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 $Filt(X)$ on a set $X$, which are needed in the sequel.
The poset of filters on a set $X$ (i.e., in the Boolean algebra $P X$), ordered by inclusion, is a frame, in fact isomorphic to the frame of open sets of $\beta X$.
The topology of $\beta X$ is the Zariski topology of the Boolean ring $P X$, with ideals/filters corresponding to closed sets; we thus have a Galois correspondence between filters and closed sets in $\beta X$, so that the poset of filters under inclusion is isomorphic to the poset of open sets of $\beta X$ under inclusion.
Proposition 1 can also be proven in a “point-free” way, removing all reference to ultrafilters = points of $\beta X$. In fact, for any distributive lattice $L$, the poset of filters in $L$ ordered by inclusion, isomorphic to $Lex(L, \mathbf{2})$, 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 $Lex(L, \mathbf{2})$ is a frame. The proof is constructive and may be internalized in any topos; see Corollary 1 below.
In the next proposition, $\vee$ denotes the join in the lattice $Filt(X)$, and $\top$ denotes the top element (the improper filter $P X$).
The map $\Phi_X: Filt(X) \to P(Filt(X))$ mapping $F \mapsto \{G: F \vee G \neq \top\}$ is injective.
We have that $F = \bigcap (\beta(X) \cap \Phi_X(F))$, because every filter is the intersection of the ultrafilters that contain it. More elementarily (without using AC): we have $(\forall A: P X)\; A \in F \Leftrightarrow prin(\neg A) \notin \Phi_X(F)$.
The last proposition suggests a Chu space (see also Pratt) with carrier $Filt(X)$, given by a triple
where $R_X(F, G)$ is the truth value of $F \vee G \neq \top$. The previous proposition says that $Ch(X)$ is a separable Chu space (Pratt, p. 4). It is also clearly a self-dual Chu space, since $R$ is a symmetric relation.
$R_X(F \wedge F', G) = R_X(F, G) \vee R_X(F', G)$.
By distributivity in the frame $Filt(X)$, we have $\top \leq (F \wedge F') \vee G = (F \vee G) \wedge (F' \vee G)$ iff both $\top \leq F \vee G$ and $\top \leq F' \vee G$. Taking negations on both sides of the “iff” gives the result.
Given diagram $\phi: D \to Filt(X)$, with $D$ filtered, $R_X(colim_{d\in D}\; \phi(d), G) = \forall_{d \in D} R_X(\phi(d), G)$.
Since $D$ is inhabited, we have $(colim_d\; \phi(d)) \vee G = colim_d\; (\phi(d) \vee G)$. As $\beta X$ is compact, the frame $Filt(X) = \mathcal{O}(\beta(X))$ is compact in the sense that $\hom(\top, -): Filt(X) \to \mathbf{2}$ preserves filtered colimits, i.e.,
Taking negations on both sides of this “iff” shows that $R_X(colim_{d\in D}\; \phi(d), G)$ holds iff $(\forall_{d\in D})\; R_X(\phi(d), G)$ holds.
Let $X, Y$ be sets. A funcoid from $X$ to $Y$ is a Chu space morphism $Ch(X) \to Ch(Y)$.
This is not how Porton phrases it, but it’s the same as his notion. In more detail: a funcoid $f$ from $X$ to $Y$ consists of a pair of functions $\alpha: Filt(X) \to Filt(Y)$ and $\beta: Filt(Y) \to Filt(X)$ such that for all $F \in Filt(X), G \in Filt(Y)$ the following condition is satisfied:
In other words, $R_Y(\alpha(F), G) = R_X(F, \beta(G))$.
Evidently there is a category of sets and funcoids between them. Because each Chu space $Ch(X)$ is self-dual, this actually gives a $\dagger$-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 $Ch(X)$).
For $f = (\alpha, \beta)$ a funcoid $X \to Y$, we follow Porton’s notation and let $[f]$ be the relation $R_Y(\alpha(-), -) = R_X(-, \beta(-))$.
Each component $\alpha, \beta$ of a funcoid $f$ determines the other. Either is determined uniquely from $[f]$.
