nLab
equilogical space

Context

Topology

Constructivism, Realizability, Computability

Equilogical spaces

Definition

An equilogical space is a Kolmogorov (T 0T_0) topological space TT 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,)(T, {\equiv}) and (U,)(U, {\cong}) is a continuous function f:TUf\colon T \to U such that xyx \equiv y implies f(x)f(y)f(x) \cong f(y), for all points xx and yy in TT. Two such morphisms ff and gg are considered equal if for all points xx in TT, f(x)g(x)f(x) \cong g(x).

Properties

The category EquEqu of equilogical spaces obviously contains the category of T 0T_0 topological spaces as a full subcategory (by using the trivial equivalence relation of equality on points). Moreover, as opposed to the latter, EquEqu is in fact cartesian closed; this can be seen using the equivalence of EquEqu and the category of partial equivalence relations over algebraic lattices.

On the other hand, EquEqu can be identified with a reflective exponential ideal in the ex/lex completion of the category Top 0Top_0 of T 0T_0 topological spaces. This provides an alternative proof of the cartesian closure of EquEqu, since an exponential ideal in a cartesian closed category is cartesian closed, and (Top 0) ex/lex(Top_0)_{ex/lex} is cartesian closed (in fact, locally cartesian closed) since Top 0Top_0 is weakly locally cartesian closed.

Moreover, in this way we can see that the embedding Top 0EquTop_0 \to Equ preserves all existing exponentials, since the embedding CC ex/lexC \to C_{ex/lex} does so, and EquEqu is closed under exponentials in (Top 0) ex/lex(Top_0)_{ex/lex} and contains the image of Top 0Top_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.

  • Dana Scott, A New Category? Domains, Spaces, and Equivalence Relations,

It is then discussed in more detail in

Revised on March 4, 2014 06:16:08 by Urs Schreiber (89.204.137.33)