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Precategory objects of toposes. (English) Zbl 0569.18006

By a precategory object of toposes is meant a diagram D of elementary toposes \(E_ 2\), \(E_ 1\) and \(E_ 0\), where \(E_ 0\) is a subtopos of \(E_ 1\), and geometric morphisms \(\pi_ 0,\pi_ 1,\gamma: E_ 2\to E_ 1\) and \(\partial_ 0,\partial_ 1: E_ 1\to E_ 0\), where \(\partial_ 0\circ id\cong 1_{E_ 0}\cong \partial_ 1\circ id\), \(\partial_ 0\circ \pi_ 1\cong \partial_ 1\circ \pi_ 0\), \(\partial_ 0\circ \gamma \cong \partial_ 0\circ \pi_ 0\) and \(\partial_ 1\circ \gamma \cong \partial_ 1\circ \pi_ 1.\)
The notion is readily relativized over a base topos B. The category \(D_ T\) is constructed, with objects being pairs (S,\(\xi)\), where S is an \(E_ 0\)-object and \(\xi\) : \(\partial^*_ 0S\to \partial^*_ 1S\) in \(E_ 1\) such that \(id^*\xi =1_ S\) and \(\pi^*_ 1\xi \circ \pi^*_ 0\xi =\gamma^*\xi\), and morphisms being \(E_ 0\)-morphisms \(\alpha\) : \(S\to S'\) such that \(\xi '\circ \partial^*_ 0\alpha =\partial^*_ 1\alpha \circ \xi\). It is shown that, with a base topos B with a natural numbers object and axiom of choice, \(D_ T\)- called the category of internal functors on D - is a bounded B-topos, provided \(E_ 0\) is bounded over B.
Reviewer: H.Engenes

MSC:

18B25 Topoi
18F20 Presheaves and sheaves, stacks, descent conditions (category-theoretic aspects)
03G30 Categorical logic, topoi
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