CatDat

Implication Details

Claim: If a category is Malcev and has a regular subobject classifier, then it is regular-subobject-trivial.

Proof: The regular subobject classifier $\Omega$ is an internal poset (cf. Mac Lane & Moerdijk, IV.8). Concretely, since the category is Malcev, it has binary products, so the intersection of regular subobjects is again a regular subobject. Namely, the intersection of the equalizer of $f_1, g_1 : X \rightrightarrows Y_1$ with the equalizer of $f_2, g_2 : X \rightrightarrows Y_2$ is the equalizer of $(f_1, f_2), (g_1, g_2) : X \rightrightarrows Y_1 \times Y_2$. This intersection operation on regular subobjects yields a morphism $\wedge : \Omega \times \Omega \to \Omega$, and the internal relation ${\leq_{\Omega}} \subseteq \Omega \times \Omega$ is defined as 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 regular monomorphism must be an isomorphism. (From here, we can infer that the category is thin – or, if the category has a subobject classifier, that the category is trivial.)