In this paper, we aim to revive interest in what we call the Sierpinski object in the Scott realizability topos. We show that it is of fundamental importance in studying the subcategory of order-discrete objects (section 3), arithmetic in the topos (section 4), the Sierpinski object as a dominance (section 5) and a notion of homotopy based on it (section 6). Section 1 deals with preliminaries and establishes notation.
Identifying with and giving the Sierpinski topology (with the only nontrivial open set), we endow with the product topology. Concretely, basic opens of are of the form
for a finite subset of . A function is continuous for this topology if and only if it has the property that
Such continuous maps are encoded by elements of in the following way. Consider the following coding of finite subsets of :
By convention, . So: , , , , etc.
Consider also a bijection , which we don’t specify.
Now we define a binary operation (written as multiplication) on :
For every function we have
so that, when is continuous, for all . Note that the operation is a continuous map of two variables.
We think of the operation as an application operation: apply the continuous function coded by to the argument .
The structure together with this application function is denoted . The fact that is a -model is worked ouit in detail in . We shall use -terms to denote elements of according to this interpretation. We use repeated juxtaposition and associate to the left when discussing iterated application: means . For continuous functions of several variables we let
So with these conventions, .
In , special combinators for important operations are available:
Pairing and unpairing combinators: satisfying , . In fact we can take
for , so that
Booleans: , . We have an “if … then …, else…” operator , satisfying and .
Natural numbers: we write for the element .
As usual we have a category of assemblies over , denoted and the realizability topos , which we call the Scott realizability topos (unfortunately, the term “Scott topos” is already in use). We briefly introduce these categories.
An assembly over is a pair where is a set and gives, for each , a nonempty subset of . A morphism of assemblies is a function for which there is some element of which tracks , meaning that for all and all we have .
The category is a quasitopos. Actually, has countable products: given a countable family of assemblies over , one has the assembly where
The category has several subcategories of interest. First we recall that there is an adjunction
where , with for all .
An assembly is partitioned if is always a singleton. It is not hard to see that the object is isomorphic to where for all , so all objects are isomorphic to partitioned assemblies. Up to isomorphism, the partitioned assemblies are exactly the regular projective objects (the objects for which the representable functor into Sets preserves regular epimorphisms), which, in the topos , coincide with the internally projective objects (the objects for which the endofuctor preserves epimorphisms); see 3.1 below.
An assembly is modest if whenever . The modest assemblies are, up to isomorphism, the ones for which the diagonal map is an isomorphism (here, 2 denotes any two-element set); such objects are called discrete in .
A special object of is the object of realizers where ; one sees that is both partitioned and modest. Every object which is both partitioned and modest is a regular subobject of : that is, isomorphic to an object where and is the restriction to of the map defining .
Another partitioned and modest object is the natural numbers object , where .
As Andrej Bauer observed in his thesis (), one can embed the category of countably based -spaces into the subcategory of partitioned and modest assemblies over . To be precise, by “countably based -space” we mean a -space together with a chosen enumeration of a subbasis. If is such, then the map given by
defines an embedding of into . Moreover, given countably based spaces and with associated embeddings , there is for every continuous map a continuous extension satisfying . This means that if we see the space with embedding as assembly where , then every continuous map is tracked (as a morphism of assemblies by .
Finally, we have the Scott realizability topos, the realizability topos over , denoted . Its objects are pairs where for , is a subset of (so is a -valued relation on ), which satisfies certain conditions (we refer to  for details). Every assembly is an object of if we define
As subcategory of , the category is equivalent to the category of -stable objects of , that is: the objects satisfying . We shall not rehearse the structure of or its internal logic, based on realizability over , any further, as for this theory now standard texts are available (e.g., ).
The following small proposition will be of use later on. Given an assembly let be where and . Consider also where and . Clearly, we have a projection and two projections , and these maps form a coequalizer diagram in .
Let be an arbitrary object of and let be an assembly over . The exponential can be presented as follows: its underlying set is the set of functions from to . Given two such functions , the set consists of those coded triples (coded in the sense of ) of elements of , which satisfy the following three conditions:
Observe that the objects and are partitioned assemblies, hence projective objects in . Also note that the contravariant functor sends coequalizers to equalizers. Finally, use the explicit description of exponentials where the exponent is projective, from , pp. 136–7.∎
3 The Sierpiński object and order-discrete objects
The Sierpiński object is the image of Sierpiński space under the embedding . Concretely, is the modest set with and . The object of realizers is the modest set .
The two objects above are related, as follows. The object of realizers is isomorphic to the exponential , where is the natural numbers object of .
We prove that the underlying set of is . Since any morphism from to is in particular a function from to , one inclusion is clear. Conversely, if , then is tracked by , where is the continuous function defined by:
Thus, is the assembly where is the (non-empty) set of trackers of .
