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Constructions on regular and exact categories
The Giraud Theorems Characterizing toposes
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addition algebras apply assume axiom called clear cocomplete coequalizer colim colimits commutes complete composition condition congruence consider construction containing Conversely Corollary covering defined definition denote diagram direct system directed directed colimits easily embedding equalizer equivalence relation evidently exact exact category exact sequence examples exists fact factorization finite limits follows functor given gives hence holds implies induced intersections inverse isomorphism ker f kernel pair least lemma means monomorphism morphism natural Note object obtain particular presentable preserves projections Proof properties Proposition prove pullback reflects regular category regular decompositions regular epimorphism relation result satisfies sheaf sheaves similar split square subobjects sufficient supp suppose terminal theorem theory tion topology unions unique universal variety yields