Posites

Idea

The notion of posite (or $(0,1)$-site, less properly $0$-site) is a decategorification of the notion of site (which may also be called a $1$-site or $(1,1)$-site). For categorifications thereof, see (∞,1)-site and 2-site.

Just as a site is a category with a coverage, whose (set-valued) sheaves form a Grothendieck topos, so a posite is a poset with a coverage, whose $(0,1)$-sheaves (truth value-valued sheaves) form a locale. On the other hand, we can also consider a posite as a site in its own right, in which case the $1$-sheaves on it form a localic topos.

As locale theory serves as an approach to topology in which locales take the role traditionally held by topological spaces, so posites take the role traditionally held by topological bases.

A formal topology may be generated by a posite equipped with a positivity predicate.

Definitions

In other words, a posite is a poset (or more generally a preordered set) equipped with a Grothendieck topology or a coverage. (The name is a portmanteau of ‘poset’ and ‘site’.)

The definition of coverage may be simplified a little in this case.

If $S$ is a meet-semilattice, then there is another alternative defintion. (Actually, it is not necessary that the poset $S$ have all finitary meets but only bounded? ones. That is, if $v, w \leq u$ for some fixed $u$, then $v \wedge w$ must exist, but not otherwise. Such a poset $S$ is a semilattice if and only if it has a top element. Compare the notion of locally cartesian category.)

It is a theorem that every posite whose underlying poset has all bounded binary meets is equivalent (in the sense that there is an isomorphism of sites as defined below) to a locally cartesian posite.

In any case, given an element $u$ of $S$ and a subset $V$ of $S$, if

then we say that $V$ is a basic cover of $u$.

Clauses (1–3) simply state that $f$ is a flat functor; when $S$ and $T$ are semilattices, we can replace this with the requirement that $f$ preserves meets (including the top element). Clause (4) requires $f$ to respect covers.

Ideals and sheaves

One often looks at sheaves on sites. On posites, one can either look at sheaves or at ideals.

In other words, it is a sheaf valued in truth values (rather than in sets as is the default).

A subset $I$ that satisfies condition (1) alone states that $I$ is a lower subset (a (0,1)-presheaf).

The ideals on a posite form a frame $Id(S)$ (or $Sh_{0,1}(S)$) under inclusion, which may alternately be interpreted as a locale.

Every element of $S$ may be interpreted as an ideal on $S$; given two elements $u,v$ of $S$, $u$ belongs to the ideal represented by $v$ if and only if $u \leq v$; that is, $v$ represents its down set.

The frame of ideals is given by a universal property. If $L$ is a frame, we make $L$ into a posite using its canonical coverage (see the examples below). Then the operation from elements of $S$ to ideals on $S$ is a morphism of posites; furthermore, given any morphism of posites from $S$ to a frame $L$ (with its canonical coverage), there exists a unique frame homomorphism from $Id(S)$ to $L$ that makes the obvious triangle commute.

If instead we treat a posite as a special kind of site and look at its $1$-sheaves, we have this result:

The relevant locale is precisely the locale of ideals.

Interpretations

Given a posite $S$, we may think of the elements of $S$ as basic opens of a space which is generated from $S$ using its coverage, to produce a locale. Thus a posite is precisely a base for a locale. See the example coming from a topological space below to see how this can work precisely for a topological locale.

A coverage can be seen as a sequent calculus; the interpretation of

in topology is analogous to the interpretation of

in logic. Note that we have a single proposition on the left but a set of propositions (interpreted disjunctively) on the right, the reverse of an intuitionistic sequent?.

Examples

Using ideals

Any poset (or proset) $S$ has a canonical coverage, in which

• $u \lhd V$ if and only if, for any $u' \leq u$, $u'$ is a join of some $V'$ such that, for every $v' \in V'$, there is a $v \in V$ such that $v' \leq v$.

If $S$ is a frame, then we can simplify this condition:

• $u \lhd V$ if and only if $u$ is a join of $V$;

and then the canonical coverage is cartesian. In this case, $Id(S)$ is naturally isomorphic to $S$ itself. (Analogously, if $E$ is a Grothendieck topos with its canonical coverage, then $Sh(E)$, the topos of sheaves on $E$, is naturally equivalent to $E$.)

Let $X$ be a topological space, and let $S$ be a topological base of $X$. Then $S$ is a semilattice under intersection; we make $S$ into a posite by defining

• $u \lhd V$ if and only if $u$ is the union of $V$ in $X$ (which is not necessarily the same as a join in $S$).

The frame generated by this posite is naturally isomorphic to the frame of opens of $X$. If $X$ is a sober space, then this frame, interpreted as a locale, may be identified with $X$.

The locale of real numbers is generated from the locally cartesian poset $S \coloneqq \mathbb{Q}^op \times \mathbb{Q}$, where $\mathbb{Q}$ is the usual poset of rational numbers, equipped with the cartesian coverage given by

• $(x,y) \lhd \empty$ whenever $x \geq y$;
• $(w,z) \lhd \{(w,x),(y,z)\}$ whenever $w \leq y \lt x \leq z$; and
• $(w,z) \lhd \{ (x,y) \;|\; w \lt x \lt y \lt z \}$.

Often you will see $\pm\infty$ added to $\mathbb{Q}$ in a description of this locale, which adds a top element to $S$ and so makes it into a semilattice, but that makes no difference to the generated locale in this case.

Using sheaves

The double negation topology used in the sheaf-theoretic approach to forcing is a posite. More generally, Classical set-theoretic forcing is done exclusively on posites. One “reason” for this (in the sense of a reason why set-theorists have not been forced to look beyond such models) is explained in the paper by Freyd below.

The semilattice of commutative subalgebras of a $C^*$-algebra $A$ is a posite (with trivial topology) internal to whose sheaf topos we have a constructive Gelfand duality theorem for $A$, even if $A$ was externally non-commutative.

The classifying topos of any propositional geometric theory is localic, and can be presented as sheaves on a posite of propositions.

Use in predicative mathematics

We call a frame (or locale) $L$ accessible if it is isomorphic to $Id(S)$ for some small posite $S$. (Compare the concept of accessible category, in the case of a cocomplete pretopos.)

In classical mathematics, an accessible frame must be small, but this fails in predicative mathematics. (Conversely, any small frame is trivially accessible; take $S$ to be $L$ with its canonical coverage.) Since many predicativists have philosophical objections to working with large objects at all, they may prefer to work with small posites directly. Whatever the philosophy, we may use small posites in place of accessible locales, or at any rate use this to prove that the category of the latter is essentially moderate.

Formal topology is a programme for topology which is based on using small posites. However, formal topologists also require a positivity predicate on their posites; the intended interpretation is that $u$ is positive iff its corresponding open subspace is inhabited. This seems to be needed for a good theory of compact spaces (and related notions) in a predicative constructive framework.

References

• Most of this article is based on Stone Spaces, but with a different presentation.

On forcing, see:

• Peter Freyd, All topoi are localic or why permutation models prevail, Journal of Pure and Applied Algebra 46 (1987) pp 49–58.

• This MO question and its answer.