We have a canonical bijection
is the characteristic function of) with inverse
It remains to prove that these functions are tracked. For , consider the continuous function given by Then is tracked by graph(). Indeed, if and , then and this is if and otherwise.
For the inverse of , we define continuous by . (This is continuous, because the application of is continuous.) We claim that is tracked by . Indeed, for and , we have if and only if , since tracks . ∎
3.1 Order-discrete objects
An object of is order-discrete if it is orthogonal to , viz. the diagonal is an isomorphism.
We wish to characterize the order-discrete objects in terms of their realizers. To this end, we first present some useful lemmas. An object is order-discrete if and only if the diagonal is an epimorphism.
Of course, is epic if is order-discrete. Conversely, suppose is an epi. Clearly, the unique morphism is an epimorphism. Hence, the diagonal is monic. Thus, is an iso, as desired. ∎
If is a partitioned assembly and is any object, then the exponential is isomorphic to the object where
See , pp. 136–137. ∎
An object is order-discrete if and only if there is such that for any : if and with , then .
Suppose first that is order-discrete. We construct the desired element . Assume we have and with . By 3.1 we have . Define by and . The map defined as
is easily seen to be continuous. Write . We claim that . Indeed, , so that . Moreover, , since and and . Thus, . Now let be a realizer of the fact that is epic. Then is an element of for some . Thus, and . Finally, let respectively realize transitivity and symmetry of . Then, we see that
is the desired element .
Conversely, suppose we have an as in the proposition. Let be arbitrary. By 3.1, it suffices to show that from an element of , we can continuously find an and an element of . Let . Then , and , so and . Hence, if we set , then the graph of the continuous function
is the desired element. ∎
An assembly is order-discrete if and only if the existence of realizers and with implies that and are equal.
The natural numbers object (c.f. the proof of 3) is easily seen to be order-discrete using the corollary above.
Recall from the section on preliminaries that an object is called discrete if it is orthogonal to . For an assembly , this means that the realizing sets are disjoint. Thus, Corollary 3.1 shows that any order-discrete assembly is also discrete (alternatively, one might observe that the map from to is an epi in , and therefore the map is monic. From this one also easily deduces that order-discrete implies discrete, for assemblies). In fact, this holds for all objects, not just assemblies. Any order-discrete object is discrete.
One might expect the availability of a simpler proof of the proposition above, i.e. a proof that does not rely on the characterization of the (order-)discrete objects. However, note that although we have a morphism , it is not a regular epimorphism, so one cannot apply , Lemma 2.2.
In , it it shown that the class of objects orthogonal to an object satisfies some nice properties if is well-supported and internally projective. The following lemma is therefore quite useful.
An object is internally projective in if and only if it is isomorphic to a partitioned assembly.
This follows from , pp. 135–137. The proofs generalize to an arbitrary realizability topos. ∎
The order-discrete objects are closed under all existing limits, subobjects and quotients in . Moreover, they form an exponential ideal. Finally, they form a reflective subcategory of .
The first statement is true, because any collection of orthogonal objects is closed under all existing limits (see , p. 1). Similarly, any collection of orthogonal objects forms an exponential ideal. The other closure properties follow from Lemmas 2.3 and 2.8 of , and Lemma 3.1 above. The final claim is proven as a corollary right after Lemma 2.3 of . ∎
For comparison with the notion of homotopy to be defined below, we give a concrete representation of the order-discrete reflection of an arbitrary object . It has the same underlying set , and we let consist of those coded (in the sense of ) -tuples for which there is a sequence of elements of , such that the following conditions hold:
or must hold.
4 Arithmetic in
The order-discrete modest sets already make a disguised appearance in . Herein, Lietz singles out a class of so-called well-behaved modest sets. It is not hard to show that a modest set is well-behaved if and only if it is order-discrete and moreover, it has the join-property, which we define now. An assembly has the join-property if it is isomorphic to an assembly whose realizing sets are closed under binary joins, viz. if and , then for all . The natural numbers object has the join-property. The following proposition is a very slight generalization of , Theorem 2.3. Let and be modest sets. If has the join-property and is order-discrete, then the following form of the Axiom of Choice holds in :
See the proof of ,Theorem 2.3. ∎
We have already seen that the order-discrete objects form an exponential ideal and are closed under binary products. In , Theorem 2.2, it is shown that this also holds for the modest objects that have the join-property. Since is order-discrete and has the join-property, it follows that the Axiom of Choice holds for all finite types. Consequently, the following two principles
|where is short for|
|(Weak Continuity for Numbers)|
are equivalent in . Furthermore, they are both false in , c.f. , Section 2.3.
