As of May 31, 2023, we have updated our Code of Conduct.

# Questions tagged [epimorphisms]

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

102 questions
Filter by
Sorted by
Tagged with
17 views

### About superfluous epimorphism in exact rows between R modules [Ker g is superfluous submodule]

I'm doing the following exercices about R-modules and superfluous epimorphism Consider the following conmutative diagram in cathegory of R-modules, asume that both rows are exact: Exact Rows like this ...
1 vote
83 views

1 vote
56 views

### 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.
81 views

### 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 ...
1 vote
82 views

1 vote
43 views

35 views

### 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 ...
50 views

30 views

### 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. $f(1)=1$.
62 views

### 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. ...
34 views

### One of a math problems [closed]

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 ...
1 vote
156 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 ...
70 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 ...
1 vote
376 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 ...
135 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 ...
1k 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? [duplicate]

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 ...
As a well-known example of a ring homomorphism which is monic and epic, but not a ring isomorphism, serves the inclusion map $\iota:\mathbb Z\hookrightarrow\mathbb Q$. While the monocity follows ...