# 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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### 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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### 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 ...
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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?

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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### 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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### 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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### 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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### 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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### 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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### 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 ...
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### 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 ...
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### 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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### 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 ...
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### 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 ...
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### 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 ...
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### 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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### 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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### 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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### 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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### 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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### 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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### 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 ...
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### 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 ...
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### 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 ...
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### 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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### 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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### 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 $f:V \rightarrow W$ ...
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### 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 $n$, $G$ ...
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 ...