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On the Grothendieck topologies in the toposes of presheaves. (English) Zbl 0554.18003

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Notation from P. T. Johnstone’s book ”Topos Theory” (1977; Zbl 0368.18001) is heavily used throughout. \({\mathcal C}\) is a category in a topos \({\mathcal E}\). A category \({\mathcal C}\) is pseudoantisymmetric if \[ \begin{tikzcd} \operatorname{Aut}(\mathcal C)\ar[r]\ar[d] & \mathcal C \ar[d,"{(d_0,d_1)}"]\\ \mathcal C\ar[r,"{(d_0,d_1)}" '] & C_0\times C_0\end{tikzcd} \] is a pullback diagram. The authors shows via the notion of Karoubian-closed subobjects that if \(\mathcal C\) is pseudoantisymmetric then \((\mathrm{Sub} C_0)^{op}\) is a reflective sublattice in \(\mathrm{GT}(\mathcal E^{\mathcal C^{op}})\) via some functor \(tops\). Furthermore \(\mathcal C\) is fairly ordered if for any \(D_ 0\hookrightarrow C_ 0\) there is some \(\sigma\) : \(W\hookrightarrow D_ 0\) such that \[ \mathcal E\vDash \quad ((d_ 0f\in D_ 0 \wedge d_ 1f\in W)\quad \Rightarrow \quad d_ 0f=d_ 1f) \] and \(\pi_ 2: W\times D_ 0\to D_ 0\) is epi.
It is then shown that \(tops\) has an inverse \(spot\) if and only if \(\mathcal C\) is pseudoantisymmetric, fairly ordered and \(\mathcal E/C_0\) is Boolean. Some applications to “real world” problems are hinted at.
Reviewer: P.Cherenack

MSC:

18B25 Topoi
18F10 Grothendieck topologies and Grothendieck topoi
18F20 Presheaves and sheaves, stacks, descent conditions (category-theoretic aspects)
18A25 Functor categories, comma categories

Citations:

Zbl 0368.18001
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