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On the characterization of monadic categories over \(\mathcal {S}et\). (English) Zbl 0824.18004

Presheaf categories were characterized by the reviewer [Relative functor categories and categories of algebras, J. Algebra 11, 64-101 (1969; Zbl 0165.329)] as monadic and comonadic over slices of \({\mathcal S}et\). A characterization involving exactness conditions and a small generating family of tiny objects was then given by her [Internal presheaves toposes, Cah. Topologie Géom. Différ. 18, 291-330 (1977; Zbl 0365.18017)] for internal presheaves toposes over a base topos \(S\). In the paper being reviewed, monadicity over \({\mathcal S}et\) is recast in terms of exactness conditions which is seen to extend to the characterization of presheaf categories given by the reviewer (loc. cit.). The new ingredient is to view \([{\mathcal C}^{\text{op}},{\mathcal S}et]\) as the exact completion of the coproduct completion \(\text{Fam }{\mathcal C}\) of the small category \(\mathcal C\).
Reviewer: M.Bunge (Genova)

MSC:

18C15 Monads (= standard construction, triple or triad), algebras for monads, homology and derived functors for monads
18B25 Topoi
18F20 Presheaves and sheaves, stacks, descent conditions (category-theoretic aspects)
18E10 Abelian categories, Grothendieck categories
18A35 Categories admitting limits (complete categories), functors preserving limits, completions
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References:

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