Questions tagged [epimorphisms]

For questions related to epimorphisms, which are categorical generalizations of surjective functions.

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Categories in which there is a mono $A \to B$ iff there is an epi $B \to A$

Consider the property $P$ of a category $\mathcal{C}$ that for two objects $A$, $B$ in $\mathcal{C}$ there exists a monomorphism $A \to B$ iff there exists an epimorphism $B \to A$. Does the property $...
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Is the pullback functor $q^*$ essentially injective when $q$ is a regular epimorphism?

Let $q : W \rightarrow V$ be a regular epimorphism in a category $\mathcal{C}$, and consider the pullback functor $q^* : \mathcal{C}/V \rightarrow \mathcal{C}/W$. Is $q^*$ necessarily essentially ...
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Is every additive map $R$-linear if $ℤ \to R$ is an epimorphism? [duplicate]

Question. Let $R$ be a ring for which the unique homomorphism of rings $ℤ \to R$ is an epimorphism. Is every additive map between $R$-modules already $R$-linear? Context. Part of Exercise 1.5.xi in ...
Jendrik Stelzner's user avatar
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Meaning of projection onto one factor in $0\to A^{r-1}\to A^r\to A\to 0$

The following is taken from: $\textit{Partial Differential Control Theory Vol 1: Mathematical tools}$ by J F. Pommaret $\color{Green}{Background:}$ $\textbf{Definition 1.50.}$ $M$ is call a $\textit{...
Seth's user avatar
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Are monos in epis and epis in monos also monic or epic in the entire category?

Given any category $C$, let $\mathrm{Epi}(C)$ and $\mathrm{Mono}(C)$ denote the (generally non-full) subcategories of $C$ consisting of the epimorphisms and monomorphisms respectively. Then, is any ...
Geoffrey Trang's user avatar
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In the category Haus, a continuous map $f:A \rightarrow B$ is an epimorphism iff $f(A)$ is dense in $B$

I am just getting started with category theory and I am trying to prove that In the category Haus, a continuous map $f:A \rightarrow B$ is an epimorphism iff $f(A)$ is dense in $B$ I found a partial ...
some_math_guy's user avatar
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Linearly Compact Module in $R-Mod$

Definition: A module $M$ is called linearly compact if for a family of cosets $\{x_{i}+M_{i}\}_{\triangle}$, $x_{i}\in M$, $\triangle$ is a directed set, and submodules $M_{i}\subset M$ (with $M/M_{i}$...
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Does an epimorphism factor any morphism with the same codomain? [duplicate]

Suppose that $f: A \to B$ is an epimorphism and $x: X \to B$ is a morphism. Is it true that there exists a morphism $y: X \to A$ such that $f \circ y = x$? Is it necessary that the category is abelian?...
Harry Partridge's user avatar
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What does "factoring" mean in the context of ring epimorphisms?

Let $f\colon A \to B$, $g\colon A \to C$ be two ring epimorphisms. What does it mean when one says that these two factor into another ring epimorphism $h\colon B \to C$? I'm guessing that it means ...
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How to prove that a group affords no epimorphism in $S_3$

Good evening, I want to prove that the group $G = \langle a, d \mid a d a^{-1} d a = d a d^{-1} a d \rangle$ affords no epimorphism in $S_3$ (the group of the permutations of $\{1, 2, 3\}$). I tried ...
Arthur Filippi's user avatar
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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 ...
raisinsec's user avatar
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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 ...
Arthur's user avatar
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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 ...
Arthur's user avatar
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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 $(...
Arthur's user avatar
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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 ...
Margaret's user avatar
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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 $(\mathbb{Z}_6,+)$....
Arthur's user avatar
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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}...
Arthur's user avatar
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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 ...
Selena J's user avatar
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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 ...
Jendrik Stelzner's user avatar
1 vote
2 answers
176 views

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 \...
fusheng's user avatar
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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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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-...
Simone Fiorio's user avatar
2 votes
1 answer
54 views

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 ($\twoheadrightarrow$),...
user829347's user avatar
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3 votes
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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 $\...
MKSar's user avatar
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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),...
Luke's user avatar
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2 votes
1 answer
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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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4 votes
1 answer
406 views

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 ...
user46484's user avatar
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1 answer
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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=...
Nulhomologous's user avatar
1 vote
0 answers
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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 ...
ZYX's user avatar
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1 vote
1 answer
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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 ...
Michal Dvořák's user avatar
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291 views

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 ...
user46484's user avatar
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1 vote
0 answers
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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 ...
user46484's user avatar
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5 votes
1 answer
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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 ...
user46484's user avatar
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1 answer
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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....
user122424's user avatar
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-1 votes
1 answer
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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.
user122424's user avatar
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10 votes
1 answer
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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 $\mathsf{Vect}_k$...
Elías Guisado Villalgordo's user avatar
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1 answer
223 views

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. ...
sanguine's user avatar
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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 ...
Sajad's user avatar
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0 answers
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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 ...
user850424's user avatar
3 votes
0 answers
50 views

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 \...
user373827's user avatar
4 votes
1 answer
322 views

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?
user850424's user avatar
2 votes
0 answers
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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 ...
Tegiri Nenashi's user avatar
2 votes
1 answer
70 views

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? ...
user850424's user avatar
1 vote
2 answers
168 views

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 ...
Elise's user avatar
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2 votes
3 answers
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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 ...
nelichu's user avatar
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1 answer
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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,...
judie's user avatar
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1 answer
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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. $f(1)=1$.
JerryCastilla's user avatar