Left exact presheaves on a small pretopos

Borceux, Francis;Pedicchio, MC.
(1999) Journal of Pure and Applied Algebra — Vol. 135, n° 1, p. 9-22 (1999)

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Authors
  • Borceux, FrancisUCLouvain
    Author
  • Pedicchio, MC.
    Author
Abstract
Given a small exact category E with finite colimits, we prove that the category Lex(E) of left exact presheaves on E is exact precisely when in E, the equivalence relation generated by a reflexive symmetric relation R is a finite iterate of R. This is in particular the case when E is Noetherian, that is, every ascending chain of subobjects is stationary. When this condition is satisfied and moreover E is a pretopos, Lex(E) becomes a topos. Various examples are given, distinguishing the possible situations. (C) 1999 Elsevier Science B.V. All rights reserved.
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Borceux, F., & Pedicchio, MC. (1999). Left exact presheaves on a small pretopos. Journal of Pure and Applied Algebra, 135(1), 9-22. https://doi.org/10.1016/S0022-4049(97)00149-7 (Original work published 1999)