# Why are Regular Categories assumed to be finitely complete?

Regular categories may equivalently defined as those with:

• finite limits
• coequalizers of kernel pairs
• pulback stable regular epis

or

• finite limits
• pullback stable regular epi/mono factorization

When carefully proving the equivalence, the only limits required are pullbacks i.e. in a category with pullbacks:

coequalizers of kernel pairs & stable regular epis $\iff$ stable regular epi/mono factorization.

Is there a compelling reason to require all finite limits?

• Possibly the link to the logic of the internal language suggests one might want a terminal object. Feb 12, 2018 at 20:35
• Perhaps one wants equalizers and products (finite ones) and so we might as well require all finite limits, since their existence follows from those ? Feb 12, 2018 at 20:38
• David: could you elaborate? Feb 12, 2018 at 20:47
• @Tyler Mike said it better in his answer. Feb 13, 2018 at 0:44

There are many reasons one might give for why the notion of "regular category" includes a terminal object (and hence all finite products), but I think one fairly compelling one is that, as David said in a comment, one wants the internal logic of a regular category to be regular logic, and one needs all finite products in order to define a type theory and internal logic: a term $x:A, y:B \vdash t:C$ is a morphism $A\times B\to C$, and a term $\cdot \vdash t:C$ is a morphism $1\to C$. (One can make do with a cartesian multicategory instead, but a locally regular category doesn't have an underlying one of those either.)
• Thanks @mike-shulman. I see you had a hand in writing the relevant nlab locally regular category. Is the definition in your answer here equivalent to that given there? Specifically, are there equalizers in a category with stable reg. epi/mono factorization and pullbacks? Side note: pullbacks + reg. epi/mono factorization (not necessarily stable) $\implies$ reg. epi = strong epi = extremal epi. so the extremal epi/mono factorization will be reg. epi/mono already. Feb 12, 2018 at 22:48