An equilogical space is a Kolmogorov () topological space along with an arbitrary equivalence relation on its points (of note, the equivalence relation need not match the topological structure in any way). A morphism between equilogical spaces and is a continuous function such that implies , for all points and in . Two such morphisms and are considered equal if for all points in , .
The category of equilogical spaces obviously contains the category of topological spaces as a full subcategory (by using the trivial equivalence relation of equality on points). Moreover, as opposed to the latter, is in fact cartesian closed; this can be seen using the equivalence of and the category of partial equivalence relations over algebraic lattices.
On the other hand, can be identified with a reflective exponential ideal in the ex/lex completion of the category of topological spaces. This provides an alternative proof of the cartesian closure of , since an exponential ideal in a cartesian closed category is cartesian closed, and is cartesian closed (in fact, locally cartesian closed) since is weakly locally cartesian closed.
Moreover, in this way we can see that the embedding preserves all existing exponentials, since the embedding does so, and is closed under exponentials in and contains the image of . This embedding also preserves all limits, but it does not in general preserve colimits.
It is then discussed in more detail in