nLab Scott adjunction

Draft – page under development. Help in completing it is welcome.

Motivations and idea

The original motivations and applications for the Scott adjunction come from model theory [Henry 2019], but a lot of its scientific placing and naming choices have a geometric flavour.

Geometric intuition

Rudiments on the Scott topology

The set of points of a locale has both a canonical topology and canonical partial order. The first of these structures gives raise to the sometimes called Isbell adjunction [cite towards higher topology], which related locales to topological spaces.

Theorem

Assigning to a topological space its locale of opens provides a left adjoint to taking the space of points.

$O: Top \leftrightarrows Loc: pt$.

The duality between sober spaces and locales with enough points is a very celebrated result in the theory of dualities, and many other dualities of topological kind happen as sub dualities of this one.

For the second structure, let us recall a couple of properties of such partial order. Let $O$ be a locale. Then its poset of points has directed joins, and every morphism of locales induces a directed join preserving morphism of posets. Thus we obtain a (2-) functor from the (2-)category of locales to that of posets with directed joins.

It is natural to wonder wether this process can be reversed, i.e. whether we can associate a locale to a directed poset with joins, hoping for assigning a topology to each reasonable poset.

In the 1980s Scott defined the Scott topology of a good enough poset, which assigns to a poset with directed joins a topological space. Yet, an analysis of this construction yields precisely what we are looking for. For $P$ a locale, and $T$ the booleans, the poset $Pos(P,T)$ is precisely the opens of the Scott topology over $P$, and provides a locale associated to our posets.

Theorem (See the intro of Di Liberti 2022)

Assigning to a poset with directed colimits its Scott locale provides a left adjoint to taking the poset of points.

$S: Pos_{\omega} \leftrightarrows Loc: pt$.

Higher Topology

The full fledged Scott adjunction offers a categorification of the result above, generalising posets with directed colimits to accessible categories with directed colimits, and locales to topoi. The general idea (and even its proof structures) carries by replacing the poset of truth values with the category of sets. Of course some additional care must be taken to handle correctly size issues and other technical aspects.

In particular, one defines the Scott topos $S(A)$ of an accessible category with directed colimits $A$ to be $Acc_{\omega}(A, Set)$, where the latter is the category of functors preserving directed colimits into $Set$.

Theorem (Di Liberti and Henry)

Assigning to an accessible category with directed colimits its Scott topos provides a left adjoint to taking the category of points of a topos.

$S: Acc_{\omega} \leftrightarrows Topoi: pt$.

It is also possible to categorify the adjunction between topological spaces and locales via the theory of ionads, which was generalised by to deliver precisely this result.

Theorem (Di Liberti)

Assigning to an accessible category with directed colimits its Scott topos provides a left adjoint to taking the category of points of a topos.

$O: BIon \leftrightarrows Topoi: pt$.

It would be nice to drop here the diagram of interaction between S and O in Towards Higher Topology, but I do not know how to do that.

Logical aspects

To be written. Free geometric theory associated to an accessible category with directed colimits. Recap of the intro of Formal Model Theory & Higher Topology.

Mention formal model theory and Create separate entries for it.

Applications

To be written. Focus on Henry 2019.

References

• Simon Henry: An abstract elementary class non-axiomatizable in $L_{\infty, \kappa}$, The Journal of Symbolic Logic 84 3 (2019) 1240-1251 [arXiv:1812.00652, doi:10.1017/jsl.2019.25]

• Ivan Di Liberti: The Scott adjunction [arXiv:2009.07320]

• Ivan Di Liberti: General facts on the Scott Adjunction, Applied Categorical Structures 30 (2022) 569–591 [arXiv:2009.14023, doi:10.1007/s10485-021-09666-6]

• Ivan Di Liberti: Towards higher topology, Journal of Pure and Applied Algebra 226 3 (2022) 106838 [arXiv:2009.14145, doi:10.1016/j.jpaa.2021.106838]

• Ivan Di Liberti: Formal model theory and higher topology, Mathematical Logic Quarterly 70 1 (2024) [arXiv:2010.00319, doi:10.1002/malq.202300006]