Questions tagged [epimorphisms]

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

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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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1answer
32 views

One of a math problems

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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1answer
42 views

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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1answer
47 views

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 ...
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2answers
68 views

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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1answer
39 views

$R$ 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 ...
0
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2answers
73 views

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?

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 ...
3
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1answer
41 views

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 epi, but not a ring isomorphism, serves the inclusion map $\iota:\Bbb Z\hookrightarrow\Bbb Q$. While the monocity follows immediately ...
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1answer
104 views

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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1answer
196 views

Is an epimorphic endomorphism of a noetherian commutative ring necessarily an isomorphism?

Let $A$ be noetherian commutative ring with one, and let $f:A\to A$ be an epimorphic endomorphism of $A$. Is $f$ necessarily an isomorphism? ("An epimorphic endomorphism" means of course "an ...
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1answer
79 views

Monomorphism & epimorphism in the category of schemes

Is there a morphism in the category of schemes which is simultaneously a monomorphism and an epimorphism yet is not an isomorphism? "Nicer" examples are preferred (e.g. with integral Noetherian ...
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1answer
75 views

Example of a NON-effective epimorphism

I'm reading Introduction to Étale Cohomology by Tamme and I'm confused by the notion of effective epimorphism (page 25, section 1.3.1). Recall that an epimorphism (in a category $\mathbf{C}$) is a ...
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1answer
58 views

Epimorphism and generator of Mod-$ R $

I am stuck with a question that should be very simple, because ALL the books and references that I have read say the same: ''is obvious and we will not prove it''. Is the following statement: ''Let $...
2
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2answers
271 views

Why is epimorphism not defined as follows?

Epimorphism in category theory, can be seen as a generalization of "surjective function" in Set. But we could also have characterized surjective functions in Set as follows: Definition (*). $f:A\...
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2answers
105 views

Fundamental homomorphism theorem (epimorphism)

Let φ : R → S be a ring epimorphism. Prove that R/kerφ ∼= S. Is this the fundamental homomorphism theorem? I thought the FHT started with a ring homomorphism and not an epimorphism. Does this change ...
2
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1answer
192 views

Morphism=monomorphism•epimorphism?

Is it true that any morphism in any category can be written as a combination of monomorphism and epimorphism? In SET and categories where monomorphism is an injective function and epimorphism is a ...
7
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1answer
173 views

Strict Epimorphism of Schemes

I am reading Milne's Etale Cohomology and ran across this problem which has so-far eluded me. According to Milne, in any category with fiber products, we say that a morphism $f:Y \to X$ is a strict ...
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2answers
540 views

The inclusion of $\mathbb{Z}\to\mathbb{Q}$ is not and epimorphism

I have to prove this only in basic ring theory, I have read something in category theory, but its too complex. The definition of epimorphism that I have is that the function can cancel other functions ...
10
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3answers
784 views

Epimorphism and monomorphism explained without math?

I'm trying to understand category theory to increase my coding skills and epimorphism and monomorphism aren't clear to me. Unfortunately, my last formal education was when I was 12 due to ...
4
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1answer
236 views

Epimorphism, in the category of commutative rings with unity , with domain a field is an isomorphism?

Let $R$ be a commutative ring with unity and $k$ be a field. Let $f: k \to R$ be an "epimorphism" of commutative rings (https://en.wikipedia.org/wiki/Epimorphism) i.e. $f$ is a ring homomorphism ...
4
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1answer
490 views

Is surjection an epimorphism without the axiom of choice?

If a map of sets $f:A\to B$ is surjective then it is an epimorphism. Is it possible to prove this without the Axiom of Choice? I know that in order to prove that surjective maps have right inverses we ...
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2answers
105 views

If $f:G→ℤ_{8}$ be a group epimorphism , then what can be said about $G$?

I was trying to answer the following problem : Let $G$ be a finite group and $ f : G \to \Bbb Z_8$ be a group epimorphism, then which of the following must be true ? (a) $G$ is isomorphic to $Z_8$ ....
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1answer
107 views

In a category with binary products, every strong epimorphism is an epimorphism.

