# If there are initial and terminal objects, does the category have zero morphisms?

I think the answer is yes because of the following immediate (and easy) facts:

1. The morphisms associated with initial (terminal) objects are right (left) zeroes.
2. If $$r$$ is a right zero and $$l$$ a left zero, then $$lr$$, if defined, is a zero morphism.
3. If $$z$$ is a zero morphism, then $$gzf$$, whenever defined, is also a zero morphism.

Now, if we have an initial object $$I$$ and a terminal object $$T$$, we can define $$0_{X, Y} := X\to I\to T\to I\to Y$$ which thereby form a compatible family of zero morphisms.

Is my reasoning correct?

In your definition of $$0_{X,Y}$$, you compose maps $$X\to I$$ and $$T\to I$$. Why should these maps exist?

For example, in the category of sets, there is no function from the terminal object $$1$$ (a one-point set) to the initial object $$\varnothing$$ (the empty set), so this category does not have zero morphisms: there is no possible choice for the zero morphism $$0_{1,\varnothing}$$.

In fact, if $$C$$ is a category with an initial object $$0$$ and a terminal object $$1$$, then $$C$$ has zero morphisms if and only if $$1\cong 0$$.

In one direction, if $$1\cong 0$$, then there is an arrow $$z_{1,0}\colon 1\to 0$$, and then for any objects $$X$$ and $$Y$$, we can define $$z_{X,Y} = {{!}\circ z_{1,0}\circ {!}}\colon X\to 1\to 0\to Y$$ and check that this gives a compatible family of zero morphisms.

Conversely, if $$C$$ has zero morphisms, then in particular there is a (zero) morphism $$1\to 0$$, but any arrow from a terminal object to an initial object is an isomorphism (e.g., see here).

• Oof, how embarrassing! My bad.
– Atom
Dec 10, 2023 at 5:21