These are notes that were made in response to a reading of Victor Porton’s online work titled Algebraic General Topology. This work is a study of general notions of “space” (proximity spaces, pretopological spaces, and the like) from Porton’s idiosyncratic point of view, centering on the concepts that he calls “funcoid” and “reloid”.
Despite the poor reception of this work (some of it justifiable, as Porton employs a heavy symbolism, and some key results are deeply buried), there is some real structural content to his work. In particular, the central concept of “funcoid” is in fact connected with such varied notions as Chu spaces, coherence spaces, syntopogenous spaces, flat profunctors, interchange of finite limits and filtered colimits, … The purpose of these notes is to bring out some of these connections in a concise way, so that some points of his work might be better appreciated, especially by members of the categorical community who deal in abstract topology.
I have titled this page “topogeny”, both as a proposed catch-all term for doing topology from a point of view that takes nearness relations as primitive, and also as a euphonious and grammatically versatile term that can cover various senses of “funcoid” as they recur in various cryptomorphically equivalent ways.
In this section we collect various facts about the set of filters on a set , which are needed in the sequel.
The poset of filters on a set (i.e., in the Boolean algebra ), ordered by inclusion, is a frame, in fact isomorphic to the frame of open sets of .
The topology of is the Zariski topology of the Boolean ring , with ideals/filters corresponding to closed sets; we thus have a Galois correspondence between filters and closed sets in , so that the poset of filters under inclusion is isomorphic to the poset of open sets of under inclusion.
Proposition 1 can also be proven in a “point-free” way, removing all reference to ultrafilters = points of . In fact, for any distributive lattice , the poset of filters in ordered by inclusion, isomorphic to , is itself a distributive lattice in which there is also interchange between finite limits and filtered colimits, so that finite meets distribute over arbitrary joins and is a frame. The proof is constructive and may be internalized in any topos; see Corollary 1 below.
In the next proposition, denotes the join in the lattice , and denotes the top element (the improper filter ).
The map mapping is injective.
We have that , because every filter is the intersection of the ultrafilters that contain it. More elementarily (without using AC): we have .
where is the truth value of . The previous proposition says that is a separable Chu space (Pratt, p. 4). It is also clearly a self-dual Chu space, since is a symmetric relation.
By distributivity in the frame , we have iff both and . Taking negations on both sides of the “iff” gives the result.
Given diagram , with filtered, .
Since is inhabited, we have . As is compact, the frame is compact in the sense that preserves filtered colimits, i.e.,
Taking negations on both sides of this “iff” shows that holds iff holds.
Let be sets. A funcoid from to is a Chu space morphism .
This is not how Porton phrases it, but it’s the same as his notion. In more detail: a funcoid from to consists of a pair of functions and such that for all the following condition is satisfied:
In other words, .
Evidently there is a category of sets and funcoids between them. Because each Chu space is self-dual, this actually gives a -category.
The modes of argument that Porton employs to establish some basic results may be cast in a form familiar to those who have worked a little with Chu spaces. A key fact is separability (see Proposition 2 and the remark that follows on the Chu space ).
For a funcoid , we follow Porton’s notation and let be the relation .
Each component of a funcoid determines the other. Either is determined uniquely from .
Suppose and are funcoids. Then for all we have since both sides equal . Hence by separability, for all ; i.e., . Similarly, if and are funcoids, then . The same argument shows both the functions and are uniquely determined from the relation .
For a funcoid from to , we have that preserves finite meets.
whence by separability. A similar argument shows that .
For a funcoid from to , we have that preserves filtered colimits.
whence by separability.
We state now a key theorem which gives a cryptomorphically equivalent notion of funcoid.
A map preserves finite limits and filtered colimits if and only if is the component of a funcoid from to .
We prove this in the next section. Since just one component uniquely determines a funcoid , the theorem says we may equivalently define a funcoid from to as a map that preserves finite limits and filtered colimits.
This formulation is vaguely reminiscent of the notion of stable function between coherence spaces, due to Girard. I’m not sure how much to make of this however.
Let , be sets. The following definition is adapted from syntopogenous space.
A topogeny from to is a relation that preserves coproducts in each separate argument. More concretely:
and are false, for all ;
if and only if ( or ), and if and only if ( or ).
