A significant problem of the dominant web search model is the lack of a realistic way to acquire user search context. Search engines use implicit feedback, which is extremely sparse and does not allow users to properly define what they want to know, or what they think of search results. In our proposed "web exploration engine", which we implemented as a prototype, documents have been automatically pre-classified into a large number of categories representing a hierarchy of search contexts. Users can browse this structure or search within a particular category (context) by explicitly selecting it. Keyword relevance is not global but specific to a category. The main innovation we propose is the "floating" query resulting from this feature: the original search query is re-evaluated and the importance of its features re-calculated for every context the user explores. This allows users to search or browse in a truly local (context-dependent) way with a minimum of effort on their part.