I can't prove the title proposition. So let $f$ be a strong epimorphism of a category with binary products. Let $u,v$ be morphisms such that uf=vf. I want to prove that $u=v$. I consider the ...
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1answer
64 views

Is there a group epimorphism $\mathbb{R} \to S_3$?

I had to found some group epimorphisms and I stucked with this example. Is there an epimorphism from a group $(R,+)$ - real numerbes with addition onto a group $(S_3,\circ)$?
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1answer
34 views

Show that a continuous epimorphism pushes forward connectedness

Let $f:X\to Y$ be a continuous epimorphism where $X$ is connected. Show that $Y$ is connected. Now I know the concept of epimorphism and connectedness, just not sure how those two are linked with ...
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2answers
147 views

Epimorphism and surjection

I have seen on http://mathworld.wolfram.com/Epimorphism.html A morphism $f:Y \to X$ in a category is an epimorphism if, for any two morphisms $u,v:X\to Z, uf=vf$ implies $u=v$. In the categories ...
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2answers
376 views

A monomorphism and epimorphism that is neither injective nor surjective(set-theoretically)

Is it possible to construct a category such that there is a monomorphism and epimorphism that is neither injective nor surjective(set-theoretically). I notice that there is a solution in which a ...
2
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1answer
695 views

A morphism of sheaves is an epimorphism if and only if it is surjective on the level of stalks

Let $\phi: \mathscr{F} \to \mathscr{G}$ be a morphism of sheaves. Recall that $\phi$ is an epimorphism if given $\psi_1, \psi_2: \mathscr{G} \to \mathscr{H}$, then $\psi_1 \circ \phi = \psi_2 \circ \...
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4answers
855 views

Why does the name “epimorphism” refer to a surjective homorphism?

The wikipedia page talks about epimorphisms with category theory in mind, but I have no experience with this and ask this question from a group theory point of view (answers from any point of view are ...
3
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2answers
147 views

Epimorphisms and faithful functors in a rigid abelian tensor category

Let $\mathsf{C}$ be a rigid abelian tensor category in the sense of Deligne and Milne's notes (p.9). Let $\mathbf{1}$ denote the identity object in $\mathsf{C}$ with respect to $\otimes$. The abelian ...
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5answers
4k views

Morphism epimorphism if and only if surjective

In the category of sets, I want to prove that a morphism is an epimorphism if and only if it is surjective. In both directions, I'm having a hard time approaching this problem. This is how far I got....
8
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1answer
218 views

Is I-adic completion a ring epimorphism?

Let $R$ be a commutative ring and let $I \subset R$ be an ideal. For any $n \ge 1$, the ring homomorphism $R \rightarrow R/I^n$ is surjective, hence an epimorphism in the category of rings. What about ...
3
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1answer
290 views

Counterexample for commutative diagram of epimorphism.

$f:X\rightarrow Y$ is an epimorphism iff first diagram on the picture is a pushout. Can I say "there is a pushout along the morphism h"? Is my example of non-epimorphism (below on the picture) ...
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1answer
53 views

Let $h: G \to H$ an epimorphism. If N is a normal subgroup of H, show that $G/h^{-1}(N) \simeq H/N.$

Let $h: G \to H$ an epimorphism. If N is a normal subgroup of H, show that $$G/h^{-1}(N) \simeq H/N.$$ I have the good intuition that I have to use the first isomorphism theorem. I'm not able how ...
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0answers
68 views

Projective cover and epimorphism

Let $C$ be an abelian category and let $X$ be an object with finite length. Thus there is a composition series $0=X_0 \stackrel{\iota_0}{\rightarrow}X_1\stackrel{\iota_1}{\rightarrow}\cdots X_{n-1} \...
3
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0answers
167 views

Intuitive idea to consider strict epimorphism

In a category with fiber products, a morphism $Y\to X$ is said to be strict epimorphism if the sequence is exact: $Y\times_XY\xrightarrow{p_1,p_2}Y\to X$, (here the first arrow should be a double ...
2
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2answers
557 views

