nLab regular functor

Definition

A functor $F : C \to D$ between regular categories is called regular if it preserves finite limits and the canonical covers: regular epimorphisms.

Properties

• For $\mathcal{T}$ a regular theory and $\mathcal{C}_{\mathbb{T}}$ its syntactic category, regular functors $\mathcal{C}_{\mathbb{T}} \to \mathcal{E}$ into some topos $\mathcal{E}$ are precisely models of the theory in $\mathcal{E}$.

  For $\mathcal{C}_{\mathbb{T}}$ equipped with the structure of the syntactic site (the regular coverage), this is in turn equivalent to geometric morphisms $\mathcal{E} \to Sh(\mathcal{C}_{\mathbb{T}})$ into the sheaf topos over $\mathcal{C}_{\mathbb{T}}$ (the classifying topos for the theory).

Related concepts

• cartesian functor

• regular functor

• coherent functor

• geometric functor