Suppose $(\alpha, \beta)$ and $(\alpha, \beta')$ are funcoids. Then for all $G \in Filt(Y)$ we have $R_X(-, \beta(G)) = R_X(-, \beta'(G))$ since both sides equal $R_Y(\alpha(-), G)$. Hence $\beta(G) = \beta'(G)$ by separability, for all $G$; i.e., $\beta = \beta'$. Similarly, if $(\alpha, \beta)$ and $(\alpha', \beta)$ are funcoids, then $\alpha = \alpha'$. The same argument shows both the functions $\alpha$ and $\beta$ are uniquely determined from the relation $[f]$.
For a funcoid $f = (\alpha, \beta)$ from $X$ to $Y$, we have that $\alpha: Filt(X) \to Filt(Y)$ preserves finite meets.
(Compare Porton, Proposition 6.13.) With the help of Lemma 1, we have
whence $\alpha(F \wedge F') = \alpha(F) \wedge \alpha(F')$ by separability. A similar argument shows that $\alpha(\top) = \top$.
For a funcoid $f = (\alpha, \beta)$ from $X$ to $Y$, we have that $\alpha: Filt(X) \to Filt(Y)$ preserves filtered colimits.
(Compare Porton, Theorem 6.25.) For a filtered diagram $\phi: D \to Filt(X)$, we have with the help of Lemma 2 the following calculation:
whence $\alpha(colim_d\; \phi(d)) = colim_d\; \alpha(\phi(d))$ by separability.
We state now a key theorem which gives a cryptomorphically equivalent notion of funcoid.
A map $\alpha: Filt(X) \to Filt(Y)$ preserves finite limits and filtered colimits if and only if $\alpha$ is the component of a funcoid from $X$ to $Y$.
We prove this in the next section. Since just one component $\alpha$ uniquely determines a funcoid $(\alpha, \beta)$, the theorem says we may equivalently define a funcoid from $X$ to $Y$ as a map $Filt(X) \to Filt(Y)$ 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 $X$, $Y$ be sets. The following definition is adapted from syntopogenous space.
A topogeny from $X$ to $Y$ is a relation $\xi: P X \times P Y \to \mathbf{2}$ that preserves coproducts in each separate argument. More concretely:
$\xi(A, \emptyset)$ and $\xi(\emptyset, B)$ are false, for all $A \in P X, B \in P Y$;
$\xi(A, C \cup D)$ if and only if ($\xi(A, C)$ or $\xi(A, D)$), and $\xi(A \cup B, C)$ if and only if ($\xi(A, C)$ or $\xi(B, C)$).
Note that if $C \subseteq D$, then $\xi(A, C)$ implies $\xi(A, D)$ since $D = C \cup D$; similarly, if $A \subseteq B$, then $\xi(A, C)$ implies $\xi(B, C)$. Thus the “if” clauses of condition 2. in the definition of topogeny are equivalent to the monotonicity of $\xi: P X \times P Y \to \mathbf{2}$.
For a basic and instructive example of a topogeny, consider a metric space $(X, d)$, and define a “nearness” relation $\xi$ where two subsets $A, B$ are near (or $\xi(A, B)$) if the distance between them is zero, viz. $\inf\; \{d(x, y): x \in A, y \in B\} = 0$.
There is a natural bijection between topogenies from $X$ to $Y$ and maps $\alpha: Filt(X) \to Filt(Y)$ 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).
$Filt(X)$ is the filtered colimit completion (in the $2$-category of preorders or $\mathbf{2}$-enriched categories) of $(P X)^{op}$; otherwise put, $Filt(X) \cong Lex(P X, \mathbf{2})$. The “Yoneda embedding” $prin: (P X)^{op} \to Lex(P X, \mathbf{2})$, which is the universal map from $(P X)^{op}$ to filtered-cocomplete posets, preserves finite limits. Thus, for any $\alpha: Filt(X) \to Filt(Y)$ that preserves finite limits and filtered colimits, the restriction
preserves finite limits.
In the other direction, because finite limits commute with filtered colimits in $Filt(Y) = Lex(P Y, \mathbf{2})$, the filtered-cocontinuous extension of any left exact $(P X)^{op} \to Filt(Y)$ is also left exact. Thus we have a natural bijection
We also have an isomorphism $Lex(P Y, \mathbf{2}) \cong Rex(P Y, \mathbf{2})^{op}$ that takes $\phi: P Y \to \mathbf{2}$ to $(\lambda B: P Y)(\neg\phi(\neg B))$. Left exact maps $(P X)^{op} \to Lex(P Y, \mathbf{2})$ are thus in bijection with right exact maps $P X \to Rex(P Y, \mathbf{2})$, i.e., with maps $P X \times P Y \to \mathbf{2}$ 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 $Filt(X) \to Filt(Y)$. In detail, the identity topogeny on $X$ is derived from a map $P X \to Rex(P X, \mathbf{2})$, mapping
where the condition on $B$ says $A \subseteq \neg B$ is false, or $A \cap B \neq \emptyset$. Thus the identity topogeny is the relation consisting of pairs $(A, B) \in P X \times P X$ such that $A \cap B$ is inhabited.
