Homotopy Theory of Computable Spaces
Alyssa Renata
Abstract:
In this thesis, we develop the homotopy theory of equilogical spaces and QCB (Quotients of Countably-Based) spaces. An equilogical space is a T0 countably-based space equipped with an equivalence relation, while a QCB space is the quotient of some equilogical space by its relation. We show that from any QCB space X we can reconstruct an equilogical space inducing X under quotienting, thus exhibiting the QCB spaces as a reflective subcategory of the equilogical spaces.
We construct a Quillen model structure for QCB spaces in which the weak equivalences are homotopy equivalences. We then seek a corresponding homotopy theory for equilogical spaces, but the notion of homotopy for equilogical spaces induced by the unit interval [0, 1] is not transitive. Hence, we instead study a notion of homotopy corresponding to taking the transitive closure of [0, 1]-paths.
To accomodate this study, we prove that the category of equilogical spaces can be viewed as a homotopy category induced by a computational notion of homotopy. In fact, since the category of equilogical spaces embeds into a realizability topos, this result is a special case of an existing result that realizability toposes are homotopy categories [Ber20]. From this point of view, we sketch a proof for obtaining a path object corresponding to the aforementioned transitive-closure homotopy.