terminal, initial and zero object in an Ab-enriched category

1. Proposition

Let \(\mathcal{C}\) be a Ab-enriched category category and \(A \in \mathrm{Ob}(\mathcal{C})\).

\(A\) is an initial object
\(A\) is a terminal object
\(A\) is a zero object

2. Proof

2.1. 1), 2) \(\implies\) 3)

By assumption of being initial/terminal and construction of a category, there exists only one morphism \(\mathrm{id}_{A} = 0\). Hence \(\mathrm{Hom}_{\mathcal{C}}(A,A)\) is the trivial abelian group.

Suppose \(B \in \mathrm{Ob}(\mathcal{C})\), then

2.1.1. 1) \(\implies\) 3)

for \(f \in \mathrm{Hom}_{\mathcal{C}}(A,B)\) and the tensor product

\begin{align*} C: \mathrm{Hom}_{\mathcal{C}}(A,A) \times \mathrm{Hom}_{\mathcal{C}}(A,B) \rightarrow \mathrm{Hom}_{\mathcal{C}}(A,B) \end{align*}

we get

\begin{align*} f =& \\ =& f \circ \mathrm{id}_{A} =& C(0,f) \\ =& 0 \\ \end{align*}

Therefore the morphism is unique and by construction of the trivial map also exists

2.1.2. 2) \(\implies\) 3)

for \(f \in \mathrm{Hom}_{\mathcal{C}}(B,A)\) and the tensor product

\begin{align*} C: \mathrm{Hom}_{\mathcal{C}}(B,A) \times \mathrm{Hom}_{\mathcal{C}}(B,B) \rightarrow \mathrm{Hom}_{\mathcal{C}}(B,A) \end{align*}

we get

\begin{align*} f =& \mathrm{id}_{A} \circ f \\ =& C(f,0) \\ =& 0 \end{align*}

Therefore the morphism is unique and by construction of the trivial map also exists

2.2. 3) \(\implies\) 1), 2)

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