While it is true that topogenies are closed under relational composition, this isn’t the composition that corresponds to composing maps $Filt(X) \to Filt(Y)$ (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 $\xi: P X \times P Y \to \mathbf{2}$ correspond to “flat profunctors” $\phi: P X^{op} \times P Y \to \mathbf{2}$ (where $\phi$ takes coproducts $A \cup A'$ in the argument $P X$ to products and products $B \cap B'$ in the argument $P Y$ to products) under the correspondence $\phi(A, B) = \neg \xi(A, \neg B)$. The composition of flat profunctors is relational composition; working through the correspondence, one arrives at the following definition.
If $\nu: X \to Y$ and $\xi: Y \to Z$ are topogenies, then their topogenic composite $\xi \circ \nu: X \to Z$ is defined by the rule
Proposition 3, together with the symmetry of the notion of topogeny (i.e., the fact that each topogeny from $X$ to $Y$ is tantamount to a topogeny from $Y$ to $X$), enables us to associate to each lex filtered-cocontinuous $\alpha: Filt(X) \to Filt(Y)$ a lex filtered-cocontinuous $\beta: Filt(Y) \to Filt(X)$. Tracing through the correspondences above, and using the fact that a filter $G \in Filt(Y)$ is a filtered colimit of principal filters, here is the formula:
Given a lex filtered-cocontinuous map $\alpha: Filt(X) \to Filt(Y)$, and with $\beta$ as given above, we have $\top \leq \alpha(F) \vee G$ if and only if $\top \leq F \vee \beta(G)$. In other words, the pair $(\alpha, \beta)$ is a funcoid from $X$ to $Y$.
We have $\top \leq \alpha(F) \vee G$ precisely when there is $B \in \alpha(F)$ and $B' \in G$ such that $B \cap B' = \emptyset$. We may write $F$ as a filtered colimit of principal filters $prin(A)$ with $A \in F$; since $\alpha$ preserves filtered colimits, the condition may be rewritten
For such $B, B'$ we have $B' \subseteq \neg B$, so by upward closure of $G$ the condition may be condensed to the equivalent
Coming from the other direction: we have $\top \leq F \vee \beta(G)$ precisely when there is $A \in F$ and $A' \in \beta(G)$ such that $A \cap A' = \emptyset$. Spelling this out, this may be rewritten as
For such $A, A'$ we have $A \subseteq \neg A'$, so the last condition may be condensed down to
which matches (1) since we can swap $B$ and $\neg B$, 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 $Ch(X) \to Ch(Y)$,
As left exact Scott-continuous maps $Filt(X) \to Filt(Y)$,
As flat or meet-preserving profunctors of type $P X^{op} \to P Y$,
As topogenies or separately join-preserving maps $P X \times P Y \to \mathbf{2}$.
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 $X$ to a set $Y$ 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 $Lex((P X)^{op}, Filt(Y))$, or equivalently $Lex(P X, Filt(Y))$ since negation provides an isomorphism $(P X)^{op} \cong P X$. Now $Filt(Y) = \mathcal{O}(\beta Y)$ (cf. Proposition 1) is a sober spatial frame, so that (abbreviating $\mathcal{O}(\beta Y)$ to $\mathcal{O}$) 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 $j \dashv j_\ast$ of left exact maps, and $Lex(P X, -)$ will preserve that adjunction). The codomain is a power of the frame $Lex(P X, \mathbf{2}) \cong Filt(X)$ and is thus a frame itself; the domain $Lex(P X, Filt(Y))$, being closed under finite meets and arbitrary joins, is a subframe, as was to be shown.
We prove a more general fact: that if $L$ is any frame, then $Lex((P X)^{op}, L)$ is also a frame. Then apply this to $L = Filt(Y)$.
