0
$\begingroup$

I am unable to convince myself the what is said in the title is true. I found it https://www.encyclopediaofmath.org/index.php/Monomorphism here. Where they have given the definition of monomorphism.

Let me explain my confusion; Suppose I take $\mathcal{C}$ to the the category of abelian groups . The title states that if I construct a subcategory $\mathcal{S}$ of $\mathcal{C}$ with objects as all abelian groups and morphisms as only injective group homomorphisms then $\mathcal{S}$ is a subcategory.

But subcategory is itself a category that means any two objects in the subcategory must have a morphism between them, but does taking only injective gp homomorphisms serves the purpose? What I mean is any two ablian groups might not have any injective map between them at all. Then how are we considering them as a subcategory?

Any hint or help is much appreciated. Thanks in advance.

$\endgroup$
  • 3
    $\begingroup$ All this means is that a composite of two injective homomorphisms is an injective homomorphism. $\endgroup$ – Angina Seng Oct 24 '19 at 6:41
  • 2
    $\begingroup$ "But subcategory is itself a category that means any two objects in the subcategory must have a morphism between them" Why would it mean that? $\endgroup$ – Eric Wofsey Oct 24 '19 at 6:46
  • 1
    $\begingroup$ The class of morphisms between two different objects is also allowed to be empty. You only demand the existence of at least one endomorphism for every object. $\endgroup$ – TMO Oct 24 '19 at 6:51
  • 2
    $\begingroup$ In $\mathbf{Set}$ there are no morphisms $A\to\emptyset$ when $A$ is nonempty. $\endgroup$ – Oscar Cunningham Oct 24 '19 at 7:03
  • 1
    $\begingroup$ A category definitely does not need to have an arrow between any two objects: see this question (Oscar already gave an example where there is no arrow in one direction, but there is of course still an arrow in the other direction). $\endgroup$ – Mark Kamsma Oct 24 '19 at 10:47

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.