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Components, complements and the reflection formula
Theory and applications of categories, CT2006, Tome 19 (2007), pp. 19-40 Cet article a éte moissonné depuis la source Theory and Applications of Categories website

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We illustrate the formula $ (\downarrow p)x = \Gamma_!(x/p) $, which gives the reflection $\downarrow p$ of a category $p : P \to X$ over $X$ in discrete fibrations. One of its proofs is based on a ``complement operator" which takes a discrete fibration $A$ to the functor $\neg A$, right adjoint to $\Gamma_!(A\times-):Cat/X \to Set$ and valued in discrete opfibrations.Some consequences and applications are presented.

Classification : 18A99
Keywords: categories over a base, discrete fibrations, reflection, components, tensor, complement, strong dinaturality, limits and colimits, atoms, idempotents, graphs and evolutive sets 
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     author = {Claudio Pisani},
     title = {Components, complements and the reflection formula},
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     volume = {19},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/TAC_2007_19_a1/}
}
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Claudio Pisani. Components, complements and the reflection formula. Theory and applications of categories, CT2006, Tome 19 (2007), pp. 19-40. http://geodesic.mathdoc.fr/item/TAC_2007_19_a1/