Indeed, a frame $L$ is precisely a lex-total poset, i.e., a poset with the property that the Yoneda-Dedekind embedding $y: L \to [L^{op}, \mathbf{2}]$ has a left exact left adjoint $\sigma: [L^{op}, \mathbf{2}] \to L$. Applying the functor $Lex((P X)^{op}, -)$ to this adjunction $\sigma \dashv y$ (both left exact maps), we get an induced adjunction with right adjoint
Since $Filt(X)$ is a frame, so is $[L^{op}, Filt(X)]$. We thus have realized $Lex((P X)^{op}, L)$ as the category of algebras (or poset of fixed points) of a left exact monad (aka a nucleus) on a frame $[L^{op}, Filt(X)]$. By a well-known result in locale theory (see for example Mac Lane-Moerdijk, section IX.4, Proposition 3), this implies $Lex((P X)^{op}, L)$ is a frame, as was to be shown.
Slightly more generally, if $D$ is a distributive lattice and $L$ is a frame, then $Lex(D, L)$ is a frame. The same method of proof applies, by exhibiting $Lex(D, L)$ as the lattice of fixed points of a nucleus on the frame $[L^{op}, Filt(D)]$, where $Filt(D) \cong Lex(D, \mathbf{2})$ 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 $n$, given sets $X_1, \ldots, X_n$, an ($n$-)staroid on these sets is a function $\phi: P X_1 \times \ldots \times P X_n \to \mathbf{2}$ which preserves finite joins in separate arguments.
Porton also has a notion of $n$-staroid for infinite $n$, one which is apparently stronger than the obvious generalization of the previous definition.
The poset of $n$-staroids under the pointwise order is a co-frame. More generally, if $L_1, \ldots, L_n$ are distributive lattices, then the poset of maps $\phi: L_1 \times \ldots \times L_n \to \mathbf{2}$ 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 $\psi: L \coloneqq L_1 \otimes \ldots \otimes L_n \to \mathbf{2}$. This tensor product $L$ is a distributive lattice by Proposition 8. The poset of such $\psi$ is dual to the poset of ideals in $L$, or to the poset of filters in $L^{op}$, where $L^{op}$ is also a distributive lattice. Then apply Corollary 1.
The description of staroids can be considerably sharpened in fact:
If $X_1, \ldots, X_n$ are sets, then the poset of staroids $\psi: P X_1 \times \ldots \times P X_n \to \mathbf{2}$ is isomorphic to the co-frame of closed subsets of $\beta X_1 \times \ldots \times \beta X_n$.
This poset is dual to the frame of ideals of the distributive lattice $P X_1 \otimes \ldots \otimes P X_n$ which, by Proposition 9, is the coproduct $\sum_i P X_i$ in the category of Boolean algebras. By Stone duality, the poset of ideals in the Boolean algebra $\sum_i P X_i$ is isomorphic to the topology of its spectrum $\beta X_1 \times \ldots \times \beta X_n$.
This gives yet another view on funcoids or topogenies from $X$ to $Y$ (cf. Remark 3): they are tantamount to
From this point of view, a topogeny from $X$ to $Y$ is precisely a relation from $\beta X$ to $\beta Y$ 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 $X, Y$ is that they are equivalent to closed subsets $C \hookrightarrow \beta X \times \beta Y$: 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 $X$ that come equipped with an endotopogeny $C \hookrightarrow \beta X \times \beta X$, i.e., a topogeny from $X$ to itself. The morphisms are continuous maps:
Let $(X, C_X)$ and $(Y, C_Y)$ be sets equipped with topogenies. A continuous map between them is a function $f: X \to Y$ such that for ultrafilters $F, G \in \beta X$, we have $C_X(F, G)$ implies $C_Y(\beta(f)(F), \beta(f)(G))$.
For various reasons, we will be particularly interested in the full subcategory whose objects are sets $X$ with reflexive topogenies, i.e., endotopogenies on $X$ that contain the diagonal of $\beta X \times \beta X$. 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 $Set$, are topological over $Set$.
If $f: X \to Y$ is a function and $C$ is an endotopogeny on $Y$, then there is an induced endotopogeny $f^\ast C \coloneqq (\beta(f) \times \beta(f))^{-1}(C) \subseteq \beta X \times \beta X$ by pulling back. This $f^\ast C$ is the largest endotopogeny on $X$ that renders $f$ continuous, and is the initial structure that lifts $f: X \to Y$ seen as a source diagram.
