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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.

Divya
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    All this means is that a composite of two injective homomorphisms is an injective homomorphism. – Angina Seng Oct 24 '19 at 06:41
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    "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? – Eric Wofsey Oct 24 '19 at 06:46
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    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. – Con Oct 24 '19 at 06:51
  • @EricWofsey for every pair of objects (A,B) shouldn't there be a morphism if it is a category? – Divya Oct 24 '19 at 06:53
  • @ThorWittich ok, so that means in my subcategory $\mathcal{S}$ if there is an injective map between two ab gps I will take that morphism otherwise I can take the class to be empty. Say Mor($\mathbb{Z_n}$, $\mathbb{Z}$) is empty in $\mathcal{S}$ – Divya Oct 24 '19 at 06:56
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    In $\mathbf{Set}$ there are no morphisms $A\to\emptyset$ when $A$ is nonempty. – Oscar Cunningham Oct 24 '19 at 07:03
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    A category definitely does not need to have an arrow between any two objects: see [this question](https://math.stackexchange.com/questions/3193852/semigroups-with-no-morphisms-between-them) (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). – Mark Kamsma Oct 24 '19 at 10:47
  • Thanks for all the help everyone. To conclude since there is no guarantee of a morphism between every pair of objects in it. In the question whenever there is no injective map between two ab groups we will just call its morphism class to be empty. So another interesting subcategory will be all ab groups with only isomorphisms. – Divya Oct 24 '19 at 17:36

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