The topos corresponding to forcing in computability theory, where generics meet all computable (e.g., ) dense sets, is generally understood within the framework of sheaf topoi over a posetal site with a restricted topology. Here's a structured explanation:

Key Idea

In classical forcing, the topos of sheaves over the forcing poset equipped with the dense topology (where...

Short answer Yes. If you want “generics that meet only the computable dense sets” for a given forcing poset P, take the topos of sheaves on P for the Grothendieck topology generated by those computable dense sets. This is the standard topos-theoretic way to restrict forcing to a chosen family of dense sets.

More precisely

• Start with a (typically countable, computably presented) forcing poset P ...

Toposes and Forcing in Computability Theory

Your question touches on the fascinating intersection of category theory and logic in a computability-theoretic context. I'll try to provide some insights and potential directions.

Basic Framework

In the standard mathematical setting, forcing via topos theory works as follows:

• A forcing notion P (a preorder) corresponds to a site (P with an ap...

Yes – the “effective” versions of the forcing notions that are used in recursion-/computability-theory can be described by exactly the same categorical recipe that one uses for ordinary set–theoretic forcing, the only difference being that one performs the construction

 “topos of sheaves on the forcing poset P”

not over the base topos Sets but over the effective topos Eff.
When the forcing...

One can show very directly that nothing “mystical’’ is hiding here – the computable‐forcing topos is just the sheaf‐topos on your countable forcing poset once you equip it with the Grothendieck coverage whose covering sieves are exactly the c.e. (downward closed) dense subsets. Equivalently it is the subtopos of the ordinary presheaf‐topos Set^(Pᵒᵖ) determined by the Lawvere–Tierney topology j su...

Excellent question. This is a very natural line of inquiry that connects two deep and beautiful subjects. The short answer is yes, there is a canonical topos, and it is Hyland's effective topos, along with its variations.

Let's break down how the connection works, building from the standard picture of forcing to the effective one.

1. The Standard Picture: Forcing and Sheaf Toposes

A...

This is an excellent question that sits at the intersection of computability theory and topos theory. While there isn't a single canonical topos for forcing in computability theory in the literature, I can outline the key ideas and point to relevant work.

The Basic Framework

In classical set-theoretic forcing, we have:

• A poset (P, ≤) of forcing conditions
• The topos is Sh(P), sheaves on P ...

Yes. ‟Effective forcing’’ is obtained from the usual forcing topos simply by changing the Grothendieck topology:
instead of declaring every dense sieve to be covering we declare only the computably-enumerable dense ones to be covering.

Once that is done the rest of the usual construction goes through unchanged and one obtains a perfectly good Grothendieck topos that contains a *generic filt...