Reduction replaces a mathematical object with a simpler model that retains the relevant information. We introduce left and right vector fields on small categories as tools for reducing finite acyclic categories while preserving their directed homotopical information. We relate these fields to directed deformation retracts and beat-object reductions, and show that right vector-field reductions preserve the directed sectional category of right directed fibrations and the global sections of functorial databases. We also extend initial scaffolds from posets to acyclic categories. These provide smaller indexing categories that preserve limits and, in particular, globally coherent selections in databases. Finally, we prove that initial scaffolds are preserved by right directed deformation retracts and hence by right vector-field reductions.

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