CatDat

Implication Details

Assumptions: Malcev, subobject classifier

Conclusions: subobject-trivial

Proof: The subobject classifier $\Omega$ is an internal poset (cf. Mac Lane & Moerdijk, IV.8). Concretely, the intersection of subobjects yields a morphism $\wedge : \Omega \times \Omega \to \Omega$, and the internal relation ${\leq_{\Omega}} \subseteq \Omega \times \Omega$ is the equalizer of $\wedge, p_1 : \Omega \times \Omega \rightrightarrows \Omega$. The relation ${\leq_{\Omega}}$ is reflexive, hence symmetric by assumption. Since it also antisymmetric and has a largest element $\top$, every monomorphism must be an isomorphism. (From here, we can infer that the category is trivial.)