# Remarks on punctual local connectedness

## Peter Johnstone

We study the condition, on a connected and locally connected geometric morphism $p : \cal E \to \cal S$, that the canonical natural transformation $p_*\to p_!$ should be (pointwise) epimorphic - a condition which F.W. Lawvere called the Nullstellensatz', but which we prefer to call punctual local connectedness'. We show that this condition implies that $p_!$ preserves finite products, and that, for bounded morphisms between toposes with natural number objects, it is equivalent to being both local and hyperconnected.

Keywords: axiomatic cohesion, locally conected topos

2000 MSC: Primary 18B25, secondary 18A40

Theory and Applications of Categories, Vol. 25, 2011, No. 3, pp 51-63.

Published 2011-02-10.

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