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.
Keywords: categories over a base, discrete fibrations, reflection, components, tensor, complement, strong dinaturality, limits and colimits, atoms, idempotents, graphs and evolutive sets
2000 MSC: 18A99
Theory and Applications of Categories,
Vol. 19, 2007,
No. 2, pp 19-40.
http://www.tac.mta.ca/tac/volumes/19/2/19-02.dvi
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