Mathematics Stack Exchange
Mathematics Stack Exchange

Questions tagged [monomorphisms]

For questions related to monomorphisms, which are categorical generalizations of injective functions.

Definition

A morphism $\varphi\colon X \to Y$ is a monomorphism (monic morphism) if for any object $W$ and morphisms $f,g \colon W \to X$, if $\varphi f=\varphi g$ then $f=g$. This is a categorical generalization of the idea of an injective function of sets.

diagram corresponding to a monic morphism

Helpful Resources

133 questions
0
votes
0 answers

Mapping of a finite set onto itself

I'm studying Herstein's Topics in Algebra and stumbled upon the following problem (Chapter 0, section 2, problem 8): If the set S has a finite number of elements, prove the following: (a) If $\sigma$ maps S onto S, then $\sigma$ is one-to-one. (b)…
Alejandro Aguilar
  • 13
  • 4
0
votes
0 answers

Define the image of a composition of maps

I just asked myself the following question and couldn't see if or where there is an issue. Let $f\colon A\to B$ an arrow and let $i_{1}\colon A_{1}\rightarrowtail A$ and $i_{2}\colon A_{2}\rightarrowtail A$ two subobjects of $A$. Assume there is a…
Dominique Fosse
  • 1
0
votes
3 answers

How do kernels prove how a function fails to be injective

I know that for $f\colon X \to Y$, where $e_Y$ is the identity of $Y$: $$ \ker(f) = \left\{x \in X \, \middle| \, f(x) = e_Y \right\} $$ I've learnt that kernels imply how much a homomorphism fails to be injective. But why is a function…
hgiesel
  • 1,257