  (Pre)closure spaces are a generalization of topological spaces covering also
the notion of neighbourhood in discrete structures, widely used to model and
reason about spatial aspects of distributed systems.
  In this paper we introduce an abstract theoretical framework for the
systematic investigation of the logical aspects of closure spaces. To this end,
we introduce the notion of closure (hyper)doctrines, i.e. doctrines endowed
with inflationary operators (and subject to suitable conditions). The
generality and effectiveness of this concept is witnessed by many examples
arising naturally from topological spaces, fuzzy sets, algebraic structures,
coalgebras, and covering at once also known cases such as Kripke frames and
probabilistic frames (i.e., Markov chains). Then, we show how spatial logical
constructs concerning surroundedness and reachability can be interpreted by
endowing hyperdoctrines with a general notion of paths. By leveraging general
categorical constructions, we provide axiomatisations and sound and complete
semantics for various fragments of logics for closure operators.
  Therefore, closure hyperdoctrines are useful both for refining and improving
the theory of existing spatial logics, but especially for the definition of new
spatial logics for new applications.