More generally, given a source diagram $f_i: X \to X_i$ of sets where $X_i$ have given endotopogenies $C_i$, the initial lift is the meet of the endotopogenies $f_i^\ast C_i$ on $X$. This shows that the forgetful functor is topological. The case for topogenic spaces is wholly similar.
Similarly, if $C$ is an endotopogeny on $X$ and $f: X \to Y$, then the direct image $f_\ast C \coloneqq (\beta(f) \times \beta(f))(C) \subseteq \beta Y \times \beta Y$ is the smallest endotopogeny on $Y$ that renders $f$ continuous. This is the final structure that lifts $f: X \to Y$ seen as a sink diagram. Final lifts of sink diagrams $f_i: Y_i \to Y$ are obtained by taking joins of the endotopogenies $(f_i)_\ast C_i$.
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 $Filt(L) \cong Lex(L, \mathbf{2})$ in a distributive lattice $L$ is a frame.
Let $L$ be a distributive lattice. Then $Lex(L, \mathbf{2})$ is also a distributive lattice.
If $F, F' \subseteq L$ are two filters, then their meet may be calculated as
and their join as
The inclusion $(F \wedge G) \vee (F \wedge H) \leq F \wedge (G \vee H)$ holds in any lattice, and we have
which completes the proof.
For any poset $L$, finite meets distribute over filtered joins in $Lex(L, 2)$.
Certainly finite meets distribute over arbitrary joins in $[L, \mathbf{2}]$. Since the inclusion $Lex(L, \mathbf{2}) \hookrightarrow [L, \mathbf{2}]$ is closed under finite meets and filtered joins, the result follows.
If $L$ is a distributive lattice, then $Lex(L, \mathbf{2})$ is a frame.
The preceding lemmas show that $F \wedge -$ preserves finite joins and filtered joins, and therefore arbitrary joins.
Let $P, Q$ be join-semilattices, considered as commutative monoids. Then the commutative monoid $P \otimes Q$ is idempotent.
Certainly each element $a \otimes b$ is idempotent, since $a \otimes b = (a + a) \otimes b = a \otimes b + a \otimes b$.
For an idempotent commutative monoid $P$, we define $\leq$ so as to make $P$ a join-semilattice, viz. $p \leq q$ iff $p + q = q$. Thus $P \otimes Q$ 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 $P$ 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 $b \leq c$, then $a b \leq a c$ since $a c = a(b + c) = a b + a c$. If the unit $1$ is maximal, it follows that $a b \leq a 1 = a$ and similarly $a b \leq b$. On the other hand, if $x \leq a$ and $x \leq b$, then $x = x x \leq a b$. This shows $a b$ is the meet of $a$ and $b$.
Let $P, Q$ be distributive lattices, considered as commutative rigs with addition given by join and multiplication given by meet. Then the commutative rig $P \otimes Q$, under the order $\leq$ when considered as a join-semilattice, is also a distributive lattice.
Certainly $P \otimes Q$ is a commutative monoid in the symmetric monoidal category of join-semilattices. By the lemma it suffices to show that the unit ${\top_P} \otimes {\top_Q}$ is maximal and that multiplication is idempotent.
First we show $\top_P \otimes \top_Q$ is the maximal element of $P \otimes Q$. Indeed,
and similarly $\top_P \otimes b \leq \top_P \otimes \top_Q$, so $a \otimes b \leq \top_P \otimes \top_Q$. Since any element of $P \otimes Q$ is a join $\sum_i a_i \otimes b_i$ of elements $a_i \otimes b_i$ which all have $\top_P \otimes \top_Q$ as upper bound, any element has $\top_P \otimes \top_Q$ as upper bound.