Note that if , then implies since ; similarly, if , then implies . Thus the “if” clauses of condition 2. in the definition of topogeny are equivalent to the monotonicity of .
For a basic and instructive example of a topogeny, consider a metric space , and define a “nearness” relation where two subsets are near (or ) if the distance between them is zero, viz. .
There is a natural bijection between topogenies from to and maps that preserve finite limits and filtered colimits.
This gives another cryptomorphically equivalent view on funcoids (for Porton, perhaps the most important one, as various general notions of space can be viewed as special types of topogenous relations on sets).
is the filtered colimit completion (in the -category of preorders or -enriched categories) of ; otherwise put, . The “Yoneda embedding” , which is the universal map from to filtered-cocomplete posets, preserves finite limits. Thus, for any that preserves finite limits and filtered colimits, the restriction
preserves finite limits.
In the other direction, because finite limits commute with filtered colimits in , the filtered-cocontinuous extension of any left exact is also left exact. Thus we have a natural bijection
We also have an isomorphism that takes to . Left exact maps are thus in bijection with right exact maps , i.e., with maps that preserve finite colimits in each variable, which are topogenies.
This proposition enables one to define a category of sets and topogenies between them, simply by associating identity and composite topogenies with the corresponding maps . In detail, the identity topogeny on is derived from a map , mapping
where the condition on says is false, or . Thus the identity topogeny is the relation consisting of pairs such that is inhabited.
While it is true that topogenies are closed under relational composition, this isn’t the composition that corresponds to composing maps (noting in particular that the identity topogeny doesn’t behave as an identity under relational composition). To get the “correct” composition, we use the fact that topogenies correspond to “flat profunctors” (where takes coproducts in the argument to products and products in the argument to products) under the correspondence . The composition of flat profunctors is relational composition; working through the correspondence, one arrives at the following definition.
If and are topogenies, then their topogenic composite is defined by the rule
Proposition 3, together with the symmetry of the notion of topogeny (i.e., the fact that each topogeny from to is tantamount to a topogeny from to ), enables us to associate to each lex filtered-cocontinuous a lex filtered-cocontinuous . Tracing through the correspondences above, and using the fact that a filter is a filtered colimit of principal filters, here is the formula:
Given a lex filtered-cocontinuous map , and with as given above, we have if and only if . In other words, the pair is a funcoid from to .
We have precisely when there is and such that . We may write as a filtered colimit of principal filters with ; since preserves filtered colimits, the condition may be rewritten
For such we have , so by upward closure of the condition may be condensed to the equivalent
Coming from the other direction: we have precisely when there is and such that . Spelling this out, this may be rewritten as
For such we have , so the last condition may be condensed down to
which matches (1) since we can swap and , and we are done.
We now have all we need to establish Theorem 1:
We now have at least four ways in which to view “funcoids”:
As Chu space morphisms ,
As left exact Scott-continuous maps ,
As flat or meet-preserving profunctors of type ,
As topogenies or separately join-preserving maps .
Compare Porton, Theorem 6.28. So we have plenty of formulations to choose from, some more convenient than others for a given purpose.
As an example, let us prove the following lemma (compare Porton, Theorem 6.86). In fact we give two proofs; the second proof does not use the axiom of choice, settling a query of Porton.
The poset of funcoids from a set to a set is a frame.
(This proof is a reformulation of Porton’s proof, and uses the axiom of choice, specifically the fact that every proper filter is contained in an ultrafilter, which is slightly weaker than AC.) As in the proof of Proposition 3, the poset of funcoids is isomorphic to the poset , or equivalently since negation provides an isomorphism . Now (cf. Proposition 1) is a sober spatial frame, so that (abbreviating to ) there is a frame embedding
and this induces an embedding
that preserves finite meets and arbitrary joins (note this embedding is a left adjoint because we have an adjunction of left exact maps, and will preserve that adjunction). The codomain is a power of the frame and is thus a frame itself; the domain , being closed under finite meets and arbitrary joins, is a subframe, as was to be shown.
We prove a more general fact: that if is any frame, then is also a frame. Then apply this to .
