A Chu space, named for mathematician Po-Hsaing Chu who developed them as a student, can be thought of as a table specifying which of a number of "objects" have which of a number of "attributes". Hence the following table of objects (rows) and attributes (columns) specifies a (fairly trivial) Chu space:

Chu spaces were developed to provide examples of a rather arcane construction in category theory (see below); their history can be found here. The wikipedia page for Chu spaces is very informative.

Formally, a (finite) Chu space is a triple [{O_i}, {A_j}, ⊩], where {O_i} is a (finite) set of objects, {A_j} is a (finite) set of attributes, and ⊩ is a satisfaction relation ⊩ ⊆ {O_i} X {A_j}, with X the Cartesian product. Only finite Chu spaces will be considered here. Chu spaces as described here can be generalized by treating ⊩ not as a binary yes/no decision as the term "satisfaction relation" suggests, but as a map {O_i} X {A_j} → S to some arbitrary set S of "values" of ⊩; however, this generalization will not be pursued here. A wealth of additional material on Chu spaces can be found here.

Finite Chu spaces provide a natural model of observations and their outcomes. As pointed out by E. F. Moore (Gedankenexperiments on sequential machines. In Autonoma Studies; Shannon, C.W., McCarthy, J., Eds.; Princeton University Press: Princeton, NJ, USA, 1956; pp. 129–155), any sequence of observations of a system by an experimenter can be regarded as a bidirectional exchange of symbols and hence represented by a table relating input symbols to output symbols. Any such table specifies a Chu space. As shown here (pdf), any physical interaction can be regarded as an exchange of information at finite resolution; hence any physical interaction can be represented by a Chu space.

Chu spaces are closely related to "Classifications" as defined by Barwise and colleagues.