# Questions tagged [epimorphisms]

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

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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 ...
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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 ...
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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?...
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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 ...
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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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### 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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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 $\Pi _P : \mathcal{E} /P \... • 388 4 votes 1 answer 322 views ### Do pull backs preserve epi morphisms in a topos? Do pull backs preserve epi morphisms in a topos? I know epi morphisms are not always preserved by pull back, but what if the category is a topos? • 849 2 votes 0 answers 105 views ### Skewed duality between mono- and epi- morphisms The duality between monomorphisms and epimorphisms is well understood in terms of the opposite category (reversing the arrows). However, it is much less obvious on intuitive level. To cite Notes on ... • 1,030 2 votes 1 answer 70 views ### Relation between epi and mono in a category Let$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$is the initial object? ... • 849 1 vote 2 answers 168 views ### The axiom of choice for a category I am currently studying the counterpart of axiom of choice in ETCS which is the axiom that every surjective function has a right inverse. In category of sets the surjective functions are epimorphsims ... • 183 2 votes 3 answers 212 views ### Unique maximal ideal and ring epimorphism kernel with prime numbers equivalence Let$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}$a prime number such ... • 91 0 votes 1 answer 153 views ### Determine if the function is epimorphism Let$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,...
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. $f(1)=1$.