Indeed, a frame is precisely a lex-total poset, i.e., a poset with the property that the Yoneda-Dedekind embedding has a left exact left adjoint . Applying the functor to this adjunction (both left exact maps), we get an induced adjunction with right adjoint
Since is a frame, so is . We thus have realized as the category of algebras (or poset of fixed points) of a left exact monad (aka a nucleus) on a frame . By a well-known result in locale theory (see for example Mac Lane-Moerdijk, section IX.4, Proposition 3), this implies is a frame, as was to be shown.
Slightly more generally, if is a distributive lattice and is a frame, then is a frame. The same method of proof applies, by exhibiting as the lattice of fixed points of a nucleus on the frame , where is a frame by Corollary 1, given in an appendix below.
As a second example, we consider a generalization of Lemma 6, involving something Porton calls a “staroid” which generalizes the notion of topogeny.
(Porton) For a finite cardinal , given sets , an (-)staroid on these sets is a function which preserves finite joins in separate arguments.
Porton also has a notion of -staroid for infinite , one which is apparently stronger than the obvious generalization of the previous definition.
The poset of -staroids under the pointwise order is a co-frame. More generally, if are distributive lattices, then the poset of maps which preserve joins in separate arguments is a co-frame.
(See also this MO discussion.) This poset is isomorphic to the poset of join-preserving maps . This tensor product is a distributive lattice by Proposition 8. The poset of such is dual to the poset of ideals in , or to the poset of filters in , where is also a distributive lattice. Then apply Corollary 1.
The description of staroids can be considerably sharpened in fact:
If are sets, then the poset of staroids is isomorphic to the co-frame of closed subsets of .
This poset is dual to the frame of ideals of the distributive lattice which, by Proposition 9, is the coproduct in the category of Boolean algebras. By Stone duality, the poset of ideals in the Boolean algebra is isomorphic to the topology of its spectrum .
This gives yet another view on funcoids or topogenies from to (cf. Remark 3): they are tantamount to
From this point of view, a topogeny from to is precisely a relation from to in the pretopos of compact Hausdorff spaces. Composition of funcoids or topogenies corresponds to ordinary relational composition within this pretopos (to be checked carefully).
At this point I am going to switch over to using the word ‘topogeny’, which I much prefer (aesthetically) to ‘funcoid’. At the same time, the most useful point of view on topogenies between sets is that they are equivalent to closed subsets : this allows us to bring the considerable literature on ultrafilter theory to bear on problems.
Apparently Porton is interested in the category whose objects are sets that come equipped with an endotopogeny , i.e., a topogeny from to itself. The morphisms are continuous maps:
Let and be sets equipped with topogenies. A continuous map between them is a function such that for ultrafilters , we have implies .
For various reasons, we will be particularly interested in the full subcategory whose objects are sets with reflexive topogenies, i.e., endotopogenies on that contain the diagonal of . We will call such topogenic spaces. Clearly, the lattices of endotopogenies and of reflexive topogenies are complete lattices (where meets are given by set-theoretic intersections of closed sets, and joins are closures of set-theoretic unions).
The category of endotopogenies and the category of topogenic spaces, taken with their obvious forgetful functors to , are topological over .
If is a function and is an endotopogeny on , then there is an induced endotopogeny by pulling back. This is the largest endotopogeny on that renders continuous, and is the initial structure that lifts seen as a source diagram.
More generally, given a source diagram of sets where have given endotopogenies , the initial lift is the meet of the endotopogenies on . This shows that the forgetful functor is topological. The case for topogenic spaces is wholly similar.
Similarly, if is an endotopogeny on and , then the direct image is the smallest endotopogeny on that renders continuous. This is the final structure that lifts seen as a sink diagram. Final lifts of sink diagrams are obtained by taking joins of the endotopogenies .
The category of endotopogenies and the category of topogenic spaces is complete and cocomplete. In fact, each of those categories is both total and cototal.
Here we collect some order-theoretic and topological results that were used in the account above.
Here we prove that the poset of filters in a distributive lattice is a frame.
Let be a distributive lattice. Then is also a distributive lattice.
If are two filters, then their meet may be calculated as
and their join as
The inclusion holds in any lattice, and we have
which completes the proof.