Second we show multiplication is idempotent. Clearly $a \otimes b$ is idempotent for any $a \in P$ and $b \in Q$, by idempotency of $a$ and $b$. More generally
which is bounded above by $\sum_i a_i \otimes b_i$ since $a_i a_j \leq a_i$ and $b_i b_j \leq b_i$, but is also bounded below by $\sum_i a_i a_i \otimes b_i b_i = \sum_i a_i \otimes b_i$. 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 $P, Q$ are Boolean algebras, then every element in the distributive lattice $P \otimes Q$ has a complement (necessarily unique by distributivity). For $a \in P$ and $b \in Q$, let $\neg a$ and $\neg b$ denote their complements in $P$ and $Q$. Then an easy calculation shows that the complement of $a \otimes b$ in $P \otimes Q$ is $\neg a \otimes b + a \otimes \neg b + \neg a \otimes \neg b$. Moreover, any element in $P \otimes Q$ is a finite join of elements of the form $a_i \otimes b_i$, so we just need the auxiliary lemma that the join $x + y$ of two complemented elements in a distributive lattice is also complemented. Indeed, $\neg x \cdot \neg y$ is that complement (for instance, $1 \leq x + y + (\neg x)(\neg y)$ since $1 = (x + \neg x)(y + \neg y) = x y + x(\neg y) + (\neg x)y + (\neg x)(\neg y)$ and $x y + x(\neg y) + (\neg x)y = x + (\neg x)y \leq x + y$).
Recall that the space of ultrafilters $\beta X$ on a set $X$ may be regarded as the free compact Hausdorff space generated by $X$, so that we have a monad $\beta: Set \to Set$. It is well-known that the category of compact Hausdorff spaces is equivalent to the category of $\beta$-algebras, and that the category of topological spaces is equivalent to the category of so-called “relational $\beta$-modules”. Here we collect a few more results on $\beta$.
If $f: X \to Y$ is a function, then $\beta(f): \beta(X) \to \beta(Y)$ is an open map.
For a subset $A \subseteq X$, let $[A]$ denote the corresponding basic clopen of $\beta(X)$:
It suffices to show that the image $\beta(f)([A])$ is open in $\beta(Y)$; in fact we will show $\beta(f)([A]) = [f(A)]$. For $U \in \beta(X)$ we have
and
so a necessary condition for $V \in \beta(f)([A])$ is $A \cap f^{-1}(B) \neq \emptyset$ for all $B \in V$. Note that $A \cap f^{-1}(B) \neq \emptyset$ iff $f(A \cap f^{-1}(B)) \neq \emptyset$; using Frobenius reciprocity, this is the same as saying $f(A) \cap B \neq \emptyset$. But $f(A) \cap B \neq \emptyset$ for all $B \in V$ if and only if $f(A) \in V$ (the “if” is clear, and moreover either $f(A) \in V$ or $\neg f(A) \in V$, where the latter clearly negates $f(A) \cap B \neq \emptyset$ for all $B \in V$, and so the “only if” must hold as well).
We have thus shown that $V \in \beta(f)([A])$ implies $V \in [f(A)]$. In the other direction, if $V \in [f(A)]$, then $f(A) \cap B \neq \emptyset$ for all $B \in V$, which as we saw is equivalent to $A \cap f^{-1}(B) \neq \emptyset$ for all $B \in V$. Hence the sets $A \cap f^{-1}(B)$ generate a filter on $X$, which may be extended to an ultrafilter $U$ on $X$, for which $U \in [A]$ and $V \subseteq \beta(f)(U)$ by construction, whence $V = \beta(f)(U)$ by maximality of ultrafilters. Thus $V \in [f(A)]$ implies $V \in \beta(f)([A])$.
The ultrafilter monad $\beta: Set \to Set$ preserves weak pullbacks, i.e., satisfies the Beck-Chevalley condition.
A proof may be found here.
Let $X_j$ be a family of sets. Then the disjoint sum $\sqcup_j \beta(X_j)$ (the coproduct in $Top$) is open and dense in $\sum_j \beta(X_j) = \beta(\sum_j X_j)$ (the coproduct in $CompHaus$).
Put $X = \sum_j X_j$ in $Set$. Each inclusion $\beta(X_j) \hookrightarrow \beta(X)$ induced by the inclusion $i_j: X_j \to X$ 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 $Tych$ of Tychonoff spaces, where coproducts are formed the same as they are in $Top$, an inclusion $i: X \to Y$ is dense if and only if it is epic. But it is obvious that $\sqcup_j \beta(X_j) \to \sum_j \beta(X_j)$ is epic in $Tych$: if $Z$ is Tychonoff and $i: Z \to \bar{Z}$ its Stone-Cech compactification, then a map $f: X \to Z$ is uniquely determined by a map $i f: X \to \bar{Z}$ in $CompHaus$, which in turn is uniquely determined by the family of their restrictions to the summands $\beta(X_j)$, 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)