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Category with zero morphisms

Given two objects $X,Y$ in a category, can there be more than one zero morphism $X\rightarrow Y$?

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marked as duplicate by Zhen Lin, sdcvvc, William, Thomas, J. M. Sep 24 '12 at 10:30

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In an answer to your previous question, Qiaochu already indicated that there is a unique zero morphism between any two objects. –  Zhen Lin Sep 2 '12 at 5:37
@Zhen Lin, I think what Qiaochu showed was that for a given category, if there is a "collection" of zero morphisms satisfying certain property then that collection is unique. Here I'm talking about "a" zero morphism in the sense that it is both constant morphism and coconstant morphism. –  ashpool Sep 2 '12 at 9:04

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up vote 7 down vote accepted

The definition of a zero morphism as a morphism that is both constant and coconstant is not a very useful one. Zero morphisms make much more sense in a category with a zero object – since in that case they can be defined to be the unique morphism that factors through the zero object. Under that definition, they are unique.

In the absence of a zero object, stupid things can happen. For example, consider the category $\{ \bullet \rightrightarrows \bullet \}$ with exactly two parallel non-identity arrows. It is essentially vacuous that both of the non-identity arrows are constant and coconstant – but they are distinct by hypothesis.

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