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Connected limits, familial representability and Artin glueing. (English) Zbl 0849.18002
Let \(T : {\mathcal E} \to {\mathcal F}\) be a functor. The category obtained by Artin glueing along \(T\) is the comma category \(({\mathcal F}, T) = ({\mathcal F} \downarrow T)\). The paper develops a general technique for associating properties of \({\mathcal F} \downarrow T\) with properties of \({\mathcal E}, {\mathcal F}\) and \(T\). It establishes how various properties of \({\mathcal F} \downarrow T\) correspond exactly to the same properties of \({\mathcal E}\) and \({\mathcal F}\) together with some property of \(T\). This is done for the following properties: cartesian closed, locally cartesian closed, quasitopos, topos, presheaf topos. For example, if \({\mathcal E}\) and \({\mathcal F}\) are presheaf toposes, then \({\mathcal F} \downarrow T\) is a presheaf topos if and only if \(T\) preserves connected limits. If moreover \({\mathcal F} = {\mathcal S}et\), then this is equivalent to: \(T\) is familially representable. There is also a syntactic characterization of those monads on \({\mathcal S}et\) whose functor part is familially representable.

MSC:
18A25 Functor categories, comma categories
18B25 Topoi
18A40 Adjoint functors (universal constructions, reflective subcategories, Kan extensions, etc.)
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