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If we look at it as a full subcategory SPt(C) of Pt(C), it gives rise to a subfibration S-cod of the fibration of points.
The full subcategory SC of S-special objects is protomodular.
The full subcategory of C determined by the abelian objects is denoted Ab(C) and called the additive core of C.
0] is the full subcategory of connected manifolds, and the latter is essentially small), or by the category of affine schemes of finite type over an algebraically closed field K (the dual of the category of finitely-presented K-algebras).
This means that we can identify the 'petit topos' associated with an object A as a full subcategory of [epsilon]/A, whose objects we may think of as the 'fibrewise discrete' morphisms with codomain A.
e) By (a), we may regard D as a subcategory of [epsilon]; moreover, for any object A the slice category D/A is a full subcategory of [epsilon]/A.
As a full subcategory of [epsilon]/A, D/A is closed under arbitrary coproducts, by (d) and (e).