Ring monomorphism and epimorphism

$$\large{\Phi: \mathbb Z[X] \rightarrow \mathbb R \; s \; \Phi(p(X)) \; := \; p(\sqrt{5})}$$ Hello guys so I have the following problem: I have to prove whether the following mapping is ring ...
3
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1answer
792 views

The pushout of an epimorphism is an epimorphism

I'm reading the book "Handbook of Categorical Algebra: Volume 1, Basic Category Theory" by Francis Borceux, and in page 52 he states that "..."the pullback of a monomorphism is a monomorphism". ...
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1answer
122 views

Quotients and regular epimorphism

In category theory, is a quotient the same as a regular (or extremal?) epimorphism? (Just like a subobject corresponds to a regular mono.)
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2answers
60 views

Existence of an epimorphism

Suppose $V$ and $W$ are finite-dimensional vector spaces over $F$ such that dim$_F(V)>$dim$_F(W)$. Is it true that an epimorphism $f:V \rightarrow W$ exists? a monomorphism $f:V \rightarrow W$ ...
8
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2answers
728 views

Motivation behind the definition of monomorphism and epimorphism in category theory

Let $\mathcal{C}$ be a (locally small) category and let $X$ and $X'$ be objects in $\mathcal{C}$. (Definition $1$) We say that $f\colon X\to X'$ is a monomorphism if for any object $Y$ in $\mathcal{...
1
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1answer
292 views

Invariant dimension property and a ring epimorphism

In Hungerford's Algebra, p. 186, the Proposition 2.11 says Let $f:R\to S$ be a nonzero epimorphism of rings with identity. If $S$ has the invariant dimension property (IDP), then so does $R$. It is ...
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1answer
324 views

Let $F$ be a field and $f: \mathbb{Z} \to F$ be a ring epimorphism.

Here we have $F$ a field and $f: \mathbb{Z} \to F$ a ring epimorphism. We are to prove that $F$ is a finite field with non-zero characteristic. I know that since $f$ is an epimorphism, we have that $...
1
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1answer
114 views

Epimorphism from G to Z

I've got a problem with this exercise, I'd be thankful if someone could help. Let $G$ be a group and let $f$ be an epimorphism from $G$ to $\mathbb{Z}$. Show that for every positive integer $n$, $G$ ...
3
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1answer
158 views

split epimorphism,equalizer,universal property

Let $g:B\to A$ be a split epimorphism with $f:A\to B$, $g\circ f=\operatorname{id}_A$. Why is $g$ a coequalizer of $f\circ g$ and $\operatorname{id}_B$? Commutativity is clear,but the universal ...
3
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1answer
158 views

Epimorphism in Category

Let A and B are two objects in a category $\mathcal{C}$ and $u \in Hom(A,B)$. We say $u $ is a monomorphism if and only if for any $X \in Obj(\mathcal{C})$ and $v \in Hom(X,A)$, the map $f:Hom(X,A)\...
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2answers
159 views

Epimorphism affect on Ideals

Let $f:R\rightarrow S$ be an epimorphism of commutative rings with unit $1$ and let $I\unlhd R$ be a maximal ideal that does not contain $ker(f)$. I'm trying to show if it's true or not that $f(I)$ is ...
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1answer
188 views

For an abelian group $G=G'$, the endomorphism $x \mapsto x^{-1}$ is both monic and epic

Let $G=G'$ be any abelian group. Define $\phi : G \rightarrow G'$ by $\phi(x) = x^{-1}$. Show that $\phi$ is a monomorphism and an epimorphism.
5
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1answer
541 views

Definition of strict epimorphism

Following is the definition of strict epimorphism from the paper I'm reading. However, I have some confusion. What does $$ \mathcal{C}\xrightarrow{x}X $$ mean? I think $\mathcal{C}$ is a category ...
4
votes
2answers
385 views

Monomorphisms and epimorphisms in the category of Boolean algebras

A Boolean algebra is a ring with unity all of whose elements are idempotent. We regard a zero ring $0$ as a Boolean algebra. Let $\mathcal{B}$ be the category of Boolean algebras. A morphism in $\...