For any poset , finite meets distribute over filtered joins in .
Certainly finite meets distribute over arbitrary joins in . Since the inclusion is closed under finite meets and filtered joins, the result follows.
If is a distributive lattice, then is a frame.
The preceding lemmas show that preserves finite joins and filtered joins, and therefore arbitrary joins.
Let be join-semilattices, considered as commutative monoids. Then the commutative monoid is idempotent.
Certainly each element is idempotent, since .
For an idempotent commutative monoid , we define so as to make a join-semilattice, viz. iff . Thus in the previous proposition is endowed with a partial order, making it a join-semilattice. In this way the category of join-semilattices inherits a symmetric monoidal structure from the symmetric monoidal category of commutative monoids containing it.
For a commutative monoid in the symmetric monoidal category of join-semilattices, the monoid multiplication coincides with conjunction iff the unit is the top element and multiplication is idempotent.
Note that if , then since . If the unit is maximal, it follows that and similarly . On the other hand, if and , then . This shows is the meet of and .
Let be distributive lattices, considered as commutative rigs with addition given by join and multiplication given by meet. Then the commutative rig , under the order when considered as a join-semilattice, is also a distributive lattice.
Certainly is a commutative monoid in the symmetric monoidal category of join-semilattices. By the lemma it suffices to show that the unit is maximal and that multiplication is idempotent.
First we show is the maximal element of . Indeed,
and similarly , so . Since any element of is a join of elements which all have as upper bound, any element has as upper bound.
Second we show multiplication is idempotent. Clearly is idempotent for any and , by idempotency of and . More generally
which is bounded above by since and , but is also bounded below by . This completes the proof.
By the preceding proposition, distributive lattices form a full subcategory of the category of commutative rigs that is closed under coproducts (which is the tensor product of commutative rigs).
The category of Boolean algebras, as a full subcategory of the category of distributive lattices, is closed under coproducts.
We must show that if are Boolean algebras, then every element in the distributive lattice has a complement (necessarily unique by distributivity). For and , let and denote their complements in and . Then an easy calculation shows that the complement of in is . Moreover, any element in is a finite join of elements of the form , so we just need the auxiliary lemma that the join of two complemented elements in a distributive lattice is also complemented. Indeed, is that complement (for instance, since and ).
Recall that the space of ultrafilters on a set may be regarded as the free compact Hausdorff space generated by , so that we have a monad . It is well-known that the category of compact Hausdorff spaces is equivalent to the category of -algebras, and that the category of topological spaces is equivalent to the category of so-called “relational -modules”. Here we collect a few more results on .
If is a function, then is an open map.
For a subset , let denote the corresponding basic clopen of :
It suffices to show that the image is open in ; in fact we will show . For we have
so a necessary condition for is for all . Note that iff ; using Frobenius reciprocity, this is the same as saying . But for all if and only if (the “if” is clear, and moreover either or , where the latter clearly negates for all , and so the “only if” must hold as well).
We have thus shown that implies . In the other direction, if , then for all , which as we saw is equivalent to for all . Hence the sets generate a filter on , which may be extended to an ultrafilter on , for which and by construction, whence by maximality of ultrafilters. Thus implies .
The ultrafilter monad preserves weak pullbacks, i.e., satisfies the Beck-Chevalley condition.
A proof may be found here.
Let be a family of sets. Then the disjoint sum (the coproduct in ) is open and dense in (the coproduct in ).
Put in . Each inclusion induced by the inclusion is open, by Lemma 10, and hence their union is open. The subspace topology on the union is the same as the disjoint sum topology. In the category of Tychonoff spaces, where coproducts are formed the same as they are in , an inclusion is dense if and only if it is epic. But it is obvious that is epic in : if is Tychonoff and its Stone-Cech compactification, then a map is uniquely determined by a map in , which in turn is uniquely determined by the family of their restrictions to the summands , or by the restriction along the inclusion of the disjoint sum. This completes the proof.
Victor Porton, Algebraic General Topology, version dated June 10, 2014. (web)
Vaughan Pratt, Chu Spaces, Notes for the School on Category Theory and Applications, University of Coimbra (July 13-17, 1999). (web)