nLab
strict ind-object

Contents

Context

Category theory

Limits and colimits

Contents

Definition

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

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

Dually, strict pro-objects (alias essentially epimorphic pro-objects) are limits of small cofiltered diagrams involving only epimorphisms.

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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