adjoint functor theorem
adjoint lifting theorem
small object argument
Freyd-Mitchell embedding theorem
relation between type theory and category theory
sheaf and topos theory
enriched category theory
higher category theory
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A functor F:C→D between regular categories is called regular if it preserves finite limits and the canonical covers: regular epimorphisms.
For 𝒯 a regular theory and 𝒞 𝕋 its syntactic category, regular functors 𝒞 𝕋→ℰ into some topos ℰ are precisely models of the theory in ℰ.
For 𝒞 𝕋 equipped with the structure of the syntactic site (the regular coverage), this is in turn equivalent to geometric morphisms ℰ→Sh(𝒞 𝕋) into the sheaf topos over 𝒞 𝕋 (the classifying topos for the theory).