4.1 Second-order arithmetic in
We close this section by some brief considerations on second-order arithmetic in realizability toposes and in particular. Let be any partial combinatory algebra. Write for the -th Curry numeral in and let be a pairing combinator in with projections and .
The power object of the natural numbers object in can be given as the object where is the set
We now turn our attention to -stable subsets of , viz. subsets such that is true in . We will abbreviate this formula by .
We formulate the following triviality as a proposition for comparison with 4.1 below, in the case of .
If is nontrivial, then not every subset is -stable, i.e. the sentence is not valid in .
Note that implies (where is a variable of type ), which fails in as soon as has two disjoint nonempty subsets. If is trivial then , and obviously holds. ∎
For any , we have a realizer of . Hence, the sentence is true in .
Let . Note that there is a function such that
Moreover, there is an element such that . Now note that realizes . ∎
First-order arithmetic in is the same as first-order arithmetic in Set.
Just as for ordinary natural numbers, we write for a definable coding of to in . The sentence is known as Shanin’s Principle (SHP). It holds in the Effective Topos (, p. 127). Internally, it says that every subset of is covered by a stable subset of . If , then Shanin’s Principle does not hold in .
Assume for the sake of contradiction that it does. Then we have a realizer
Let us write and . For each , define the element by:
For each , pick some such that .
From , we effectively obtain such that for every , we have:
for some . As , there must be two different such that
for the same .
From , we effectively obtain witnessing the stability of and . Thus, by , we can use to get a common realizer:
Finally, using and , we find a realizer in the intersection . But this is impossible, because and are different, so that and are disjoint. ∎
Shanin’s Principle does not hold in and .
5 The Sierpiński object as a dominance
One can also study the Sierpiński object from a synthetic domain theoretic standpoint. This was done by Wesley Phoa (, Chapter 12 and , Proposition 3.1), who worked in the category of modest sets over the r.e. graph model, John Longley (, 5.3.7) and Alex Simpson (). Our treatment is based on Proposition 3. Define a relation between and (the object of realizers) by taking the following pullback
where is the morphism induced by the function and . Observe that is given by
In particular, is -stable.
Let be a arbitrary total pca and let be its realizability topos. Let be the object of realizers in the topos. The scheme
with not free in is valid in .
It is not hard to verify that realizes the scheme. ∎
The subobject of given by
with ranging over , is a dominance in . Moreover, it is -separated, i.e. . Furthermore, .
Double negation separation is immediate by 5. Further, it is clear that (take ).
We have already remarked that the object from 5 is -separated. Indeed, it is isomorphic to an assembly. The object is isomorphic to .
First of all, observe that is isomorphic to the object where
Define a continuous function by
Next, define by
We show that is a functional relation from to . Strictness is immediate. For single-valuedness, suppose we have and . We show that . By definition, we have
and similarly, . Thus, , as desired. Suppose and . We must effectively obtain an element of . But if realizes , then one easily sees that is an element of . So, is relational. For totality, suppose , then for some , by construction of and . We conclude that is a functional relation.
Moreover, (the arrow represented by) is easily seen to be epic. For, if , then with and by construction of and definition of .
Finally, we prove that is monic and hence that represents an isomorphism, as desired. Suppose we have and . It suffices to effectively provide an element of , since is an element of already. But is easily seen to do the job. ∎
Proposition 5 implies that we have a notion of -subobjects of any object in : a mono represents a -subobject of
if and only if its classifying map factors through.
The following proposition characterizes these -subobjects for assemblies . Let be an assembly. There is a bijective correspondence between morphisms and subsets for which there is an open with the following properties:
Moreover, an assembly is a -subobject of if and only if is isomorphic to some assembly where is as above and is the restriction of to .
Let be a morphism from to that is tracked by . Set
We show that is open. Let . Recall the notation . By continuity of the application, one can show that . Thus, is an open of .
From the definition of and the fact that tracks , it is immediate that has the desired properties.
For the converse, assume we are given an open and a subset with the properties stated. Define by if and if . We claim that it is tracked by where . That this is continuous follows from the assumption that is open. Now if and , then either , in which case , so that ; or , in which case , so that . So is tracked, as desired.
That the operations above are each other’s inverses is readily verified. The final claim follows immediately from the construction above and the description of pullbacks in the category of assemblies. ∎
The following proposition describes the lift functor for the dominance when restricted to . The lift functor on is given by on objects by:
where is some element not in and
Given an arrow , we define as the unique extension of satisfying . The natural transformation is defined as .
Given a morphism of tracked by , note that is tracked, as
is a continuous map . Verifying that is indeed a functor is routine. Also, note that is tracked by . That is natural is easily checked.
Finally, suppose we have a morphism and . By 5, we may assume that we have and an open such that for