CW-complex, Hausdorff space, second-countable space, sober space
connected space, locally connected space, contractible space, locally contractible space
constructive mathematics, realizability, computability
propositions as types, proofs as programs, computational trinitarianism
An equilogical space is a Kolmogorov ($T_0$) topological space $T$ along with an arbitrary equivalence relation $\equiv$ on its points (of note, the equivalence relation need not match the topological structure in any way). A morphism between equilogical spaces $(T, {\equiv})$ and $(U, {\cong})$ is a continuous function $f\colon T \to U$ such that $x \equiv y$ implies $f(x) \cong f(y)$, for all points $x$ and $y$ in $T$. Two such morphisms $f$ and $g$ are considered equal if for all points $x$ in $T$, $f(x) \cong g(x)$.
The category $Equ$ of equilogical spaces obviously contains the category of $T_0$ topological spaces as a full subcategory (by using the trivial equivalence relation of equality on points). Moreover, as opposed to the latter, $Equ$ is in fact cartesian closed; this can be seen using the equivalence of $Equ$ and the category of partial equivalence relations over algebraic lattices.
On the other hand, $Equ$ can be identified with a reflective exponential ideal in the ex/lex completion of the category $Top_0$ of $T_0$ topological spaces. This provides an alternative proof of the cartesian closure of $Equ$, since an exponential ideal in a cartesian closed category is cartesian closed, and $(Top_0)_{ex/lex}$ is cartesian closed (in fact, locally cartesian closed) since $Top_0$ is weakly locally cartesian closed.
Moreover, in this way we can see that the embedding $Top_0 \to Equ$ preserves all existing exponentials, since the embedding $C \to C_{ex/lex}$ does so, and $Equ$ is closed under exponentials in $(Top_0)_{ex/lex}$ and contains the image of $Top_0$. This embedding also preserves all limits, but it does not in general preserve colimits.
The concept was originally introduced for domain theory in a privately circulated manuscript by Dana Scott.
It is then discussed in more detail in
Andrej Bauer, section 4 of The Realizability Approach to Computable Analysis and Topology, PhD thesis CMU (2000) (pdf)
Andrej Bauer, Lars Birkedal, Dana Scott, Equilogical Spaces, 2001 (ps, pdf)