# Components, complements and the reflection formula

## Claudio Pisani

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.

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