nLab
equilogical space

Definition

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 $f:T\to U$ 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 $T$, $f(x)\cong g(x)$.

Properties

The category $\mathrm{Equ}$ 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, $\mathrm{Equ}$ is in fact cartesian closed; this can be seen using the equivalence of $\mathrm{Equ}$ and the category of partial equivalence relations over algebraic lattices.

On the other hand, $\mathrm{Equ}$ can be identified with a reflective exponential ideal in the ex/lex completion of the category ${\mathrm{Top}}_0$ of $T_0$ topological spaces. This provides an alternative proof of the cartesian closure of $\mathrm{Equ}$, since an exponential ideal in a cartesian closed category is cartesian closed, and $({\mathrm{Top}}_0{)}_{\mathrm{ex}/\mathrm{lex}}$ is cartesian closed (in fact, locally cartesian closed) since ${\mathrm{Top}}_0$ is weakly locally cartesian closed.

Moreover, in this way we can see that the embedding ${\mathrm{Top}}_0\to \mathrm{Equ}$ preserves all existing exponentials, since the embedding $C\to C_{\mathrm{ex}/\mathrm{lex}}$ does so, and $\mathrm{Equ}$ is closed under exponentials in $({\mathrm{Top}}_0{)}_{\mathrm{ex}/\mathrm{lex}}$ and contains the image of ${\mathrm{Top}}_0$. This embedding also preserves all limits, but it does not in general preserve colimits.

References

The concept was originally introduced for domain theory in a privately circulated manuscript by Dana Scott.

• “A New Category? Domains, Spaces, and Equivalence Relations”, Dana S. Scott

An early article on equilogical spaces:

• “Equilogical Spaces”, Andrej Bauer, Lars Birkedal, Dana S. Scott (ps)