# A note on discrete Conduché fibrations

## Peter Johnstone

The class of functors known as discrete Conduché fibrations forms a common generalization of discrete fibrations and discrete opfibrations, and shares many of the formal properties of these two classes. F. Lamarche conjectured that, for any small category $\cal B$, the category ${\bf DCF}/{\cal B}$ of discrete Conduché fibrations over $\cal B$ should be a topos. In this note we show that, although for suitable categories $\cal B$ the discrete Conduché fibrations over $\cal B$ may be presented as the `sheaves' for a family of coverings on a category ${\cal B}_{tw}$ constructed from $\cal B$, they are in general very far from forming a topos.

Keywords:

1991 MSC: Primary 18A22, Secondary 18B25.

Theory and Applications of Categories, Vol. 5, 1999, No. 1, pp 1-11.

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