# Questions tagged [epimorphisms]

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

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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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### is there an epimorphism $\pi_1(\mathbb{U}(n))\twoheadrightarrow \pi_1(\partial( [ -1,1 ]\times [ -1,1 ]))$?

We are considering the set $\mathbb{U}(n)= \{X \in \mathbb{C}^{n \times n}: X^{\ast}X = XX^{\ast} = I_n\}$ as a topological subspace of $C^{n \times n}$ with respect to the Frobenius metric ...
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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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### 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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### 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 ...
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### 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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### 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....
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### 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 ...
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### 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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### 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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Let $C$ be an abelian category and let $X$ be an object with finite length. Thus there is a composition series $0=X_0 \stackrel{\iota_0}{\rightarrow}X_1\stackrel{\iota_1}{\rightarrow}\cdots X_{n-1} \... 0answers 189 views ### Intuitive idea to consider strict epimorphism In a category with fiber products, a morphism$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 ...
$$\large{\Phi: \mathbb Z[X] \rightarrow \mathbb R \; s \; \Phi(p(X)) \; := \; p(\sqrt{5})}$$ Hello guys so I have the following problem: I have to prove whether the following mapping is ring ...