# diagonal and codiagonal morphism in additive category

Let $$\mathcal{C}$$ be an additive category. I define diagonal and codiagonal morphisms as follows:

Diagonal: Let $$A \in Ob(\mathcal{C})$$. Then the diagonal $$\Delta_A: A \longrightarrow A \oplus A$$ is the unique morphism with $$p \circ \Delta_A = id_A$$ and $$q \circ \Delta_A = id_A$$ where p,q are projections from $$A \oplus A$$ to $$A$$.

Codiagonal: Let $$B \in Ob(\mathcal{C})$$. Then the codiagonal $$\nabla_B: B \oplus B \longrightarrow B$$ is the unique morphism with $$\nabla_B \circ i= id_B$$ and $$\nabla_B \circ j = id_B$$ where i,j are injections from $$B$$ to $$B \oplus B$$.

I would like to know how can I justify the existence of these objects and morphisms. Any help?

The objects I must verify their existence are: $$A$$, $$A \oplus A$$. The morphisms I must verify: $$id_A$$, projections and injections.

• $A$ is the input from which all the rest is to be produced. By definition of category, product, and coproduct, you have the identity morphism, the projections, and the injections that you want. What remains to check is whether the product and coproduct coincide to give you the $\oplus$ that you want. They coincide in abelian categories but is mere additivity enough? – Andreas Blass Dec 4 '18 at 17:46
• Additivity suffices. In fact, in every category enriched over the category of commutative monoids, finite products and coproducts agree if they exist, i.e. they yield biproducts. – Gnampfissimo Dec 4 '18 at 18:44

## 1 Answer

All of this follows directly from the definition of an additive category, together with the definition of all of the terms in that definition. The most complicated part of that definition is understanding in detail what it means for a category to admit biproducts; for that see this blog post.