nLab
strict ind-object

Contents

Definition

If CC is a category than an ind-object xInd(C)x\in Ind(C) is a strict ind-object if it can be represented in Ind(C)Ind(C) as the (vertex of) a colimit of a small filtered diagram whose objects are in CC and morphisms are monomorphisms in CC.

(Grothendieck-Verdier 71, §8.12.1). See also Blanc 96, def. 4.1.

Dually, strict pro-objects are limits of small cofiltered diagrams involving only epimorphisms.

An ind-object isomorphic in Ind(C)Ind(C) to a strict ind-object is sometimes called essentially monomorphic. A pro-object isomorphic in Pro(C)Pro(C) to a strict pro-object is sometimes called essentially epimorphic. (This is not so good terminology unless we call strict pro-objects epimorphic which does not seem to be used.)

References

  • Alexandre Grothendieck, Jean-Louis Verdier, Prefaisceaux, in Theorie de Topos et Cohomologie Etale de Schemas 1971

  • David Blanc, Colimits for the pro-category of towers of simplicial sets, Cahiers de Topologie et Géométrie Différentielle Catégoriques (1996) Volume: 37, Issue: 4, page 258-278 (numdam)

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