Questions tagged [monomorphisms]

Question 1

Is there a category $\mathcal{C}$ with zero object and a morphism $f:A\to B$ such that $0:A\to A$ is a kernel but $f$ is not monic? If $\mathcal{C}$ is preadditive then $\ker f = 0 \iff f$ is monic. ...

Question 2

This must has been asked several times in this cite, so I apologize in advance if this turns out to be trivial... Let $A,B$ be commutative rings. We know that if $A\to B$ is an epimorphism in $\mathbf{...

Question 3

As in the title, my goal is to find the left/right invertible morphisms and mono/epimorphisms in the Rel category. I am quite new to categorical concepts, I tried to do this an exercise, I have found ...

Question 4

While extremal epimorphisms and monomorphisms need not be strong in general, extremal epimorphisms (respectively, extremal monomorphisms) are strong in any category with pullbacks (respectively, ...

Question 5

Let $A$ be a category. Let $(a_i)_{i\in I}$ and $(b_i)_{i\in I}$ be indexed families of objects of $A$. For each $i\in I$, let $$a_i\xrightarrow{f_i} b_i$$ be a morphism of $A$. Assume that the ...

Question 6

I have seen this question and I have the same problem, i.e.: Consider the slice category $\mathcal{Set}/\text{I}$. What is the subobject classifier here? In the answer it is noted, that the subobject ...

Question 7

What do we call a category in which every monomorphism is regular (i.e., an equalizer of a pair of morphisms)? Is there a standard name? For categories with a zero object (more generally, with zero ...

Question 8

Let me phrase the question in terms of elementary set theory first. Let $R \subseteq A \times B$ be a relation with the property that the image operator $$R_* : P(A) \to P(B), \quad R_*(T) := \{b \in ...

Question 9

In "A Course in Modern Mathematical Physics", Peter Szekeres gives a proof of the statement "In the case of the category of sets a morphism $A \xrightarrow{\phi} B$ is a monomorphism if ...

Question 10

Setup: In the category $\mathsf{SET}$, given a pair of functions $f_1 : U_1 \to X$ and $f_2 : U_2 \to X$, it is obvious that $\mathrm{Im}\; f_1 \cup \mathrm{Im}\;f_2 \subset X$. For the sake of ...

Question 11

Let be $R$ a ring, $\varphi:M_{1}\to M_{2}$ a isomorphism beetwen $R$-modules, $\xi_{1}:P_{1}\to M_{1}$ a epimorphism with $P_{1}$ projective and $\xi_{2}:P_{2}\to M_{2}$ a projective cover. Show that ...

Question 12

I realize there's different conventions for the use of arrows in a graph for Category Theory. Does anyone have a "semi"-comprehensive list of all the arrows?

Question 13

$\newcommand{\C}{𝓒}\newcommand{\D}{𝓓}\newcommand{\mon}{\hookrightarrow}$ Let $R: \C \to \D$ be a functor preserving monomorphisms. I define a mono $ι: A \mon B$ to be $R$-strong if for every mono $h:...

Question 14

$\require{AMScd}$ Let $i_A : A \to M$ and $i_B : B \to M$ be two split monomorphisms in an abelian category. Consider the following pullback: $$\begin{CD} P @>j_A>> A\\ @VVj_BV @VVi_AV\\ B @&...

Question 15

In a balanced category, epic monics are isomorphisms, and thus split. I am looking for a balanced category and a monic (or an epic) in it that is not a split monic (respectively epic).

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Questions tagged [monomorphisms]

Kernel zero but not monic?

When is $\operatorname{Spec}(B)\to\operatorname{Spec}(A)$ an epimorphism is the category of schemes?

Left and Right invertibility in Rel category

Is there a balanced category with a non-strong epimorphism?

Monic $\iff$ every component is monic?

How to construct the subobject classifier in the product of categories with subobject classifiers? [duplicate]

What do we call a category in which every monomorphism is regular?

Is the category $\mathbf{Rel}$ of relations balanced?

Szekeres proof that in the category of sets a morphism is a monomorphism iff it is injective [duplicate]

When is a joint image a subobject?

Proving there is a Epimorphism that Splits.

What is a comprehensive list of all the different arrows in Category Theory? (Dashed, Double Arrow, etc...)

Is there a name for this condition on monomorphisms?

Pullback of two split monomorphisms in an abelian category

A monic in a balanced category that is not split?

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