epimorphism, then $f$ is unital; i.e. $f(1)=1$.
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Function-like monos and epis
When dealing with $\textbf{Set}$ we have that if $f:A\to B$ is a monomorphism, $g:A\to A’$ is an epimorphism, and adding $f’:A’\to B$ we have a commuting triangle, then $f’$ must be a monomorphism. ...
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One of a math problems [closed]
For modules, let $M = M_1 ⊕ M_2$ and let $f :M→N$ be an epimorphism with $K = \ker f$ and $N = f(M_1) + f (M_2)$.
(1) Prove that if $K= ( K \cap M_1)+ (K \cap M_2)$, then this sum is direct.
Could ...
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156
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Monoidal adjunction whose right-adjoint functor has structure morphisms which are epimorphisms
Let $(\mathbf{C},\otimes,1)$ and $(\mathbf{D},*,e)$ be monoidal categories and let $L:\mathbf{C}\rightarrow \mathbf{D}$ and $R:\mathbf{D}\rightarrow \mathbf{C}$ be functors. Suppose that there exists ...
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Fibrations are thought of as epimorphisms
In the book More concise algebraic topology on the page 213 they write
We think of fibrations as analogous to epimorphisms.
BUT Hovey on the page 51 says
$f$ is a fibration if it is in $J-inj$.
My ...
- 3,907
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vote
2
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Monomorphisms and epimorphisms in the category of chain complexes
Let $\mathsf{C}$ be an abelian category and $\mathsf{Comp(C)}$ its category of chain complexes. Suppose that $f\colon (C,d)\to (C',d')$ is a monomorphism in $\mathsf{Comp(C)}$. I want to prove that ...
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$R$ is a commutative ring with $1$, prove that there exist epimorphism from $R[x]$ onto $R$.
$R$ is a commutative ring with $1$, prove that there exist epimorphism from $R[x]$ onto $R$.
I maybe able to show that R[x] onto R is a homomorphism but I'm not sure how to show that it is onto
and ...
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If $R$ is a PID, $S$ an integral domain and $f: R \to S$ is an epimorphism, why is it that either $f$ is an isomorphism or $S$ is a field? [duplicate]
If $R$ is a PID, $S$ an integral domain and $f: R \to S$ is an epimorphism, why is it that either $f$ is an isomorphism or $S$ is a field?
PID - Principal Ideal Domain
What I know:
If $S$ is not a ...
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1
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Proving the inclusion map of an integral domain into its quotient field is an epimorphism
As a well-known example of a ring homomorphism which is monic and epic, but not a ring isomorphism, serves the inclusion map $\iota:\mathbb Z\hookrightarrow\mathbb Q$. While the monocity follows ...
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1
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280
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In Set, why are projections not epic but injections are monic?
I'm working through Bird and DeMoor's Algebra of Programming and I have some basic gaps in my understanding. Problem 2.28 asks if projection outl is epic in Set, if inl is monic, and why the ...
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