zero object and zero morphism as identity

1. Proposition

Let \(\mathcal{C}\) be a pointed category (enriched over \(\mathrm{Set}_*\)) and \(A \in \mathrm{Ob}(\mathcal{C})\).

TFAE:

  1. \(\mathrm{id}_{A} = 0\)
  2. \(A\) is the zero object

2. Proof

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

Let \(B\) be an object, then for a morphism

\begin{align*} f: A \rightarrow B \end{align*}

it holds, that

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

hence \(\mathrm{Hom}_{\mathcal{C}}(A,B) = 1\) (since by assumption \(\mathrm{Hom}_{\mathcal{C}}(A,B) \neq 0\))

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

definition

Date: nil

Author: Anton Zakrewski

Created: 2024-10-14 Mo 09:06